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Circles

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

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CirclesAMC 10 Step by Step

Name: Date: October 28, 2026

20 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)

1.2002 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) $16$
2.2002 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) $3\pi$
3.2025 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) $\frac{3}{10}$
4.2025 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) $11$
5.2024 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) $\frac{19}{9}$
6.2023 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) $\frac{1}{9}$
7.2022 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) $136\pi$
8.2021 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) $14$
9.2019 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) $17$
10.2019 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) $\frac{87\pi}{8}$
11.2017 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) $\frac{4}{3}-\frac{4\sqrt{3}\pi}{27}$
12.2017 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) $\frac{120}{31}$
13.2016 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) $2\pi+2\sqrt{2}$
14.2016 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) $\sqrt{6}/2$
15.2016 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) $500$
16.2015 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) $32$
17.2014 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) $\dfrac{\sqrt5}3$
18.2013 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) $30$
19.2013 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) $72$
20.2011 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) $\frac{1509}{256}$

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.