Printable problems

Circles

Chords, tangents, inscribed angles, power of a point, arc length, sectors, inscribed/circumscribed circles

Add an answer page Back to the topic

CirclesAMC 10 Step by Step

Name: Date: October 28, 2026

all 93 problems, easiest first

What you need to know

Inscribed Angle Theorem. An inscribed angle (corner on the circle) is half the arc it cuts off. So an inscribed angle whose sides pass through the two ends of a diameter is a right angle. Example: an inscribed angle on a $140^\circ$ arc is $70^\circ$ .
Tangents. A tangent line is perpendicular to the radius at the point where it touches. Two Tangent Theorem: the two tangent segments from a point outside the circle are equal. Example: from a point $10$ away from the center of a circle with radius $6$ , both tangents are $\sqrt{100-36} = 8$ long.
Power of a Point. If chords $AB$ and $CD$ cross at $P$ , then $PA\cdot PB = PC\cdot PD$ . From a point $P$ outside the circle, with a tangent $PT$ and a line (a secant) meeting the circle at $A$ and then $B$ : $PT^2 = PA\cdot PB$ . Example: chords with $PA = 4$ , $PB = 6$ , $PC = 3$ give $PD = 8$ . A secant with $PA = 4$ , $PB = 9$ gives $PT = 6$ .
Arcs and sectors. A central angle of $\theta$ degrees cuts off $\frac{\theta}{360^\circ}$ of the circle: the arc length is $\frac{\theta}{360^\circ}\cdot 2\pi r$ , and the sector (pizza slice) has area $\frac{\theta}{360^\circ}\cdot \pi r^2$ . Example: with $r = 6$ and $\theta = 60^\circ$ , the arc is $2\pi$ long and the sector's area is $6\pi$ .

How AMC 10 tests it

Shaded regions (problems 3–12): circles in squares, or semicircles on the sides of a triangle. Subtract sectors and triangles.
Tangent lines from a point, or one line tangent to two circles. Radii to the points of tangency give a right triangle or a trapezoid with two right angles.
Problems 17–23: Power of a Point for a length, or a rope wrapping around a polygon, giving a sum of sectors.

Standard approaches

Draw the radius to every point of tangency and every chord end. Look for right angles, and for isosceles triangles (radii are equal).
Turn inscribed angles into arcs, add the arcs, and turn back.
When two chords, or a tangent and a secant, pass through one point, write Power of a Point.

Worked example

Point $P$ lies outside a circle with center $O$ and radius $6$ . The two tangent segments from $P$ touch the circle at $A$ and $B$ , and $\angle APB = 60^\circ$ . What is the area of the region bounded by $\overline{PA}$ , $\overline{PB}$ , and the minor arc $AB$ ?

(A) $18\sqrt3 - 6\pi$ (B) $36\sqrt3 - 6\pi$ (C) $36\sqrt3 - 12\pi$ (D) $36 - 12\pi$ (E) $72\sqrt3 - 12\pi$

Solution. Step 1. Each tangent is perpendicular to its radius, so $\angle OAP = \angle OBP = 90^\circ$ .

Step 2. Right triangles $OAP$ and $OBP$ share the hypotenuse $OP$ and have $OA = OB$ , so they are congruent, and $\angle APO = \frac12\cdot60^\circ = 30^\circ$ . Triangle $OAP$ is a $30$ - $60$ - $90$ triangle with short leg $OA = 6$ , so $PA = 6\sqrt3$ and $\angle AOP = 60^\circ$ .

Step 3. The kite $OAPB$ is two copies of triangle $OAP$ , so its area is $2\cdot\frac12\cdot 6\cdot 6\sqrt3 = 36\sqrt3$ .

Step 4. The central angle is $\angle AOB = 120^\circ$ , so sector $OAB$ has area $\frac{120}{360}\cdot 36\pi = 12\pi$ .

Step 5. The region is the kite minus the sector: $36\sqrt3 - 12\pi$ .

The answer is $\boxed{\textbf{(C)}\ 36\sqrt3 - 12\pi}$ .

Pitfalls

Putting the diameter into $\pi r^2$ instead of the radius. The area comes out $4$ times too big, and the answer choices often differ by that factor of $4$ .
Finding a sector's area with the inscribed angle instead of the central angle (which is twice as big).
In Power of a Point, using $PA\cdot AB$ instead of $PA\cdot PB$ . Both lengths start at $P$ . Example: with $PA = 4$ and $AB = 5$ , the product is $4\cdot 9 = 36$ , not $20$ .

