Circumference and Area of Circles Section 8.7

Goal Find the circumference and area of circles.

Key Vocabulary Circle Center Radius Diameter Circumference Central angle Sector

Circle A circle is a plane figure that consists of a set of points that are equidistant from a given point called the center. The circumference of a circle is the distance around it.

Identifying the Parts of a Circle A radius is a line segment that connects the outside of the circle to its center. r

A diameter is a line segment with both endpoints on the circle that passes through the center. d

Numerical Relationships A radius is exactly one-half of a diameter (r = ½ d). Therefore a diameter is twice a radius (d = 2r). 5.5 cm If the radius is 5.5 cm, then the diameter is ___________ cm. 11

Circumference

Remember, circumference is the distance around the circle. If you divide a circle’s circumference by its diameter, you always get the same irrational number – pi (symbol: π ) This is true of every circle. We estimate pi to be 3.14 or the fraction 22/7.

Circumference Formulas C = π d C = 2 π r

Example 41 m C = π d C = (3.14)(41) C = m We substitute 3.14 in for pi.

Example 1 Find the Circumference of a Circle The circumference is about 25 inches. ANSWER SOLUTION C = 2πr Formula for the circumference Substitute 4 for r. = 2π(4) Simplify. = 8π Use 3.14 as an approximation for π. ≈ 8(3.14) Multiply. = Find the circumference of the circle.

Your Turn: ANSWER 38 cm ANSWER 57 ft ANSWER 50 in. Find the circumference of the circle. Round your answer to the nearest whole number

Area of a Circle

Slices of 

Area of a Circle Slices of ½ 

Area of a Circle Slices of ¼ 

Area of a Circle Slices of ¼ 

Area of a Circle Slices of ¼ 

Area of a Circle Slices of ¼ 

Area of a Circle Slices of ¼ 

Area of a Circle Total distance along the top = ½ circumference Slices of ¼ 

Area of a Circle Total distance along the top = ½ circumference Length of side = radius Slices of ¼ 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Slices of 1 / 8 

Area of a Circle Total distance along the top = ½ circumference Slices of 1 / 8 

Area of a Circle Total distance along the top = ½ circumference Length of side = radius Slices of 1 / 8 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Slices of 1 / 16 

Area of a Circle Total distance along the top = ½ circumference Slices of 1 / 16 

Area of a Circle Total distance along the top = ½ circumference Length of side = radius Slices of 1 / 16 

Area of a Circle Slices of 1 / …  And if I had time to create a pie with many, many more slices than 16, maybe 32, or 64, or 128 or even more slices than any other power of 2 (which all these numbers are), then I would end up with a pie with an infinite (  ) number of red and yellow slices which when sliced up would look more and more like a rectangle as the top and bottom edges become less wavy and the sides become more vertical…

Area of a Circle Slices of 1 /  

Area of a Circle Total distance along the top = ½ circumference Slices of 1 /  

Area of a Circle Total distance along the top = ½ circumference Length of side = radius Slices of 1 /  

Area of a Circle Total distance along the top = ½ circumference ½ circumference = ½ (2    radius) Length of side = radius Slices of 1 / 

Area of a Circle Area = Top distance x Length of side Total distance along the top = ½ circumference ½ circumference = ½ (2    radius) Length of side = radius Slices of 1 / 

Area of a Circle Area = ½ (2    radius)  radius Total distance along the top = ½ circumference ½ circumference = ½ (2    radius) Length of side = radius Slices of 1 / 

Area of a Circle Area =   radius  radius Total distance along the top = ½ circumference ½ circumference = ½ (2    radius) Length of side = radius Slices of 1 / 

Area of a Circle Area =   radius 2 Total distance along the top = ½ circumference ½ circumference = ½ (2    radius) Length of side = radius Slices of 1 / 

Area of a Circle A = π r 2 Area = pi times radius squared r

Example 2 Find the Area of a Circle The area is about 154 square centimeters. ANSWER Substitute 7 for r. = π(7) 2 Simplify. = 49π Use a calculator. ≈ Find the area of the circle. A = πr 2 Formula for the area of a circle SOLUTION

Example 3 Use the Area of a Circle The radius is about 11 feet. ANSWER Substitute 380 for A. 380 = πr 2 Divide each side by π. Use a calculator ≈ r 2 Take the positive square root. 11 ≈ r Find the radius of a circle with an area of 380 square feet. A = πr 2 Formula for the area of a circle SOLUTION

Your Turn; ANSWER 201 in. 2 ANSWER 28 cm 2 ANSWER 113 ft 2 Find the area of the circle. Round your answer to the nearest whole number

Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) Central Angle An angle whose vertex lies on the center of the circle. Definition:

Central Angle Central angle- Its sides contain two radii of the circle. A B C  ACB is a central angle Radius

Sum of Central Angles The sum of the central angles of a circle = 360 o –as long as they don’t overlap.

Sector: An Actual Slice of Pie A sector of a circle is the region between two radii of a circle and the included arc. Because a sector is a portion of a circle, the following proportion can be used to find the area of a sector.

Area of a Sector When your central angle is in degrees, the area A of a sector is

Example 4 Find the Area of a Sector Find the area of the blue sector. SOLUTION First find the area of the circle. 1. A = πr 2 = π(9) 2 ≈ The area of the circle is about 254 square meters. Then find the area of the sector. Let x equal the area of the sector. 2. Area of sector Area of entire circle Measure of central angle Measure of entire circle =

Example 4 Find the Area of a Sector The area of the sector is about 85 square meters. ANSWER 360x = 30,480 Simplify. Divide each side by 360. = , x Simplify. x ≈ x = 254 · 120 Cross product property Substitute. 254 x = 360° 120°

Your Turn: In Exercises 7 and 8, A represents the area of the entire circle and x represents the area of the blue sector. Complete the proportion used to find x. Do not solve the proportion. 1. A = 22 m 2 x ? = 180° ? 2. A = 28 ft 2 x ? = ? 360° ANSWER = x ANSWER = x

Your Turn: ANSWER 2 ft 2 Find the area of the blue sector. Round your answer to the nearest whole number. ANSWER 20 cm 2 ANSWER 42 in

Assignment Pg #1 – 35 odd, 39 – 45 odd