www.ck12.org 1 CHAPTER 1 Circumference Here you ll learn how to find the distance around, or the circumference of, a circle. What if you were given the radius or diameter of a circle? How could you find the distance around that circle? After completing this Concept, you ll be able to use the formula for circumference to solve problems like this one. Guidance Circumference is the distance around a circle. The circumference can also be called the perimeter of a circle. However, we use the term circumference for circles because they are round. Circumference Formula: C = πd where the diameter d = 2r, or twice the radius. So C = 2πr as well. π, or pi is the ratio of the circumference of a circle to its diameter. It is approximately equal to 3.14159265358979323846... To see more digits of π, go to http://www.eveandersson.com/pi/digits/. You should have a π button on your calculator. If you don t, you can use 3.14 as an approximation for π. You can also leave your answers in terms of π for many problems. Example A Find the circumference of a circle with a radius of 7 cm. Plug the radius into the formula. C = 2π(7)=14π 44 cm Example B The circumference of a circle is 64π units. Find the diameter. Again, you can plug in what you know into the circumference formula and solve for d. 64π = πd 64 units = d Chapter 1. Circumference
2 www.ck12.org Example C A circle is inscribed in a square with 10 in. sides. What is the circumference of the circle? Leave your answer in terms of π. From the picture, we can see that the diameter of the circle is equal to the length of a side. C = 10π in. Vocabulary A circle is the set of all points that are the same distance away from a specific point, called the center. Aradius is the distance from the center to the outer rim of the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. Circumference is the distance around a circle. π, or pi is the ratio of the circumference of a circle to its diameter. Guided Practice 1. Find the perimeter of the square in Example C. Is it more or less than the circumference of the circle? Why? 2. The tires on a compact car are 18 inches in diameter. How far does the car travel after the tires turn once? How far does the car travel after 2500 rotations of the tires? 3. Find the radius of circle with circumference 88 in. Answers: 1. The perimeter is P = 4(10)=40 in. In order to compare the perimeter with the circumference we should change the circumference into a decimal. C = 10π 31.42 in. This is less than the perimeter of the square, which makes sense because the circle is inside the square. 2. One turn of the tire is the circumference. This would be C = 18π 56.55 in. 2500 rotations would be 2500 56.55 in approx141,375 in, 11,781 ft, or 2.23 miles.
www.ck12.org 3 3. Use the formula for circumference and solve for the radius. C = 2πr 88 = 2πr 44 π = r r 14 in Interactive Practice Practice Fill in the following table. Leave all answers in terms of π. TABLE 1.1: diameter radius circumference 1. 15 2. 4 3. 6 4. 84π 5. 9 6. 25π 7. 2π 8. 36 9. Find the circumference of a circle with d = 20 π cm. Square PQSR is inscribed in T. RS = 8 2. 10. Find the length of the diameter of T. 11. How does the diameter relate to PQSR? 12. Find the perimeter of PQSR. 13. Find the circumference of T. For questions 14-17, a truck has tires with a 26 in diameter. 14. How far does the truck travel every time a tire turns exactly once? What is this the same as? 15. How many times will the tire turn after the truck travels 1 mile? (1 mile = 5280 feet) 16. The truck has travelled 4072 tire rotations. How many miles is this? 17. The average recommendation for the life of a tire is 30,000 miles. How many rotations is this? Chapter 1. Circumference
4 www.ck12.org CHAPTER 2 Area of a Circle Here you ll learn how to find the area of a circle given its radius or diameter. What if you were given the radius or diameter of a circle? How could you find the amount of space the circle takes up? After completing this Concept, you ll be able to use the formula for the area of a circle to solve problems like this. Guidance MEDIA Click image to the left for more content. To find the area of a circle, all you need to know is its radius. If r is the radius of a circle, then its area is A = πr 2. We will leave our answers in terms of π, unless otherwise specified. To see a derivation of this formula, see http://w ww.rkm.com.au/animations/animation-circle-area-derivation.html, by Russell Knightley. Example A Find the area of a circle with a diameter of 12 cm. If d = 12 cm, then r = 6 cm. The area is A = π 6 2 = 36π cm 2. Example B If the area of a circle is 20π units, what is the radius? Plug in the area and solve for the radius. 20π = πr 2 20 = r 2 r = 20 = 2 5units
www.ck12.org 5 Example C A circle is inscribed in a square. Each side of the square is 10 cm long. What is the area of the circle? The diameter of the circle is the same as the length of a side of the square. Therefore, the radius is 5 cm. A = π5 2 = 25π cm 2 Vocabulary A circle is the set of all points that are the same distance away from a specific point, called the center. Aradius is the distance from the center to the outer rim of the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. Area is the amount of space inside a figure and is measured in square units. π, or pi is the ratio of the circumference of a circle to its diameter. Guided Practice 1. Find the area of the shaded region from Example C. 2. Find the diameter of a circle with area 36π. 3. Find the area of a circle with diameter 20 inches. Answers: 1. The area of the shaded region would be the area of the square minus the area of the circle. A = 10 2 25π = 100 25π 21.46 cm 2 2. First, use the formula for the area of a circle to solve for the radius of the circle. If the radius is 6 units, then the diameter is 12 units. A = πr 2 36π = πr 2 36 = r 2 r = 6 3. If the diameter is 20 inches that means that the radius is 10 inches. Now we can use the formula for the area of a circle. A = π(10) 2 = 100π in 2. Chapter 2. Area of a Circle
6 www.ck12.org Interactive Practice Practice Fill in the following table. Leave all answers in terms of π. TABLE 2.1: radius Area circumference 1. 2 2. 16π 3. 10π 4. 24π 5. 9 6. 90π 7. 35π 7 8. π 9. 60 10. 36 Find the area of the shaded region. Round your answer to the nearest hundredth. 11. 12. 13.