Traps that recur

Assuming the inner corners lie on a circle of radius 3 (the table radius minus the mat width), rather than at distance x from the center. (2008 AMC 10A #25)
Assuming the inner circle is the incircle of the triangle of centres, whose inradius is 1, or reading the picture's apparent symmetry as meaning the inner centre is equidistant from the three outer centres. (2025 AMC 10A #22)
Losing a minus sign when writing the diameter, giving (sqrt 2 - 1)(sqrt 3 + 1) and the sum 5 in (B). (2025 AMC 10B #20)
Finding only the small pipe wedged between the two and answering 1/9, forgetting that a pipe on the far side of the small one is also in contact with both. (2024 AMC 10B #21)

1.2023 AMC 10B #3
A $3-4-5$ right triangle is inscribed in circle $A$ , and a $5-12-13$ right triangle is inscribed in circle $B$ . What is the ratio of the area of circle $A$ to the area of circle $B$ ?
1. A) $\frac{9}{25}$
2. B) $\frac{1}{9}$
3. C) $\frac{1 }{5}$
4. D) $\frac{25}{169}$
5. E) $\frac{4}{25}$
2.2010 AMC 10B #6
A circle is centered at $O$ , $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB = 50^\circ$ . What is the degree measure of $\angle CAB$ ?
1. A) $20$
2. B) $25$
3. C) $45$
4. D) $50$
5. E) $65$
3.2010 AMC 10A #5
The area of a circle whose circumference is $24\pi$ is $k\pi$ . What is the value of $k$ ?
1. A) $6$
2. B) $12$
3. C) $24$
4. D) $36$
5. E) $144$
4.2006 AMC 10B #6
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $\frac{2}{\pi}$ , as shown. What is the perimeter of this region?
1. A) $\frac{4}{\pi}$
2. B) $2$
3. C) $\frac{8}{\pi}$
4. D) $4$
5. E) $\frac{16}{\pi}$
5.2006 AMC 10B #4
Circles of diameter $1$ inch and $3$ inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
1. A) $2$
2. B) $3$
3. C) $6$
4. D) $8$
5. E) $9$
6.2002 AMC 10B #5
Circles of radius $2$ and $3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.
1. A) $3\pi$
2. B) $4\pi$
3. C) $6\pi$
4. D) $9\pi$
5. E) $12\pi$
7.2002 AMC 10A #5
Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
1. A) $\pi$
2. B) $1.5\pi$
3. C) $2\pi$
4. D) $3\pi$
5. E) $3.5\pi$
8.2001 AMC 10 #4
What is the maximum number of possible points of intersection of a circle and a triangle?
1. A) $2$
2. B) $3$
3. C) $4$
4. D) $5$
5. E) $6$
9.2025 AMC 10B #5
In $\triangle ABC$ , $AB=10$ , $AC=18$ , and $\angle B=130^\circ.$ Let $O$ be the center of the circle containing $A$ , $B$ , and $C.$ What is the degree measure of $\angle CAO$ ?
1. A) $20$
2. B) $30$
3. C) $40$
4. D) $50$
5. E) $60$
10.2021 AMC 10B #7
In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell$ . Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S$ ?
1. A) $24\pi$
2. B) $32\pi$
3. C) $64\pi$
4. D) $65\pi$
5. E) $84\pi$
11.2019 AMC 10A #6
For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is equidistant from all four vertices of the quadrilateral?
- a square
- a rectangle that is not a square
- a rhombus that is not a square
- a parallelogram that is not a rectangle or a rhombus
- an isosceles trapezoid that is not a parallelogram
