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Chapter 6 – Circles

In previous chapters, you have extensively studied triangles and quadrilaterals to learn more about their sides and angles. In this chapter, you will connect your understanding of polygons with your knowledge of the area ratios of similar figures to find the area and circumference of circles of all sizes. You will explore the relationships between angles, arcs, and chords in a circle. Then you will find the equation of a circle using the Pythagorean Theorem.

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In this chapter, you will learn:

  • how to find the area and circumference of a circle and parts of circles and use this ability to solve problems in various contexts.
  • how to use the relationships between angles, arcs, and line segments within a circle to solve problems
  • how to find the measures of angles and arcs that are formed by tangents and secants
  • about the relationships between the lengths of segments created when tangents or secants intersect outside a circle.
  • how to find the equation of a circle and graph circles
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6.1

What If There Are No Sides?

Pg. 4

A Special Ratio and Circumference

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6.1 – What If There Are No Sides?

A Special Ratio and Circumference

In previous chapters you discovered the perimeter and area of polygons. Today you will discover how to find the perimeter of a shape with no sides, a circle.

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6.1 – PARTS OF A CIRCLE

Examine the picture of the circle shown. Explain how the radius and diameter are related. Then explain what the circumference of a circle measures.

Circumference is length around the circle, perimeter

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6.2 – RATIOS OF CIRCLES

Find the measure of the circumference by wrapping a string around the entire perimeter of the circle and then measuring how long the string is in centimeters. Then estimate the diameter (length across the circle). Then write the ratio. What do you notice? What value are you finding?

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6.3 – CIRCUMFERENCE FORMULA

a. Use the relationship you just discovered to find the formula for the circumference of a circle.

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b. Since the diameter is twice the size of the radius, write a new equation for the circumference of a circle when given the radius instead of the diameter by substituting in

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6.4 – MISSING INFORMATION

Using your understanding of the relationship between circumference (perimeter of a circle) and the diameter, find the missing values.

a. Find the circumference of a circle with a diameter of 12 inches.

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c. Find the circumference of a circle with a radius of 16 inches. Leave answers in terms of pi.

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d. Michael noticed a pattern in the relationship between the radius, diameter, and circumference. Complete the table below. Be sure to leave pi in your answer when finding the circumference.

3

5

10

15

24

2r

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6.5 – WHEELS GO ROUND AND ROUND

Evelyn's in-line skate wheels have a 0.42m diameter. How many meters will Evelyn travel after 5000 revolutions of the wheels on her in-line skates?

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6.6 – FENCING

Iann has a garden in the shape shown below. Find the amount of fencing required to contain the garden. Leave answers in terms of pi.

2

mm

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6.7 – LENGTH AROUND

Bethaney decides to run around the schools track one time. Determine how far she ran. Leave answers in terms of pi.

m

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6.8 – CAN LABEL

Determine the length of the label for the tuna can. Leave answers in terms of pi.