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

Geometry · 93 problems have this as their main idea

primary-topic problems (7.2% of all)

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)

Problems, easiest first

Open any of them to try it here; the worked solution is on the problem's own page.

2008 AMC 10B · #10Circles

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 $AC$ ?

2007 AMC 10B · #4Circles

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 $\angle ABC?$

2007 AMC 10A · #14Circles

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?

2006 AMC 10B · #8Circles

A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?

2006 AMC 10A · #12Circles

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?

2005 AMC 10B · #7Circles

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?

2005 AMC 10B · #8Circles

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?

2004 AMC 10B · #9Circles

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?

2004 AMC 10B · #12Circles

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 …

2002 AMC 10A · #7Circles

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?

2025 AMC 10B · #12Circles

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 …

2025 AMC 10A · #10Circles

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?

2024 AMC 10A · #14Circles

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 …

2023 AMC 10A · #13Circles

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?

2022 AMC 10A · #15Circles

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 $a+b+c?$

2021 AMC Fall 10A · #15Circles

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 $C?$

2021 AMC Fall 10B · #11Circles

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?

2021 AMC 10B · #14Circles

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?

2020 AMC 10B · #14Circles

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?

2019 AMC 10A · #16Circles

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 $1 ?$

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