11.5 Areas of Circles and Sectors Hubarth Geometry

. Theorem 11.9 Area of a Circle The area of a circle is 𝜋 times the square of the radius. . r 𝐴=𝜋 𝑟 2

a. Area r = 2.5 cm A = 113.1 cm2 b. Diameter A = πr2 A = πr2 Ex 1 Use the Formula for Area of a Circle Find the indicated measure. a. Area r = 2.5 cm A = 113.1 cm2 b. Diameter A = πr2 A = πr2 = π (2.5)2 113.1 = πr2 = 6.25π = r2 113.1 π A ≈ 19.63 r ≈ 6 D =12

. Theorem 11.10 Area of a Sector The ratio of the area of a sector of a circle to the area of the whole circle (𝜋 𝑟 2 ) is equal To the ratio of the measure of the intercepted arc to 360. 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟 𝐴𝑃𝐵 𝜋 𝑟 2 = 𝑚 𝐴𝐵 360 , or Area of sector APB= 𝑚 𝐴𝐵 360 ∙𝜋 𝑟 2 A r . P B

≈ 39.10 ≈ 161.97 Ex 2 Find Areas of Sectors Find the areas of the sectors formed by ∠UTV. Find the measures of the minor and major arcs. Because m ∠UTV = 70°, m 𝑈𝑉 = 70° and m 𝑈𝑆𝑉 = 360° – 70° = 290°. Area of small sector = 𝑚 𝑈𝑉 360 ∙𝜋 𝑟 2 Area of large sector = 𝑚 𝑈𝑆𝑉 360 ∙𝜋 𝑟 2 = 70 360 ∙𝜋 (8) 2 290 360 ∙𝜋 (8) 2 ≈ 39.10 ≈ 161.97

Ex 3 Use the Area of a Sector Theorem Use the diagram to find the area of ⊙V. Area of sector TVU= 𝑚 𝑇𝑈 360 ∙𝐴𝑟𝑒𝑎 𝑜𝑓 ⊙𝑉 35= 40 360 ∙𝑎𝑟𝑒𝑎 𝑜𝑓 ⊙𝑉 𝑎𝑟𝑒𝑎 𝑜𝑓 ⊙𝑉=315

Ex 4 Standardized Test Practice The area you need to paint is the area of the rectangle minus the area of the entrance. The entrance can be divided into a semicircle and a square. 180° = 36(26) – (π 82 ) + 162 360° = 936 – [32π + 256] ≈ 579.47 The correct answer is C.

Practice Use the diagram to find the indicated measure. 1. Area of D 615.75 ft2 2. Area of red sector 205.25 ft2 3. Area of blue sector 410.50 ft2 4. Find the area o f H. 5. Find the area of the figure. 43.74 m2 907.92 cm2