Areas of Regular Polygons and Composite Figures

11-4 Areas of Regular Polygons and Composite Figures Notes VOCABULARY Center of a Regular Polygon Radius of a Regular Polygon Apothem Central Angle of a Regular Polygon Composite Figure Example 1 – Identify Segments and Angles in Regular Polygons In the figure, pentagon PQRST is inscribed in ʘX. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. 1. Center: 2. Radius: 3. Apothem: 4. Central Angle A regular regular polygon with five sides is a pentagon. Thus, the measure of each central angle of pentagon PQRST is ________ or ________. Name __________________________ Date __________________________ Real World Example 2 – Area of a Regular Polygon The top of the table shown is a regular hexagon with a side length of 3 feet and an apothem of 1.7 feet. What is the area of the tabletop to the nearest tenth? Since the polygon has 6 sides, the polygon can be divided into ____ congruent isosceles triangles, each with a base of ____ ft and a height of ________ ft. Find the area of one triangle: = Area of a triangle = Substitution = Simplify The total area will be the product of one of the triangles multiplied by the number of triangles. Since there are six triangles, __________ ● ____ = __________ ft2. Key Concept: Area of a Regular Polygon The area A of a regular n-gon with side length s is one half the product of the apothem a and perimeter P. 1 2 Symbols 𝐴 = 𝑎(𝑛𝑠) or 𝐴 = __________ Example 3 – Use the Formula for the Area of a Regular Polygon A. Find the area of the regular hexagon. Round to the nearest tenth. Find the measure of a central angle. A regular hexagon has ____ congruent central angles, so 𝑚∠𝑄𝑃𝑅 = __________ = ________. Find the apothem. Apothem ̅̅̅̅ 𝑃𝑆 is the height of isosceles ΔQPR. It bisects QPR, so mSPR = ________. It also bisects QR, so SR = ________ meters. ΔPSR is a ____________________ right triangle with a shorter leg that measures 2.5 meters, so 𝑃𝑆 = ____________. Use the apothem and side length to find the area. = Area of a regular polygon = Substitution = Use a calculator B. Find the area of the regular pentagon. Round to the nearest tenth. A regular pentagon has 5 congruent central angles, so 𝑚∠𝐴𝐶𝐵 = ____________ = ________. Apothem ̅̅̅̅ 𝐶𝐷 is the height of ____________________ ΔBCA. It bisects BCA, so mBCD = ________. Use trigonometric ratios to find the side length and apothem of the polygon. AB = 2 ● ______ or 2(________________). So, the pentagon’s perimeter is 5 ● 2(________________) = __________________. The length of the apothem CD is ________________. = Area of a regular polygon = Substitution = Use a calculator Example 4 – Find the Area of a Composite Figure by Adding The dimensions of an irregularly shaped pool are shown. What is the area of the surface of the pool? The figure can be separated into a rectangle with dimensions ______ feet by ______ feet, a triangle with a base of ______ feet and a height of ______ feet, and two semicircles with radii of ____ feet. Area of Composite Figure = Area of Rectangle + Area of Triangle + Area of Two Semicircles = Formulae = Substitution = Simplify = Use a calculator Example 5 – Find the Area of a Composite Figure by Subtracting Find the area of the shaded portion of the figure. To find the area of the figure, subtract the area of the smaller rectangle from the area of the larger rectangle. The length of the larger rectangle is 25 + 100 + 25 or ________ feet. The width of the larger rectangle is 25 + 20 + 25 or ______ feet. Area of Shaded Region = Area of Large Rectangle – Area of Small Rectangle = Formulae = Substitution = Simplify = Combine like terms