Calculating Total Area of a Regular Polygon
Calculating the total area of a regular polygon is a Grade 6 geometry skill in Reveal Math, Course 1. A regular polygon can be divided into congruent triangles by drawing lines from the center to each vertex. The area of one triangle is (1/2) × side length × apothem, where the apothem is the perpendicular distance from the center to the middle of any side. Multiplying that triangle's area by the number of sides gives the total polygon area. This decomposition strategy shows students how complex shapes can be calculated from simpler ones they already know.
Key Concepts
Property To find the total area of a regular polygon: 1. Find the area of just ONE of the decomposed shapes (using A = 1/2 b h for a triangle, or A = 1/2 h (b1 + b2) for a trapezoid). 2. Multiply that single area by the total number of shapes. Total Area = Number of shapes Area of one shape.
Examples Example 1: A regular pentagon is decomposed into 5 identical triangles. Each triangle has a base of 8 cm and a height of 5.5 cm. Area of one triangle = 1/2 8 5.5 = 22 sq cm. Total area = 5 22 = 110 sq cm. Example 2: A regular hexagon is decomposed into 2 identical trapezoids. Each trapezoid has bases of 10 in and 20 in, and a height of 8.66 in. Area of one trapezoid = 1/2 8.66 (10 + 20) = 129.9 sq in. Total area = 2 129.9 = 259.8 sq in.
Explanation Because regular polygons are perfectly symmetrical, every "slice" you make is identical in size. You don't need to measure every single piece! Find the area of just one single slice, and then multiply it by the total number of slices you have.
Common Questions
How do you calculate the area of a regular polygon?
Divide the polygon into congruent triangles from the center. The area of one triangle = (1/2) × base × apothem, where base is one side length and apothem is the perpendicular distance from center to side. Total area = (number of sides) × (1/2 × base × apothem) = (1/2) × perimeter × apothem.
What is the apothem of a regular polygon?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any of its sides. Every regular polygon has equal apothems for all sides since all sides and angles are equal.
What is the formula for the area of a regular polygon?
Area = (1/2) × perimeter × apothem, or equivalently Area = (1/2) × n × s × a, where n is the number of sides, s is the side length, and a is the apothem. This formula works for any regular polygon.
How is calculating a regular polygon area related to triangle area?
A regular polygon is decomposed into n congruent triangles from the center. Each triangle has base = side length and height = apothem. The total area is just n times one triangle's area — the same triangle area formula applied multiple times.
What are common mistakes when finding a regular polygon's area?
Students often confuse the apothem with the radius (center to vertex distance), which is longer. Using the radius instead of the apothem overestimates the area.
When do students learn the area of regular polygons?
Regular polygon area is introduced in Grade 6 geometry as an extension of triangle and quadrilateral area formulas. It builds on the concept of decomposing shapes in Reveal Math, Course 1.
Which textbook covers calculating the total area of a regular polygon?
This skill is in Reveal Math, Course 1, used in Grade 6 math. It is part of the area chapter, which covers a range of polygon shapes.