1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 5.4 Circles

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Know the definition of a circle and its related terms. o Be able to find the circumference (perimeter) and area of a circle.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Know the definition of a circle and its related terms Important Terms and Definitions for Circles Circle: The set of all points in a plane that are equidistant from a fixed point called the center of the circle. Radius: The distance from the center of a circle to any point on the circle. (The letter r is used to represent the radius of a circle.)

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Know the definition of a circles and its related terms Important Terms and Definitions for Circles (cont.) Diameter: The distance from one point on a circle through the center to the point directly opposite it. (The letter d is used to represent the diameter of a circle.) Circumference: The perimeter of a circle.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Know the definition of a circle and its related terms Important Terms and Definitions for Circles (cont.)

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Formulas for Circumference and Area of a Circle Formulas for Circles For circumference: C = 2  r and C =  d For area:

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Formulas for Circumference and Area of a Circle Notes The Greek letter  (Pi) is the symbol used for the constant 3.1415926535…. This number is an infinite nonrepeating decimal. For our purposes, we will use  = 3.14 (accurate to hundredths). However, you should always be aware that 3.14 is only an approximation for  and that related answers are only approximations.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Finding the Circumference and Area of a Circle Find a. the circumference and b. the area of a circle with a radius of 6 ft. Solutions a.Using the formula for circumference: The circumference is 37.68 ft.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Finding the Circumference and Area of a Circle (cont.) b.Using the formula for area: The area is 113.04 ft 2.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Finding the Circumference and Area of a Circle Find a. the circumference and b. the area of a circle with a diameter of 5.2 in.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Finding the Circumference and Area of a Circle (cont.) Solutions a.Using the formula for circumference: The circumference is 16.328 in. b.Using the formula for area: (In this case ) The area is 21.2264 in 2.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Finding the Perimeter of a Semicircle Find the perimeter of a figure that is a semicircle (half of a circle) including its diameter. The diameter is 20 cm long. Solution

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Finding the Perimeter of a Semicircle (cont.) Now we find the perimeter of the figure by adding the length of the semicircle (which is half of the circumference of a circle) to the length of the diameter. Length of semicircle: Perimeter of figure: P = 31.4 + 20 = 51.4 cm.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Finding the Area of a Washer Find the area of the washer (shaded portion) with dimensions as shown in the figure.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Finding the Area of a Washer (cont.) Solution Subtract the area of the inside (smaller) circle from the area of the outside (larger) circle. area of washer

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Finding the Perimeter and Area Find a. the perimeter and b. the area of the figure shown here with a square base and a semicircle cut out of one side. One side of the square is 10 inches long.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Finding the Perimeter and Area (cont.) Solution a.The perimeter of the figure is the sum of the lengths of three sides of the square and the length of the semicircle. (Note that the diameter of the semicircle is 10 in.) Three sides of the square = 10 + 10 + 10 = 30 in. Length of semicircle = Perimeter of the figure = 30 + 15.70 = 45.70 in.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Finding the Perimeter and Area (cont.) b.Find the area of the square then subtract the area of the semicircle. Area of square Area of semicircle Area of the figure = 100 – 39.25 = 60.75 in. 2

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems 1.Find the circumference and area of a circle with a radius of 3 inches. 2.Find the circumference and area of a circle with a diameter of 14 centimeters. 3.Find the perimeter and area of a semicircle with a diameter of 8 feet.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1. 2. 3.