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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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.
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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
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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
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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)
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