1. Developing Formulas for Circles and Regular Polygons
Geometry
Developing Formulas for Circles and Regular Polygons
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2. 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
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3. 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
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4. 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.
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5. Formulas for Circle
The more pieces you divide the circle into, the more accurate the estimate will be.
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6. 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
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7. 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.
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8. 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∏.
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9. 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∏
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10. 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
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11. 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≈ in2
12 in. diameter
A = ∏(6)2 since, r = 10 2
A≈ in2
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12. 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
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13. 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.
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14. 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
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15. Formulas for a Regular Polygon
The area of a regular polygon with apothem a and perimeter P is
A = 1aP 2
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16. 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
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17. 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.
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18. 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 =
tan36°
The tangent of an angle is opp. leg
adj. leg
Solve for a.
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19. 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 = (40)
2 tan36°
A ≈ in2
Area of a regular polygon.
The perimeter is 8(5) = 40 in.
Simplify. Round to the nearest tenth.
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20. 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
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21. Now some problems for you to practice !
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22. 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.
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23. 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) in2
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24. Find the area of each regular polygon. Round to the nearest tenth:6) 5)
3 cm
Answer: 5) in2 ; 6) 32.7 cm2
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25. 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) m2
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26. 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
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27. 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
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28. 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∏
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29. 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≈ in2
12 in. diameter
A = ∏(6)2 since, r = 10 2
A≈ in2
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30. 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.
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31. Formulas for a Regular Polygon
The area of a regular polygon with apothem a and perimeter P is
A = 1aP 2
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32. 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
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33. You did a great job today!
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