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
12.2018 AMC 10B #7
In the figure below, $N$ congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. The ratio $A:B$ is $1:18$ . What is $N$ ?
1. A) $16$
2. B) $17$
3. C) $18$
4. D) $19$
5. E) $36$
13.2016 AMC 10A #15
Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the radius in inches of the scrap cookie?
1. A) $\sqrt{2}$
2. B) $1.5$
3. C) $\sqrt{\pi}$
4. D) $\sqrt{2\pi}$
5. E) $\pi$
14.2015 AMC 10B #9
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$ . What is the area of the shark's fin falcata?
1. A) $\dfrac{4\pi}{5}$
2. B) $\dfrac{9\pi}{8}$
3. C) $\dfrac{4\pi}{3}$
4. D) $\dfrac{7\pi}{5}$
5. E) $\dfrac{3\pi}{2}$
15.2014 AMC 10A #12
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?
1. A) $27\sqrt{3}-9\pi$
2. B) $27\sqrt{3}-6\pi$
3. C) $54\sqrt{3}-18\pi$
4. D) $54\sqrt{3}-12\pi$
5. E) $108\sqrt{3}-9\pi$
16.2012 AMC 10B #16
Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
1. A) $10\pi+4\sqrt{3}$
2. B) $13\pi-\sqrt{3}$
3. C) $12\pi+\sqrt{3}$
4. D) $10\pi+9$
5. E) $13\pi$
17.2012 AMC 10A #11
Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$ , respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$ . What is $BC$ ?
1. A) $4$
2. B) $4.8$
3. C) $10.2$
4. D) $12$
5. E) $14.4$
18.2011 AMC 10B #12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
1. A) $\frac{\pi}{3}$
2. B) $\frac{2\pi}{3}$
3. C) $\pi$
4. D) $\frac{4\pi}{3}$
5. E) $\frac{5\pi}{3}$
19.2009 AMC 10A #6
A circle of radius $2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
1. A) $\frac{1}{2}$
2. B) $\frac{\pi}{6}$
3. C) $\frac{2}{\pi}$
4. D) $\frac{2}{3}$
5. E) $\frac{3}{\pi}$
20.2009 AMC 10B #16
Points $A$ and $C$ lie on a circle centered at $O$ , each of $\overline{BA}$ and $\overline{BC}$ are tangent to the circle, and $\triangle ABC$ is equilateral. The circle intersects $\overline{BO}$ at $D$ . What is $\frac{BD}{BO}$ ?
1. A) $\frac {\sqrt2}{3}$
2. B) $\frac {1}{2}$
3. C) $\frac {\sqrt3}{3}$
4. D) $\frac {\sqrt2}{2}$
5. E) $\frac {\sqrt3}{2}$
21.2008 AMC 10B #10
Points $A$ and $B$ are on a circle of radius $5$ and $AB=6$ . Point $C$ is the midpoint of the minor arc $AB$ . What is the length of the line segment $AC$ ?
1. A) $\sqrt{10}$
2. B) $\frac{7}{2}$
3. C) $\sqrt{14}$
4. D) $\sqrt{15}$
5. E) $4$
22.2007 AMC 10B #4
The point $O$ is the center of the circle circumscribed about $\triangle ABC,$ with $\angle BOC=120^\circ$ and $\angle AOB=140^\circ,$ as shown. What is the degree measure of $\angle ABC?$
1. A) $35$
2. B) $40$
3. C) $45$
4. D) $50$
5. E) $60$
23.2007 AMC 10A #14
A triangle with side lengths in the ratio $3 : 4 : 5$ is inscribed in a circle with radius 3. What is the area of the triangle?
1. A) $8.64$
2. B) $12$
3. C) $5\pi$
4. D) $17.28$
5. E) $18$
24.2006 AMC 10B #8
A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?
1. A) $20\pi$
2. B) $25\pi$
3. C) $30\pi$
4. D) $40\pi$
5. E) $50\pi$
25.2006 AMC 10A #12
Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown.

