26. Circles · Mastering AMC 8

26.10%

Circle Properties

In this section you will be able to find a circle's circumference and area, and the length and area of any slice of it.

Roll a bicycle wheel one full turn. How far does it go? Lay a string around the tire and you get the answer: a bit more than 6 radii!

The edge of a circle is its circumference. It is always $2\pi$ radii long, about $6.28$ radii.

Cut a pizza into many thin slices and lay them head to toe. You get a near-rectangle, $\pi r$ long and $r$ tall. That is why the area is $\pi r^2$ .

A slice of the circle is a sector, and its curved edge is an arc. A $90^\circ$ slice is $90/360 = 1/4$ of the circle, so its arc and its area are both a quarter of the whole.

Exploreplay first, formulas later

Try this: Press Roll it out. How many radii long is the edge of the circle? Change the radius. Does that number change?
Try this: Switch to Cut into slices and add more slices. What shape do the slices make? How long and how tall is it?
Try this: Back in Unroll the edge, set the arc angle to 90°. What fraction of the whole edge rolls out now?

Name it

Theorem 26.1.1 · Area and Circumference

A circle with radius $r$ has

\text{Area} = \pi r^2

\text{Circumference} = 2\pi r

Theorem 26.1.2 · Arcs of a circle

An arc of a circle with radius $r$ and angle $a^\circ$ :

\text{Area of a sector} = \pi r^2 \times \frac{a^\circ}{360}

that is, $\pi \times \text{radius}^2 \times$ (fraction of circle in sector).

\text{Length of the arc} = 2\pi r \times \frac{a^\circ}{360}

that is, $2\pi \times \text{radius} \times$ (fraction of circle in sector).

Definition 26.1.3 · Angle of an arc

This is the angle that the arc makes at the center of the circle.

Try it

Problem 26.4.14(additional)

BmMT

A rectangle of height 10 and width 24 is inscribed in a circle. What is the circumference of that circle? Express your answer in terms of $\pi$ .

Contest cornerreal contest problems

Problem 26.4.6

AMC 8

Margie’s winning art design is shown. The smallest circle has radius 2 inches, with each successive circle’s radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?

(A) 42 (B) 44 (C) 45 (D) 46 (E) 48

Problem 26.4.7

AMC 8

Semicircles $POQ$ and $ROS$ pass through the center of circle $O$ . What is the ratio of the combined areas of the two semicircles to the area of circle $O$ ?

Checkpoint for section 26.1

Get 2 of 3 right to master this section. One try each, no hints.

26.20%

Circular Area

In this section you will be able to find the area of curvy shaded regions by adding and subtracting circles, sectors and polygons.

A tile has four yellow half-moons on it. How much of the tile is white? You can’t measure a curvy region directly. But you can build it from shapes you know!

Most shaded-region problems use one trick: big shape minus the pieces you don’t want. Squares, triangles, circles and sectors all have easy areas.

Look:Before you subtract, check that the pieces don’t overlap. If they do, the overlap gets subtracted twice.

Exploreplay first, formulas later

Try this: Turn every semicircle off. Then subtract them one at a time. Watch the area go down by the same amount each time.
Try this: Turn on Show the formula. Which number is the square, and which are the semicircles?
Try this: Press a semicircle until it says add. What happens to the area? Why is that the wrong region?

Watch itpredict, then reveal

Example 26.1

EMCC

Four congruent semicircles are drawn within the boundary of a square with side length 1. The center of each semicircle is the midpoint of a side of the square. Each semicircle is tangent to two other semicircles. Region $R$ consists of points lying inside the square but outside of the semicircles. The area of $R$ can be written in the form $a - b\pi$ , where $a$ and $b$ are positive rational numbers. Compute $a + b$ .

1
The semicircles at $X$ and $Y$ are centered at the midpoints of two sides that meet at $B$ . How far apart are $X$ and $Y$ ?

Predict first:
Your prediction

open on YouTube

Example 26.2

EMCC

Suppose points A and B lie on a circle of radius 4 with center O, such that $\angle AOB = 90^\circ$ . The perpendicular bisectors of segments OA and OB divide the interior of the circle into four regions. Find the area of the smallest region.

