1. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Areas of Circles and Sectors How do we find the areas of circles and sectors? M2 Unit 3: Day 10 Lesson 6.8 Sunday, September 13, 2015

2. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Daily Homework Quiz Find the indicated measure. 1. Circumference ANSWER about 81.68 in. Daily Homework Quiz ANSWER about 7.64 ft 2. Radius C = 48 ft

3. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Daily Homework Quiz 3. Length of AB ANSWER 8.64 cm Daily Homework Quiz 4. Find the total circumference of the circles. ANSWER 100.53 cm

4. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Daily Homework Quiz Find the indicated measure. 5. Radius ANSWER about 21.88 cm Daily Homework Quiz 6. Circumference ANSWER about 222.72 in.

5. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector.

6. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Use the formula for area of a circle Find the indicated measure. a. Area r = 2.5 cm SOLUTION Write formula for the area of a circle. = π (2.5) 2 Substitute 2.5 for r. = 6.25π Simplify. ≈ 19.63 Use a calculator. A = πr 2 The area about 19.63 square centimeters. ANSWER EXAMPLE

7. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. EXAMPLE 1 Use the formula for area of a circle Find the indicated measure. A = 113.1 cm 2 b. Diameter Write formula for the area of a circle. Substitute 113.1 for A. 113.1 = πr 2 = r 2 113.1 π Divide each side by π. 6 ≈ r Find the positive square root of each side. The radius is about 6 centimeters, so the diameter is about 12 centimeters. ANSWER A = πr 2 SOLUTION EXAMPLE

8. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Sector of a Circle: the region bounded by two radii of the circle and their intercepted arc

9. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Area of a Sector: a portion of the area of the whole circle

10. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. EXAMPLE 2 Find areas of sectors Find the areas of the sectors formed by UTV. SOLUTION STEP 1 Find the measures of the minor and major arcs. Because m UTV = 70 °, mUV = 70 ° and mUSV = 360 ° – 70 ° = 290 °. EXAMPLE

11. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. EXAMPLE 2 STEP 2 Find the areas of the small and large sectors. Area of small sector = πr 2 m UV 360 ° Write formula for area of a sector. = π 8 2 70 ° 360 ° Substitute. ≈ 39.10 Use a calculator. Find areas of sectors EXAMPLE Find the areas of the sectors formed by UTV.

12. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. EXAMPLE 2 = π 8 2 290 ° 360 ° Substitute. ≈ 161.97 Use a calculator. Find areas of sectors The areas of the small and large sectors are about 39.10 square units and 161.97 square units, respectively. ANSWER Area of large sector = πr 2 m USV 360 ° Write formula for area of a sector. EXAMPLE

13. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE 1. Area of D Use the diagram to find the indicated measure. SOLUTION A = πr 2 Write formula for the area of a circle. = π (14) 2 Substitute 14 for r. = 196π Simplify. = 617.75 Use a calculator The area of D is about 615.75 ft 2. Guided Practice

14. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE 2. Area of red sector SOLUTION STEP 1 Find the measures of major arcs. Use the diagram to find the indicated measure. Because m FDE = 120, mFE = 120 and mFGE = 360 – 120 = 140. Guided Practice

15. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE Write formula for area of a sector. = π 14 2 120 360 Substitute. = 205.25 Use a calculator. Area of red sector = π r 2 m FE 360 The area of red sector is about 205.25 ft 2. ANSWER STEP 2 Find the area of red sector. Guided Practice

16. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE 3. Area of blue sector SOLUTION STEP 1 Find the measure of the blue arc. Because m FDE = 120, MFE = 120 and mFGE = 360 – 120 = 240. Use the diagram to find the indicated measure. Guided Practice

17. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Area of blue sector = π r 2 m FEG 360 Write formula for area of a sector. = π 14 2 240 360 Substitute. = 410.50 ft 2 Use a calculator. Area of blue sector is about 410.50 ft 2. ANSWER STEP 2 Find the area of blue sector. Guided Practice

18. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Use the Area of a Sector Theorem Use the diagram to find the area of V. SOLUTION Area of sector TVU = Area of V m TU 360 ° Write formula for area of a sector. 35 = Area of V 40 ° 360 ° Substitute. 315 = Area of V Solve for Area of V. The area of V is 315 square meters. ANSWER EXAMPLE

19. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. EXAMPLE 4 Standardized Test Practice SOLUTION 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.

20. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. EXAMPLE 4 = 936 – [32π + 256] 180 ° = 36(26) – (π 8 2 ) + 16 2 360 ° ≈ 579.47 The area is about 579 square feet. Standardized Test Practice The correct answer is C. ANSWER

21. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE 4. Find the area of H. SOLUTION Area of sector FHG = Area of H m FG 360 Write formula for area of a sector. 214.37 = Area of H 85 360 Substitute. 907.92 = Area of H Solve for Area of H. The area of H is 907.92 cm 2. ANSWER Guided Practice

22. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE A = b h 1 2 Write formula for area of a triangle. = 7 7 1 2 Substitute. = 24.5 = 24.5 m SOLUTION STEP 1 Take the top as base, which is 7 m and find the area of the triangle Guided Practice 5. Find the area of the figure.

23. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE Area of figure = Area of triangle + Area of semicircle = 19.5 multiply = 24.5 + 19.5 = 43.74 m 2 STEP 2 find the area of the semicircle A = π r 2 1 2 1 = π (3.5) 2 2 Write formula for area of a sector. Substitute. STEP 3 Add the areas The area of the figure is about 43.74 m 2. ANSWER Guided Practice

24. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. GUIDED PRACTICE 6. If you know the area and radius of a sector of a circle, can you find the measure of the intercepted arc? Explain. yes; the formula for the area of sector is SOLUTION m A = and if you solve this for m, you get 360 π r 2 360A π r 2 Guided Practice

25. MM2G3 Students will understand properties of circles. MM2G3 c Use the properties of circles to solve problems involving the length of an arc and the area of a sector. Homework: Page 232 (#1-15 all)