Which of these arrangements give the dog the greater area to roam, and by how many square feet?
1. A) $I, \textrm{ by } 8\pi$
2. B) $I, \textrm{ by } 6\pi$
3. C) $II, \textrm{ by } 4\pi$
4. D) $II, \textrm{ by } 8\pi$
5. E) $II, \textrm{ by } 10\pi$
26.2005 AMC 10B #7
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smallest circle to the area of the largest square?
1. A) $\frac{\pi}{16}$
2. B) $\frac{\pi}{8}$
3. C) $\frac{3\pi}{16}$
4. D) $\frac{\pi}{4}$
5. E) $\frac{\pi}{2}$
27.2005 AMC 10B #8
An $8$ -foot by $10$ -foot bathroom floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
1. A) $80-20\pi$
2. B) $60-10\pi$
3. C) $80-10\pi$
4. D) $60+10\pi$
5. E) $80+10\pi$
28.2004 AMC 10B #12
An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$ , with $b>c$ . Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$ , and let $OY$ be the radius of the larger circle that contains $Z$ . Let $a=XZ$ , $d=YZ$ , and $e=XY$ . What is the area of the annulus?
1. A) $\pi a^2$
2. B) $\pi b^2$
3. C) $\pi c^2$
4. D) $\pi d^2$
5. E) $\pi e^2$
29.2004 AMC 10B #9
A square has sides of length $10$ , and a circle centered at one of its vertices has radius $10$ . What is the area of the union of the regions enclosed by the square and the circle?
1. A) $200+25\pi$
2. B) $100+75\pi$
3. C) $75+100\pi$
4. D) $100+100\pi$
5. E) $100+125\pi$
30.2002 AMC 10A #7
A $45^\circ$ arc of circle A is equal in length to a $30^\circ$ arc of circle B. What is the ratio of circle A's area and circle B's area?
1. A) $4/9$
2. B) $2/3$
3. C) $5/6$
4. D) $3/2$
5. E) $9/4$
31.2025 AMC 10B #12
The figure below shows an equilateral triangle, a rhombus with a $60^\circ$ angle, and a regular hexagon, each of them containing some mutually tangent congruent disks. Let $T, R,$ and $H,$ respectively, denote the ratio in each case of the total area of the disks to the area of the enclosing polygon. Which of the following is true?
1. A) $T=H=R$
2. B) $H<R=T$
3. C) $H=R<T$
4. D) $H<R<T$
5. E) $H<T<R$
32.2025 AMC 10A #10
A semicircle has diameter $\overline{AB}$ and chord $\overline{CD}$ of length $16$ parallel to $\overline{AB}$ . A smaller semicircle with diameter on $\overline{AB}$ and tangent to $\overline{CD}$ is cut from the larger semicircle, as shown below.