1
The bisectors cross at $R$ . Which of the four regions is the smallest?

Predict first:

open on YouTube

Try it

Problem 26.4.1

AMC 8

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?

Problem 26.4.3

AMC 8

A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle’s shaded area to the area between the two squares?

Contest cornerreal contest problems

Problem 26.4.4

AMC 8

Rectangle $ABCD$ has sides $CD = 3$ and $DA = 5$ . A circle with a radius of 1 is centered at $A$ , a circle with a radius of 2 is centered at $B$ , and a circle with a radius of 3 is centered at $C$ . Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

Problem 26.4.8

AMC 8

A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

Problem 26.4.11

AMC 8

In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$ . Arcs $\overset{\frown}{TR}$ and $\overset{\frown}{SR}$ are each one-sixth of a circle with radius 2. What is the area of the region shown?

Checkpoint for section 26.2

Get 2 of 3 right to master this section. One try each, no hints.

26.30%

Length inside Circles

In this section you will be able to find lengths inside circles with right triangles made from a radius, a half-chord and a distance.

A straight cut across a round pizza is a chord. How far is the cut from the center? Draw one radius and you get a right triangle!

Two facts unlock most lengths in a circle:

A line from the center perpendicular to a chord cuts the chord in half.
A radius drawn to a tangent point is perpendicular to the tangent line.

Either way you get a right triangle with a radius as one side. Then the Pythagorean theorem finishes the job.

Exploreplay first, formulas later

Try this: Make the chord longer. Does it move closer to the center or farther away?
Try this: With radius 5, set the chord to 8, then 6. Turn on Show the formula. Which right triangle gives the distance?
Try this: Make the chord as long as possible. What special chord is it, and how far is it from the center?

Watch itpredict, then reveal

Example 26.3

AMC 8

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE}$ , as shown in the figure. Let $DA = 16$ , and let $FD = AE = 9$ . What is the area of $ABCD$ ?

1
Where is the center of the semicircle, and what is its radius?

Predict first:
Your prediction

open on YouTube

Example 26.4

Science Bowl

A circle, centered at C, of radius of 25 has 2 parallel chords with lengths shown. To the nearest integer, what is the distance between these chords?

1
Drop a perpendicular from $C$ to the chord of length 48. Where does it land, and how long is each half?

Predict first:
Your prediction

open on YouTube

Try it

Problem 26.4.2

AMC 8

The two circles pictured have the same center $C$ . Chord $\overline{AD}$ is tangent to the inner circle at $B$ , $AC$ is 10, and chord $\overline{AD}$ has length 16. What is the area between the two circles?

Problem 26.4.15(additional)

BmMT

Let C1 and C2 be two circles centered at point O of radii 1 and 2, respectively. Let A be a point on C2. We draw the two lines tangent to C1 that pass through A, and label their other intersections with C2 as B and C. Let x be the length of minor arc BC, as shown. Compute x.

Contest cornerreal contest problems

Problem 26.4.5

AMC 8

Isosceles right triangle $ABC$ encloses a semicircle of area $2\pi$ . The circle has its center $O$ on hypotenuse $\overline{AB}$ and is tangent to sides $\overline{AC}$ and $\overline{BC}$ . What is the area of triangle $ABC$ ?

Problem 26.4.9

AMC 8

In the right triangle $ABC$ , $AC = 12$ , $BC = 5$ , and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

Problem 26.4.10

AMC 8

A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

Problem 26.4.12

AMC 12

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?

Checkpoint for section 26.3

Get 2 of 3 right to master this section. One try each, no hints.

26.4

Practice Problems

In this section you will be able to mix circle areas, arcs and chord lengths on contest problems.

One last challenge: two circles overlap like a Venn diagram. Add the circles, subtract the overlap, and remember how a chord cuts off a sector!

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 26.4.13(additional)

EMCC

Two circles of radius 6 intersect such that they share a common chord of length 6. The total area covered may be expressed as $a\pi + \sqrt{b}$ , where $a$ and $b$ are integers. What is $a + b$ ?

Turn the layers on and off. Red points are counted twice.