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MM2G3c

Learn to find arc lengths and sector areas using circle properties. Justify measurements geometrically and algebraically. Master Theorems 6.20 and 6.21. Practice examples and guided exercises included.

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MM2G3c

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  1. AREAS Of Circles & Sectors MM2G3c Use the properties of circles to solve problems involving the length of an arc and the area of a sector MM2G3d Justify measurements and relationships in circles using geometric and algebraic techniques Standards

  2. Sector of a Circle The region bounded by two radii of the circle and their intercepted arc

  3. Theorem 6.20Area of a Circle The area of a circle is times the square of the radius A = r2

  4. a. Area = π (2.5)2 r= 2.5 cm ANSWER The area of A is about 19.63 square centimeters. EXAMPLE 1 Find the indicated measure. SOLUTION A = πr2 Write formula for the area of a circle. Substitute 2.5 for r. = 6.25π Simplify. ≈ 19.63 Use a calculator.

  5. b. Diameter 113.1 = r2 π A = 113.1 cm2 ANSWER The radius is about 6 cm, so the diameter is about 12 centimeters. EXAMPLE 1 Find the indicated measure. SOLUTION A = πr2 Write formula for the area of a circle. 113.1 = πr2 Substitute 113.1 for A. Divide each side by π. 6 ≈ r Find the positive square root of each side.

  6. Theorem 6.21Area of a Sector The ratio of the area of a sector of a circle to the area of the whole circle ( r2) is equal to the ratio of the measure of the intercepted arc to 360°

  7. Find the areas of the sectors formed by UTV. Because m UTV = 70°, mUV = 70° and mUSV = 360° – 70° = 290°. EXAMPLE 2 SOLUTION STEP 1 Find the measures of the minor and major arcs.

  8. mUV Area of small sector = πr2 360° 70° = π 82 360° EXAMPLE 2 STEP 2 Find the areas of the small and large sectors. Write formula for area of a sector. Substitute. ≈ 39.10 Use a calculator.

  9. 290° = π 82 360° ANSWER The areas of the small and large sectors are about 39.10 square units and 161.97 square units, respectively. mUSV Area of large sector = πr2 360° EXAMPLE 2 Write formula for area of a sector. Substitute. ≈ 161.97 Use a calculator.

  10. 1. Area of D ANSWER about 615.75 ft2 GUIDED PRACTICE Use the diagram to find the indicated measure.

  11. ANSWER about 205.25 ft2 GUIDED PRACTICE Use the diagram to find the indicated measure. 2. Area of red sector

  12. ANSWER about 410.50 ft2. GUIDED PRACTICE Use the diagram to find the indicated measure. 3. Area of blue sector

  13. 315 = Area ofV Solve for Area of V. The area of Vis 315 square meters. mTU Area of sectorTVU = Area ofV 360° 40° 35 = Area ofV ANSWER 360° Example 3 Use the diagram to find the area of V. SOLUTION Write formula for area of a sector. Substitute.

  14. EXAMPLE 4 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.

  15. 180° = 36(26) – (π82 ) +162 360° ANSWER The correct answer is C. EXAMPLE 4 = 936 – [32π+ 256] ≈ 579.47 The area is about 579square feet.

  16. 4. Find the area of H. ANSWER about 907.92 cm2 GUIDED PRACTICE

  17. 5. Find the area of the figure. ANSWER about 43.74 m2 GUIDED PRACTICE

  18. You Try!! Find the indicated measure a. area of circle C

  19. You Try!! Find the indicated measure b. area of shaded sector

  20. You Try!! Find the indicated measure c. area of circle G

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