What is the area of the resulting figure, shown shaded?
1. A) $16\pi$
2. B) $24\pi$
3. C) $32\pi$
4. D) $48\pi$
5. E) $64\pi$
33.2024 AMC 10A #14
One side of an equilateral triangle of height $24$ lies on line $\ell$ . A circle of radius $12$ is tangent to line $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a \sqrt{b} - c \pi$ , where $a$ , $b$ , and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a + b + c$ ?
1. A) $72$
2. B) $73$
3. C) $74$
4. D) $75$
5. E) $76$
34.2023 AMC 10A #13
Abdul and Chiang are standing $48$ feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures $60^\circ$ . What is the square of the distance (in feet) between Abdul and Bharat?
1. A) $1728$
2. B) $2601$
3. C) $3072$
4. D) $4608$
5. E) $6912$
35.2022 AMC 10A #15
Quadrilateral $ABCD$ with side lengths $AB=7, BC=24, CD=20, DA=15$ is inscribed in a circle. The area interior to the circle but exterior to the quadrilateral can be written in the form $\frac{a\pi-b}{c},$ where $a,b,$ and $c$ are positive integers such that $a$ and $c$ have no common prime factor. What is $a+b+c?$
1. A) $260$
2. B) $855$
3. C) $1235$
4. D) $1565$
5. E) $1997$
36.2021 AMC 10B #14
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$ . What is the distance between two adjacent parallel lines?
1. A) $5\frac{1}{2}$
2. B) $6$
3. C) $6\frac{1}{2}$
4. D) $7$
5. E) $7\frac{1}{2}$
37.2021 AMC Fall 10A #15
Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$ , and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$ . What is the area of the circle that passes through vertices $A$ , $B$ , and $C?$
1. A) $24\pi$
2. B) $25\pi$
3. C) $26\pi$
4. D) $27\pi$
5. E) $28\pi$
38.2021 AMC Fall 10B #11
A regular hexagon of side length $1{ }$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs?
1. A) $\frac{5\sqrt{3}}{2} - \pi$
2. B) $3\sqrt{3}-\pi$
3. C) $4\sqrt{3}-\frac{3\pi}{2}$
4. D) $\pi - \frac{\sqrt{3}}{2}$
5. E) $\frac{\pi + \sqrt{3}}{2}$
39.2020 AMC 10B #14
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region — inside the hexagon but outside all of the semicircles?
1. A) $6\sqrt{3}-3\pi$
2. B) $\frac{9\sqrt{3}}{2} - 2\pi$
3. C) $\frac{3\sqrt{3}}{2} - \frac{\pi}{3}$
4. D) $3\sqrt{3} - \pi$
5. E) $\frac{9\sqrt{3}}{2} - \pi$
40.2019 AMC 10A #13
Let $\triangle ABC$ be an isosceles triangle with $BC = AC$ and $\angle ACB = 40^{\circ}$ . Construct the circle with diameter $\overline{BC}$ , and let $D$ and $E$ be the other intersection points of the circle with the sides $\overline{AC}$ and $\overline{AB}$ , respectively. Let $F$ be the intersection of the diagonals of the quadrilateral $BCDE$ . What is the degree measure of $\angle BFC ?$
1. A) $90$
2. B) $100$
3. C) $105$
4. D) $110$
5. E) $120$
41.2019 AMC 10A #16
The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$
1. A) $4 \pi \sqrt{3}$
2. B) $7 \pi$
3. C) $\pi\left(3\sqrt{3} +2\right)$
4. D) $10 \pi \left(\sqrt{3} - 1\right)$
5. E) $\pi\left(\sqrt{3} + 6\right)$
42.2018 AMC 10A #15
Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$ , as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?
1. A) $21$
2. B) $29$
3. C) $58$
4. D) $69$
5. E) $93$
43.2015 AMC 10A #14
The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
1. A) $\mathrm{2 o'clock}$
2. B) $\mathrm{3 o'clock}$
3. C) $\mathrm{4 o'clock}$
4. D) $\mathrm{6 o'clock}$
5. E) $\mathrm{8 o'clock}$
44.2012 AMC 10A #18
The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$ , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$ . What is the area enclosed by the curve?
1. A) $2\pi+6$
2. B) $2\pi+4\sqrt{3}$
3. C) $3\pi+4$
4. D) $2\pi+3\sqrt{3}+2$
5. E) $\pi+6\sqrt{3}$
45.2011 AMC 10B #17
In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$ , and $\overline{AB}$ is parallel to $\overline{ED}$ . The angles $AEB$ and $ABE$ are in the ratio $4 : 5$ . What is the degree measure of angle $BCD$ ?
1. A) $120$
2. B) $125$
3. C) $130$
4. D) $135$
5. E) $140$
46.2011 AMC 10A #18
Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$ . What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
1. A) $3 - \frac{\pi}{2}$
2. B) $\frac{\pi}{2}$
3. C) $2$
4. D) $\frac{3\pi}{4}$
5. E) $1+\frac{\pi}{2}$
47.2010 AMC 10B #16
A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?
1. A) $\dfrac{\pi}{3}-1$
2. B) $\dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3}$
