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Introduction to :

Central angles give rise to another concept that we call: the arc. (Not ark, as in Raiders of the Lost Ark.) More like Noah's ark…but only slightly.

It's hard to define what arcs are formally. Bowties and ball gowns really don't do it for us, anyway. However, we can describe them fairly well like this: an arc is a segment of a circle. Like a line segment, every arc has two endpoints.

Notice that points A and B define two arcs at the same time—there is the arc that directly connects A and B (arc 1, in the figure), and there is the arc that connects A and B by way of point C (arc 2, in the figure).

Every arc has a buddy, just as every central angle has a buddy. To distinguish each arc from its buddy, we usually add points to our figure. In the figure above, we would call arc 2 "arc ACB."

See? Just like the animals on Noah's ark, they come in pairs.

Buddy arcs complete each other just as central angles do. They even hold hands at their endpoints. If that's too mushy-gushy, imagine them shaking hands instead, like esteemed businessmen.

For any arc with two distinct endpoints, we can draw a central angle by drawing two rays, each starting at the center of the circle and going through one of the endpoints. We say that such an angle intercepts that arc, or that the arc subtends that angle. Likewise, any central angle intercepts an arc. Below, ∠AOB intercepts arc AB.

Example 1

In the figure below, A, B, and C are on ⊙O and m∠AOB = 38°. Is arc AB a major arc, a minor arc, or a semicircle?


Example 2

What is the circumference of a circle with radius 18 cm?


Example 3

What is the length of an arc with measure 78° and radius 5 m?


Example 4

What is the radius of an arc with measure 90° and length 31 cm?


Example 5

In the figure below, points A, B, and C are on ⊙O. m∠AOB = 80° and m∠BOC = 120°. What is the measure of arc AC?


Exercise 1

Find the exact circumference of a circle with radius 7 m.

Exercise 2

Find the exact circumference of a circle with radius 89 cm.

Exercise 3

The circumference of a circle is 25π cm. What is its exact radius?

Exercise 4

The circumference of a circle is 185 cm. What is its exact radius?

Exercise 5

Find the exact length of an arc with measure 10° and radius 3 km.

Exercise 6

Find the length of an arc with measure 137° and radius 18 cm. Round your answer to two decimal places.

Exercise 7

Suppose an arc has length 17 m and measure 256°. What must its radius be? Round your answer to two decimal places.

Exercise 8

Find the radius of an arc with length 35 km and measure 180°. Round your answer to two decimal places.

Exercise 9

What is the exact length of a semicircle with radius 5 km?

Exercise 10

In the figure below, points A, B, and C are on the circle. mABC = 135° and mBC = 103°. What is mAB?

Exercise 11

In the figure below, points W, X, Y, and Z are on the circle. mWX = 24°, mXY = 67°, and mYZ = 100°. What is mZW?

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