Circles

This chapter applies to all AMC 8/10/12 test takers.

The two most important measurements of a circle are its area and circumference. Before using the formulas, make sure these definitions are clear.

The radius is the distance from the center of the circle to any point on the circle.

The diameter is the distance across the circle passing through the center.

The circumference is the perimeter, or total distance around the circle.

The area of a circle is the space enclosed by the circle and is expressed in square units.

$$\text{Area}=\pi r^2$$

$$\text{Circumference}=2\pi r=d\pi$$

A quick way to keep these straight: area is measured in square units, so its formula includes a square ($r^2$). Circumference is a length, so it doesn’t.

Equation for circle

You can describe a circle on the coordinate plane with an equation. In the formula below:

• $(h,k)$ is the center of the circle.
• $r$ is the radius.
• $(x,y)$ represents any point on the circle.

$$(x-h{)}^2+(y-k{)}^2=r^2$$

Chords, secants, and tangents

Chords are line segments that start and end on the circle.

• Any chord lies inside the circle except at its endpoints.
• The longest possible chord is the diameter. It passes through the center.

Chords are also tied to angle facts you’ll use often:

• An inscribed angle subtended by a chord is half the angle formed at the center of the circle that reaches the same endpoints.
• Inscribed angles that subtend the same arc are equal (their location on the circle doesn’t matter).

Example: The question below is from 2018 AMC 12B

A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle?
A. $25\pi$
B. $50\pi$
C. $75\pi$
D. $100\pi$
E. $125\pi$

A secant line intersects the circle at two points. Unlike a chord, it’s an infinite line rather than a segment.

A tangent is a line that touches the circle at exactly one point.

Cyclic quadrilaterals

A cyclic quadrilateral is any four-sided polygon whose vertices all lie on a circle. In other words, each vertex of the quadrilateral lies on the circle.

This kind of quadrilateral is especially useful because it has several reliable properties.

Ptolemy’s Theorem says that in a cyclic quadrilateral, the product of the diagonals equals the sum of the products of the two pairs of opposite sides. The diagram and formula below show the relationship.

$$AC(BD)=AB(CD)+BC(AD)$$

Example: The question below is from 2004 AMC 11B

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 $\frac{AD}{CD}$?
A. $\frac{9}{8}$
B. $\frac{5}{3}$
C. $2$
D. $\frac{17}{7}$
E. $\frac{5}{2}$

You can use Brahmagupta’s Formula to find the area of a cyclic quadrilateral when the side lengths are known. Let $a,b,c,$ and $d$ be the side lengths, and let $s$ be half the perimeter (the semiperimeter).

$$\text{Area}\sqrt{(s-a)(s-b)(s-c)(s-d)}$$

$$S=\frac{a+b+c+d}{2}$$

Here are a few more important properties of cyclic quadrilaterals to keep in mind.

• Pairs of opposite angles must add up to $180$ degrees.
• All sides must be less than the diameter.
• The quadrilateral reaches its maximum area when it is a square.

Intersecting chords theorem

The intersecting chords theorem states that when two chords intersect inside a circle, the product of the two segments of one chord equals the product of the two segments of the other chord.

$$AP(BP)=CP(DP)$$

Example: The question below is from 2020 AMC 12B

Let$\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC=45^{\circ }.$ What is $C{E}^2+D{E}^2?$\

A. $96$
B. $98$
C. $44\sqrt{5}$
D. $70\sqrt{2}$
E. $100$

Sector area, arc length, and interior angle

For a sector (a “slice” of a circle), three ratios match:

• sector area compared to the circle’s area
• arc length compared to the circle’s circumference
• central angle compared to $360^{\circ }$

$$\frac{\text{sector area}}{\pi r^2}=\frac{\text{arc length}}{2\pi r}=\frac{\text{interior angle}}{360^{\circ }}$$

Example: The question below is from 2002 AMC 12A

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?
A. $\frac{4}{9}$
B. $\frac{2}{3}$
C. $\frac{5}{6}$
D. $\frac{3}{2}$
E. $\frac{9}{4}$

Common themes

• Circle diagrams may not be perfectly drawn to scale, so rely on the algebraic relationships rather than the picture.
• A circumscribed right triangle’s hypotenuse must be equal in distance to the diameter of the circle.
• The sum of opposite angles of ANY cyclic quadrilateral must add up to $180^{\circ }$. Just solve for the sum of the interior angles and divide by half of the sides to find the sum of any opposite pairs.
• Any point outside of the circle can be described by the intersection of two unique tangent lines to the circle.
• If opposite angles of a quadrilateral sum up to $180^{\circ }$, the quadrilateral must be cyclic.