10.4 areas of regular polygons geometry. objectives/assignment find the area of an equilateral...
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Objectives/Assignment
• Find the area of an equilateral triangle.
• Find the area of a regular polygon, such as the area of a square, rectangle, rhombus etc.
Finding the area of an equilateral triangle• The area of any triangle with base
length b and height h is given by
A = ½bh. The following formula for equilateral
triangles; however, uses ONLY the side length.
Area of an equilateral triangle
• The area of an equilateral triangle is one fourth the square of the length of the side times
A = ¼ s2
3
3
s s
s
A = ¼ s23
Finding the area of an Equilateral Triangle• Find the area of an equilateral
triangle with 8 inch sides.
A = ¼ s23
A = ¼ 823
A = ¼ • 643
A = • 163
A = 16 3
Area of an equilateral TriangleSubstitute values.
Simplify.
Multiply ¼ times 64.
Simplify.
Area Theorems
• Area of a Rectangle
• The area of a rectangle is the product of its base and height.
h
b
A = bh
Area Theorems
• Area of a Parallelogram• The area of a
parallelogram is the product of a base and height.
A = bh
h
b
Area Theorems
• Area of a Triangle• The area of a
triangle is one half the product of a base and height.
A = ½ bh
h
b
Justification
• You can justify the area formulas for triangles follows.
• The area of a triangle is half the area of a parallelogram with the same base and height.
Areas of Trapezoids
Area of a Trapezoid The area of a
trapezoid is one half the product of the height and the sum of the bases.
A = ½ h(b1 + b2)
b1
b2
h
Area of a Kite The area of a kite
is one half the product of the lengths of its diagonals.
A = ½ d1d2
d1
d2
Areas of Rhombuses
Area of a Rhombus
The area of a rhombus is one half the product of the lengths of the diagonals.
A = ½ d1 d2
d1
d2
Finding the Area of a Trapezoid
• Find the area of trapezoid WXYZ.
• Solution: The height of WXYZ is h=5 – 1 = 4
• Find the lengths of the bases.b1 = YZ = 5 – 2 = 3
b2 = XW = 8 – 1 = 7
W(8, 1)X(1, 1)
Z(5, 5)Y(2, 5)
Finding the Area of a Trapezoid
Substitute 4 for h, 3 for b1, and 7 for b2 to find the area of the trapezoid.
A = ½ h(b1 + b2) Formula for area of a trapezoid.
A = ½ (4)(3 + 7 ) SubstituteA = ½ (40) SimplifyA = 20 Simplify
The area of trapezoid WXYZ is 20 square units
8
6
4
2
5 10 15
W(8, 1)X(1, 1)
Z(5, 5)Y(2, 5)
Finding the area of a rhombus
• Use the information given in the diagram to find the area of rhombus ABCD.
• Solution— Use the formula for
the area of a rhombus d1 = BD = 30 and d2 = AC =40
15
15
20 20A
B
C
DE
Finding the area of a rhombus
A = ½ d1 d2
A = ½ (30)(40)A = ½ (120)A = 60 square units
15
15
20 20A
B
C
DE
A = ¼ s2 31. Find the height of one isosceles triangle by using Pythagorean Formula
H = 10² - 6² = 8²
2. A = ½ * 8 * 12 = 96 m²
1. Find the sum of bases
b1 + b2 = 24*2 = 48
2. A = ½ * 48 * 9 = 216 m²
Justification
• You can justify the area formulas for parallelograms as follows.
• The area of a parallelogram is the area of a rectangle with the same base and height.