3. C) $\dfrac{\pi}{18}$
4. D) $\dfrac{1}{4}$
5. E) $\dfrac{2\pi}{9}$
48.2009 AMC 10A #21
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
1. A) $3-2\sqrt2$
2. B) $2-\sqrt2$
3. C) $4(3-2\sqrt2)$
4. D) $\frac12(3-\sqrt2)$
5. E) $2\sqrt2-2$
49.2008 AMC 10A #16
Points $A$ and $B$ lie on a circle centered at $O$ , and $\angle AOB = 60^\circ$ . A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$ . What is the ratio of the area of the smaller circle to that of the larger circle?
1. A) $\frac{1}{16}$
2. B) $\frac{1}{9}$
3. C) $\frac{1}{8}$
4. D) $\frac{1}{6}$
5. E) $\frac{1}{4}$
50.2008 AMC 10A #17
An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle but not more than 3 units from a point on the triangle?
1. A) $36+24\sqrt{3}$
2. B) $54+9\pi$
3. C) $54+18\sqrt{3}+6\pi$
4. D) $\left(2\sqrt{3}+3\right)^2\pi$
5. E) $9\left(\sqrt{3}+1\right)^2\pi$
51.2007 AMC 10B #11
A circle passes through the three vertices of an isosceles triangle that has two sides of length $3$ and a base of length $2$ . What is the area of this circle?
1. A) $2\pi$
2. B) $\frac{5}{2}\pi$
3. C) $\frac{81}{32}\pi$
4. D) $3\pi$
5. E) $\frac{7}{2}\pi$
52.2007 AMC 10A #15
Four circles of radius $1$ are each tangent to two sides of a square and externally tangent to a circle of radius $2$ , as shown. What is the area of the square?
1. A) $32$
2. B) $22 + 12\sqrt {2}$
3. C) $16 + 16\sqrt {3}$
4. D) $48$
5. E) $36 + 16\sqrt {2}$
53.2007 AMC 10B #18
A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. What is $r$ ?
1. A) $\sqrt{2}$
2. B) $1+\sqrt{2}$
3. C) $\sqrt{6}$
4. D) $3$
5. E) $2+\sqrt{2}$
54.2007 AMC 10B #13
Two circles of radius $2$ are centered at $(2,0)$ and at $(0,2).$ What is the area of the intersection of the interiors of the two circles?
1. A) $\pi -2$
2. B) $\frac{\pi}{2}$
3. C) $\frac{\pi \sqrt{3}}{3}$
4. D) $2(\pi -2)$
5. E) $\pi$
55.2006 AMC 10B #19
A circle of radius $2$ is centered at $O$ . Square $OABC$ has side length $1$ . Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$ , respectively. What is the area of the shaded region in the figure, which is bounded by $BD$ , $BE$ , and the minor arc connecting $D$ and $E$ ?
1. A) $\frac{\pi}{3}+1-\sqrt{3}$
2. B) $\frac{\pi}{2}(2-\sqrt{3})$
3. C) $\pi(2-\sqrt{3})$
4. D) $\frac{\pi}{6}+\frac{\sqrt{3}+1}{2}$
5. E) $\frac{\pi}{3}-1+\sqrt{3}$
56.2005 AMC 10A #12
The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $2$ ?
1. A) $\frac{1}{3}\pi+\frac{\sqrt{3}}{2}$
2. B) $\frac{2}{3}\pi$
3. C) $\frac{2}{3}\pi+\frac{\sqrt{3}}{4}$
4. D) $\frac{2}{3}\pi+\frac{\sqrt{3}}{3}$
5. E) $\frac{2}{3}\pi+\frac{\sqrt{3}}{2}$
57.2004 AMC 10B #16
Three circles of radius $1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?
1. A) $\frac{2 + \sqrt{6}}{3}$
2. B) $2$
3. C) $\frac{2 + 3\sqrt{2}}{2}$
4. D) $\frac{3 + 2\sqrt{3}}{3}$
5. E) $\frac{3 + \sqrt{3}}{2}$
58.2003 AMC 10A #19
A semicircle of diameter $1$ sits at the top of a semicircle of diameter $2$ , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
1. A) $\frac{1}{6}\pi-\frac{\sqrt{3}}{4}$
2. B) $\frac{\sqrt{3}}{4}-\frac{1}{12}\pi$
3. C) $\frac{\sqrt{3}}{4}-\frac{1}{24}\pi$
4. D) $\frac{\sqrt{3}}{4}+\frac{1}{24}\pi$
5. E) $\frac{\sqrt{3}}{4}+\frac{1}{12}\pi$
59.2003 AMC 10A #17
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?
1. A) $\frac{3\sqrt{2}}{\pi}$
2. B) $\frac{3\sqrt{3}}{\pi}$
3. C) $\sqrt{3}$
4. D) $\frac{6}{\pi}$
5. E) $\sqrt{3}\pi$
60.2003 AMC 10B #19
Three semicircles of radius $1$ are constructed on diameter $\overline{AB}$ of a semicircle of radius $2$ . The centers of the small semicircles divide $\overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
1. A) $\pi - \sqrt{3}$
2. B) $\pi - \sqrt{2}$
3. C) $\frac{\pi + \sqrt{2}}{2}$
4. D) $\frac{\pi +\sqrt{3}}{2}$
5. E) $\frac{7}{6}\pi - \frac{\sqrt{3}}{2}$
61.2002 AMC 10B #18
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
1. A) $8$
2. B) $9$
3. C) $10$
4. D) $12$
5. E) $16$
62.2002 AMC 10A #19
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?
1. A) $2\pi/3$
2. B) $2\pi$
3. C) $5\pi/2$
4. D) $8\pi/3$
5. E) $3\pi$
63.2025 AMC 10A #22
A circle of radius $r$ is surrounded by three circles, whose radii are 1, 2, and 3, all externally tangent to the inner circle and externally tangent to each other, as shown in the diagram below.

