Developing Formulas for Circles and Regular Polygons Geometry Developing Formulas for Circles and Regular Polygons CONFIDENTIAL

Find the each measurement: Warm up Find the each measurement: 1) d2 of a kite if A = 14 cm2 and d1 = 20 cm. 2) the area of a trapezoid in which b1 = 3 yd, b2 = 6 yd and h = 4 yd Answer: 1) d2 = 1.4 cm; 2) 18 yd2 CONFIDENTIAL

Solving for C gives the formula C= ∏d. Formulas for Circle A circle is the locus of a point in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has a radius r= AB and diameter d = CD. The irrational number ∏ is defined as the ratio of the circumference C to the diameter d, or ∏ = C. d Solving for C gives the formula C= ∏d. Also d = 2r, so C= 2∏r. C D B A CONFIDENTIAL

Formulas for Circle You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram. ∏r r The base of the parallelogram is about half the circumference, or ∏r, and the height is close to the radius r. So A ≈ ∏r. r = ∏r2. CONFIDENTIAL

Formulas for Circle The more pieces you divide the circle into, the more accurate the estimate will be. CONFIDENTIAL

Circumference and Area of a Circle A circle with diameter d and radius r has circumference C= ∏d or C= 2∏r and area A = ∏r2. diameter radius CONFIDENTIAL

Finding measurements of of Circle Find each measurement: A) the area of P in terms of ∏. 16 cm P A = ∏r2 A = ∏(8)2 A = 64∏ cm2 Area of a circle. Divide the diameter by 2 to find the radius, 8. Simplify. CONFIDENTIAL

Finding measurements of of Circle Find each measurement: B) the radius of X in which C = 24∏ in. C = 2∏r 24A = 2∏r r = 12 in Circumference of a circle. Substitute 24∏ for C. Divide both sides by 2∏. CONFIDENTIAL

Finding measurements of of Circle C) the circumference of S in which A = 9x2∏ cm2. Step 1: Use the given area to solve for r. A = ∏r2 9x2∏ = ∏r2 9x2= 2r2 3x= r Area of a circle. Substitute 9x2 for A. Divide both sides by 2. Take square root of both sides. Step 2: Use the value of r to find the circumference. Substitute 3x for r. Divide both sides by 2. Simplify. C = 2∏r C = 2∏(3x) C = 6x∏ CONFIDENTIAL

1) Find the area of A in terms of ∏ in which C = (4x - 6)∏ m. Now you try! 1) Find the area of A in terms of ∏ in which C = (4x - 6)∏ m. ANSWER: (4x2 – 12x + 9)∏ m2 CONFIDENTIAL

Music Application A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the area of the top of each drum. Round to the nearest tenth. 10 in. diameter A = ∏(5)2 since, r = 10 2 A≈ 78.5 in2 14 in. diameter A = ∏(7)2 since, r = 12 A≈ 153.9 in2 12 in. diameter A = ∏(6)2 since, r = 10 2 A≈ 113.1 in2 CONFIDENTIAL

Now you try! 2) A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the circumference of the top of each drum. Round to the nearest tenth. ANSWER: C ≈31.4 in; C ≈ 37.7 in; C ≈ 44.0 in CONFIDENTIAL

Formulas for a Regular Polygon The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its side pass through consecutive vertices. Each central angle measure of a regular n - gon is 360°. n C H B D G F E Regular pentagon DEFGH has center C, apothem BC, and central angle /DCE. CONFIDENTIAL

Formulas for a Regular Polygon To find the area of a regular n - gon with side length s and apothem a, divide it into n congruent isosceles triangles. area of each triangle : 1as 2 total area of the polygon: A = n. 1as or A= 1aP The perimeter is P = ns CONFIDENTIAL

Formulas for a Regular Polygon The area of a regular polygon with apothem a and perimeter P is A = 1aP 2 CONFIDENTIAL

Finding the area of a Regular Polygon Find the area of each regular polygon. Round to the nearest tenth. A) a regular hexagon with side length 6 m The perimeter is 6(6)= = 36 m . 6 m 3m 3√3m The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°- 90° Triangle theorem, the apothem is 3√3. A= 1aP 2 A= 1(3√3)(36) A= 54√3≈ 93.5 m2 Area of a regular polygon Substitute 3√3 for a and 36 for P Simplify CONFIDENTIAL

B) a regular pentagon with side length 6 in. 36° 4 in a Step 1: Draw the pentagon. Draw an isosceles triangle with its vertex at the centre of the pentagon. The central angle is 360° = 72 ° . 5 Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. CONFIDENTIAL

