10.4 Areas of Regular Polygons Geometry

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.

Important!!!

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 s s s A = ¼ s2

Finding the area of an Equilateral Triangle Find the area of an equilateral triangle with 8 inch sides. A = ¼ s2 Area of an equilateral Triangle A = ¼ 82 Substitute values. A = ¼ • 64 Simplify. A = • 16 Multiply ¼ times 64. A = 16 Simplify.

Area Theorems A = bh Area of a Rectangle The area of a rectangle is the product of its base and height. h b A = bh

Area Theorems A = bh Area of a Parallelogram The area of a parallelogram is the product of a base and height. h b A = bh

Area Theorems A = ½ bh Area of a Triangle The area of a triangle is one half the product of a base and height. h b A = ½ bh

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 b2 h b1 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)

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 d2 d1

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

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 ) Substitute A = ½ (40) Simplify A = 20 Simplify The area of trapezoid WXYZ is 20 square units

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 20 20 A C 15 D E

Finding the area of a rhombus A = ½ d1 d2 A = ½ (30)(40) A = ½ (120) A = 60 square units 15 20 20 A C 15 D E

Practice

A = ¼ s2 Find the sum of bases Find the height of one isosceles b1 + b2 = 24*2 = 48 2. A = ½ * 48 * 9 = 216 m² Find the height of one isosceles triangle by using Pythagorean Formula H = 10² - 6² = 8² 2. A = ½ * 8 * 12 = 96 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.

Using the Area Theorems 9 Find the area of ABCD. Solution: Use AB as the base. So, b=16 and h=9 Area=bh=16(9) = 144 square units.