What is $r$ ?
1. A) $\frac{1}{4}$
2. B) $\frac{6}{23}$
3. C) $\frac{3}{11}$
4. D) $\frac{5}{17}$
5. E) $\frac{3}{10}$
64.2025 AMC 10B #20
Four congruent semicircles are inscribed in a square of side length $1$ so that their diameters are on the sides of the square, one endpoint of each diameter is at a vertex of the square, and adjacent semicircles are tangent to each other. A small circle centered at the center of the square is tangent to each of the four semicircles, as shown below. The diameter of the small circle can be written as $(\sqrt a+b)(\sqrt c+d)$ , where $a$ , $b$ , $c$ , and $d$ are integers. What is $a + b + c + d$ ?
1. A) $3$
2. B) $5$
3. C) $8$
4. D) $9$
5. E) $11$
65.2024 AMC 10B #21
Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$ , lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
1. A) $\frac{1}{9}$
2. B) $1$
3. C) $\frac{10}{9}$
4. D) $\frac{11}{9}$
5. E) $\frac{19}{9}$
66.2023 AMC 10A #22
Circle $C_1$ and $C_2$ each have radius $1$ , and the distance between their centers is $\frac{1}{2}$ . Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$ . Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$ . What is the radius of $C_4$ ?
1. A) $\frac{1}{14}$
2. B) $\frac{1}{12}$
3. C) $\frac{1}{10}$
4. D) $\frac{3}{28}$
5. E) $\frac{1}{9}$
67.2022 AMC 10B #22
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$ , $x^{2}+y^{2}=64$ , and $(x-5)^{2}+y^{2}=3$ . What is the sum of the areas of all circles in $S$ ?
1. A) $48\pi$
2. B) $68\pi$
3. C) $96\pi$
4. D) $102\pi$
5. E) $136\pi$
68.2021 AMC Fall 10A #19
A disk of radius $1$ rolls all the way around the inside of a square of side length $s>4$ and sweeps out a region of area $A$ . A second disk of radius $1$ rolls all the way around the outside of the same square and sweeps out a region of area $2A$ . The value of $s$ can be written as $a+\frac{b\pi}{c}$ , where $a,b$ , and $c$ are positive integers and $b$ and $c$ are relatively prime. What is $a+b+c$ ?
1. A) $10$
2. B) $11$
3. C) $12$
4. D) $13$
5. E) $14$
69.2019 AMC 10B #20
As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ lie in the same halfplane determined by line $AD$ , and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F.$ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form $\frac{a}{b}\cdot\pi-\sqrt{c}+d,$ where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$ ?
1. A) $13$
2. B) $14$
3. C) $15$
4. D) $16$
5. E) $17$
70.2019 AMC 10B #23
Points $A(6,13)$ and $B(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$ -axis. What is the area of $\omega$ ?
1. A) $\frac{83\pi}{8}$
2. B) $\frac{21\pi}{2}$
3. C) $\frac{85\pi}{8}$
4. D) $\frac{43\pi}{4}$
5. E) $\frac{87\pi}{8}$
71.2017 AMC 10A #22
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?
1. A) $\frac{4\sqrt{3}\pi}{27}-\frac{1}{3}$
2. B) $\frac{\sqrt{3}}{2}-\frac{\pi}{8}$
3. C) $\frac{1}{2}$
4. D) $\sqrt{3}-\frac{2\sqrt{3}\pi}{9}$
5. E) $\frac{4}{3}-\frac{4\sqrt{3}\pi}{27}$
72.2017 AMC 10B #22
The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$ . Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$ . Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$ . What is the area of $\triangle ABC$ ?
1. A) $\frac{120}{37}$
2. B) $\frac{140}{39}$
3. C) $\frac{145}{39}$
4. D) $\frac{140}{37}$
5. E) $\frac{120}{31}$
73.2016 AMC 10B #21
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
1. A) $\pi+\sqrt{2}$
2. B) $\pi+2$
3. C) $\pi+2\sqrt{2}$
4. D) $2\pi+\sqrt{2}$
5. E) $2\pi+2\sqrt{2}$
74.2016 AMC 10A #21
Circles with centers $P, Q$ and $R$ , having radii $1, 2$ and $3$ , respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$ , respectively, with $Q'$ between $P'$ and $R'$ . The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$ ?
1. A) $0$
2. B) $\sqrt{6}/3$
3. C) $1$
4. D) $\sqrt{6}-\sqrt{2}$
5. E) $\sqrt{6}/2$
75.2016 AMC 10A #24
A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$ . Three of the sides of this quadrilateral have length $200$ . What is the length of the fourth side?
1. A) $200$
2. B) $200\sqrt{2}$
3. C) $200\sqrt{3}$
4. D) $300\sqrt{2}$
5. E) $500$
76.2015 AMC 10B #19
In $\triangle{ABC}$ , $\angle{C} = 90^{\circ}$ and $AB = 12$ . Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$ , and $W$ lie on a circle. What is the perimeter of the triangle?
1. A) $12+9\sqrt{3}$
2. B) $18+6\sqrt{3}$
3. C) $12+12\sqrt{2}$
4. D) $30$
5. E) $32$
77.2014 AMC 10B #22
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
1. A) $\dfrac{1+\sqrt2}4$
2. B) $\dfrac{\sqrt5-1}2$
3. C) $\dfrac{\sqrt3+1}4$
4. D) $\dfrac{2\sqrt3}5$
5. E) $\dfrac{\sqrt5}3$
78.2013 AMC 10B #23
In triangle $ABC$ , $AB = 13$ , $BC = 14$ , and $CA = 15$ . Distinct points $D$ , $E$ , and $F$ lie on segments $\overline{BC}$ , $\overline{CA}$ , and $\overline{DE}$ , respectively, such that $\overline{AD} \perp \overline{BC}$ , $\overline{DE} \perp \overline{AC}$ , and $\overline{AF} \perp \overline{BF}$ . The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?