Step 2: Use the tangent ratio to find the apothem. 36° 4 in a Step 2: Use the tangent ratio to find the apothem. tan 36° = 4 a a = 4 tan36° The tangent of an angle is opp. leg adj. leg Solve for a. CONFIDENTIAL

Step 3 : Use the apothem and the given side length to find the area . 36° 4 in a Step 3 : Use the apothem and the given side length to find the area . A= 1aP 2 A = 1. 4 .(40) 2 tan36° A ≈ 101.1 in2 Area of a regular polygon. The perimeter is 8(5) = 40 in. Simplify. Round to the nearest tenth. CONFIDENTIAL

3) Find the area of a regular octagon with a side of 4 cm. Now you try! 3) Find the area of a regular octagon with a side of 4 cm. ANSWER: A ≈ 77.3 cm2 CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

Find each measurement: Assessment Find each measurement: 10 cm ∏ C 1) the circumference of C. 3x in A 2) the area of A in terms of ∏. Answer: 1) 10 cm ; 2) 9x2 in2 ; 3) 12∏ft 3) the circumference of Pin which A = 36∏ ft2. CONFIDENTIAL

4) a food pizza parlor offers pizzas with diameters of 8 in. , 10 in 4) a food pizza parlor offers pizzas with diameters of 8 in., 10 in. and 12 in. Find the area of each pizza size. Round to the nearest tenth. Answer: 4) 50.3 in2 cm ; 78.3 in2 ; 3) 113.1 in2 CONFIDENTIAL

Find the area of each regular polygon. Round to the nearest tenth: 6) 5) 3 cm Answer: 5) 258.8 in2 ; 6) 32.7 cm2 CONFIDENTIAL

Find the area of each regular polygon. Round to the nearest tenth: An equilateral triangle with an apothem length of 2 ft. 8) An regular dodecagon with an side length of 5 ft. Answer: 7) 2.3 ft2 ; 8) 279.9 m2 CONFIDENTIAL

Solving for C gives the formula C= ∏d. Let’s review Formulas for Circle A circle is the locus of a point in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has a radius r= AB and diameter d = CD. The irrational number ∏ is defined as the ratio of the circumference C to the diameter d, or ∏ = C. d Solving for C gives the formula C= ∏d. Also d = 2r, so C= 2∏r. C D B A CONFIDENTIAL

Circumference and Area of a Circle A circle with diameter d and radius r has circumference C= ∏d or C= 2∏r and area A = ∏r2. diameter radius CONFIDENTIAL

Finding measurements of of Circle C) the circumference of S in which A = 9x2∏ cm2. Step 1: Use the given area to solve for r. A = ∏r2 9x2∏ = ∏r2 9x2= 2r2 3x= r Area of a circle. Substitute 9x2 for A. Divide both sides by 2. Take square root of both sides. Step 2: Use the value of r to find the circumference. Substitute 3x for r. Divide both sides by 2. Simplify. C = 2∏r C = 2∏(3x) C = 6x∏ CONFIDENTIAL

Music Application A drum kit contains three drums with diameters of 10 in., 12 in. and 14 in. Find the area of the top of each drum. Round to the nearest tenth. 10 in. diameter A = ∏(5)2 since, r = 10 2 A≈ 78.5 in2 14 in. diameter A = ∏(7)2 since, r = 12 A≈ 153.9 in2 12 in. diameter A = ∏(6)2 since, r = 10 2 A≈ 113.1 in2 CONFIDENTIAL

Formulas for a Regular Polygon The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its side pass through consecutive vertices. Each central angle measure of a regular n - gon is 360°. n C H B D G F E Regular pentagon DEFGH has center C, apothem BC, and central angle /DCE. CONFIDENTIAL

Formulas for a Regular Polygon The area of a regular polygon with apothem a and perimeter P is A = 1aP 2 CONFIDENTIAL

Finding the area of a Regular Polygon Find the area of each regular polygon. Round to the nearest tenth. A) a regular hexagon with side length 6 m The perimeter is 6(6)= = 36 m . 6 m 3m 3√3m The hexagon can be divided into 6 equilateral triangles with side length 6 m. By the 30°-60°- 90° Triangle theorem, the apothem is 3√3. A= 1aP 2 A= 1(3√3)(36) A= 54√3≈ 93.5 m2 Area of a regular polygon Substitute 3√3 for a and 36 for P Simplify CONFIDENTIAL

You did a great job today! CONFIDENTIAL