The Basics of Circles at a Glance

Mathematicians like to define things before they start talking about them. This is a good idea in other areas of life too. Ever brought a cake pan to a baseball game because you heard there would be a batter?

Given a point O and a distance r, the circle with center O and radius r is the set of all points in a plane that are exactly r units away from O. Easy enough, right?

Examples
Exercises

Example 1

Suppose the radius of ⊙O is 7 m and OP is 3 m. Is the point P in the exterior of ⊙O, in the interior of ⊙O, or on ⊙O?
First, let's compare the distance from P to O to the radius of ⊙O: OP = 3 m and r, the radius of ⊙O, is 7 m. Since 3 < 7, we know that OP < r. By definition of interior, we know that P is in the interior of ⊙O.

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Example 2

Suppose the radius of ⊙O is 10 km and OP is 10 km. Is the point P in the exterior of ⊙O, in the interior of ⊙O, or on ⊙O?
The distance from O to P in this case is equal to the radius of ⊙O. Therefore, by definition of a circle, P is on ⊙O. It's straddling the fence.

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Example 3

Suppose the radius of ⊙O is 175 cm. OP is 213 cm. Is the point P in the exterior of ⊙O, in the interior of ⊙O, or on ⊙O?
First, let's compare the distance from P to O to the radius of ⊙O: OP = 213 cm and r, the radius of ⊙O, is 175 cm. Since 213 > 175, we know OP > r. By definition of exterior, we know that P is in the exterior of ⊙O.

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Example 4

Suppose a circle is divided into six central angles of the same measure. What is the measure of one of them?

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Exercise 1

Consider a circle with center O and radius 20 m. Is P such that OP = 8 m in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

You'll need to compare the distance between P and O with the radius of the circle.

Exercise 2

Consider a circle with center O and radius 20 m. Is Q such that OQ = 30 cm in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

The units are there for a reason.

Exercise 3

Consider a circle with center O and radius 20 m. Is S such that OS = 20 m in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

What is the distance between O and S? What is the distance of the radius?

Exercise 4

Consider a circle with center O and radius 20 m. Is O in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

Point O is pretty important, isn't it?

Exercise 5

Consider a circle with center O and radius 20 m. Is T such that OT = 100 m in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

Is the radius larger or smaller than the distance between O and T?

Exercise 6

Consider a circle with center O and radius 20 m. Is U such that OU > OS in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

Remember, OS was a distance of 20 m.

Exercise 7

Consider a circle with center O and radius 20 m. Is V such that OS < OV < OU in the interior of the circle, in the exterior of the circle, or on the circle?

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Hint

Remember, OS was a distance of 20 m and OU > OS.

Exercise 8

In the figure below, points A, B, C, and D are on ⊙O. If m∠BOC = 50°, m∠COD = 140°, and m∠DOA = 65°, what is m∠AOB?

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Hint

Remember that all central angles in a circle add up to 360°.

Exercise 9

In the figure below, suppose now that m∠BOC = 67° instead of 50°. How does the measure of m∠AOB change?

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Hint

Remember that all central angles in a circle add up to 360°.

The Basics of Circles at a Glance

Example 1

Example 2

Example 3

Example 4

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Exercise 8

Exercise 9

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