1. A) $18$
2. B) $21$
3. C) $24$
4. D) $27$
5. E) $30$
79.2013 AMC 10A #23
In $\triangle ABC$ , $AB = 86$ , and $AC=97$ . A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$ . Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$ ?
1. A) $11$
2. B) $28$
3. C) $33$
4. D) $61$
5. E) $72$
80.2011 AMC 10B #25
Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$ . For $n \ge 1$ , if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$ and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$ ?
1. A) $\frac{1509}{8}$
2. B) $\frac{1509}{32}$
3. C) $\frac{1509}{64}$
4. D) $\frac{1509}{128}$
5. E) $\frac{1509}{256}$
81.2010 AMC 10B #19
A circle with center $O$ has area $156\pi$ . Triangle $ABC$ is equilateral, $\overline{BC}$ is a chord on the circle, $OA = 4\sqrt{3}$ , and point $O$ is outside $\triangle ABC$ . What is the side length of $\triangle ABC$ ?
1. A) $2\sqrt{3}$
2. B) $6$
3. C) $4\sqrt{3}$
4. D) $12$
5. E) $18$
82.2010 AMC 10B #20
Two circles lie outside regular hexagon $ABCDEF$ . The first is tangent to $\overline{AB}$ , and the second is tangent to $\overline{DE}$ . Both are tangent to lines $BC$ and $FA$ . What is the ratio of the area of the second circle to that of the first circle?
1. A) $18$
2. B) $27$
3. C) $36$
4. D) $81$
5. E) $108$
83.2007 AMC 10A #24
Circles centered at $A$ and $B$ each have radius $2$ , as shown. Point $O$ is the midpoint of $\overline{AB}$ , and $OA = 2\sqrt {2}$ . Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$ , respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$ ?
1. A) $\frac {8\sqrt {2}}{3}$
2. B) $8\sqrt {2} - 4 - \pi$
3. C) $4\sqrt {2}$
4. D) $4\sqrt {2} + \frac {\pi}{8}$
5. E) $8\sqrt {2} - 2 - \frac {\pi}{2}$
84.2006 AMC 10A #23
Circles with centers $A$ and $B$ have radius 3 and 8, respectively. A common internal tangent intersects the circles at $C$ and $D$ , respectively. Lines $AB$ and $CD$ intersect at $E$ , and $AE=5$ . What is $CD$ ?
1. A) $13$
2. B) $\frac{44}{3}$
3. C) $\sqrt{221}$
4. D) $\sqrt{255}$
5. E) $\frac{55}{3}$
85.2006 AMC 10B #24
Circles with centers $O$ and $P$ have radii $2$ and $4$ , respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points $C$ and $D$ on the circle with center $P$ are such that $AD$ and $BC$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$ ?
1. A) $18\sqrt{3}$
2. B) $24\sqrt{2}$
3. C) $36$
4. D) $24\sqrt{3}$
5. E) $32\sqrt{2}$
86.2005 AMC 10A #23
Let $\overline{AB}$ be a diameter of a circle and $C$ be a point on $\overline{AB}$ with $2 \cdot AC = BC$ . Let $D$ and $E$ be points on the circle such that $\overline{DC} \perp \overline{AB}$ and $\overline{DE}$ is a second diameter. What is the ratio of the area of $\triangle DCE$ to the area of $\triangle ABD$ ?
1. A) $\frac{1}{6}$
2. B) $\frac{1}{4}$
3. C) $\frac{1}{3}$
4. D) $\frac{1}{2}$
5. E) $\frac{2}{3}$
87.2004 AMC 10A #22
Square $ABCD$ has side length $2$ . A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$ . What is the length of $\overline{CE}$ ?
1. A) $\frac{2+\sqrt{5}}{2}$
2. B) $\sqrt{5}$
3. C) $\sqrt{6}$
4. D) $\frac{5}{2}$
5. E) $5-\sqrt{5}$
88.2004 AMC 10B #25
A circle of radius $1$ is internally tangent to two circles of radius $2$ at points $A$ and $B$ , where $AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the picture, that is outside the smaller circle and inside each of the two larger circles?
1. A) $\frac{5}{3} \pi - 3\sqrt 2$
2. B) $\frac{5}{3} \pi - 2\sqrt 3$
3. C) $\frac{8}{3} \pi - 3\sqrt 3$
4. D) $\frac{8}{3} \pi - 3\sqrt 2$
5. E) $\frac{8}{3} \pi - 2\sqrt 3$
89.2004 AMC 10B #24
In triangle $ABC$ we have $AB=7$ , $AC=8$ , $BC=9$ . Point $D$ is on the circumscribed circle of the triangle so that $AD$ bisects angle $BAC$ . What is the value of $\frac{AD}{CD}$ ?
1. A) $\dfrac{9}{8}$
2. B) $\dfrac{5}{3}$
3. C) $2$
4. D) $\dfrac{17}{7}$
5. E) $\dfrac{5}{2}$
90.2004 AMC 10A #21
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radian is $180$ degree.)
1. A) $\frac{\pi}{8}$
2. B) $\frac{\pi}{7}$
3. C) $\frac{\pi}{6}$
4. D) $\frac{\pi}{5}$
5. E) $\frac{\pi}{4}$
91.2004 AMC 10A #23
Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$ . Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$ . What is the radius of circle $B$ ?
1. A) $\frac23$
2. B) $\frac {\sqrt3}{2}$
3. C) $\frac78$
4. D) $\frac89$
5. E) $\frac {1 + \sqrt3}{3}$
92.2002 AMC 10B #24
Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom?
1. A) $5$
2. B) $6$
3. C) $7.5$
4. D) $10$
5. E) $15$
93.2008 AMC 10A #25
A round table has radius $4$ . Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$ . Furthermore, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$ ?
1. A) $2\sqrt{5}-\sqrt{3}$
2. B) $3$
3. C) $\frac{3\sqrt{7}-\sqrt{3}}{2}$
4. D) $2\sqrt{3}$
5. E) $\frac{5+2\sqrt{3}}{2}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.