Area of Regular Polygons
Overview
A regular polygon is both equilateral and equiangular, and that regularity makes its area easy to find: the polygon divides into congruent isosceles triangles, one for each side, all meeting at the center. Find the area of one triangle and multiply — or use the shortcut formula A = ½Pa, half the perimeter times the apothem.
Vocabulary
- Regular Polygon
- A polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent).
- Center of a Regular Polygon
- The center of the circle circumscribed around the polygon; it is the same distance from every vertex.
- Radius of a Regular Polygon
- The segment from the center to a vertex.
- Apothem
- The perpendicular segment from the center to a side of the polygon. It is the height of each isosceles triangle formed at the center.
- Central (Vertex) Angle
- The angle at the center formed by two radii to consecutive vertices. Its measure is 360° divided by the number of sides.
Main Ideas
- Any regular polygon can be divided into congruent isosceles triangles; the number of triangles equals the number of sides.
- Area = (area of one triangle) × (number of sides).
- Shortcut formula: A = ½Pa, where P is the perimeter and a is the apothem.
- The apothem bisects both the central angle and the side, forming a right triangle with half the central angle — use trigonometry or a special right triangle to find a missing apothem or side.
- For a square, the apothem is half the side length.
Lesson Notes
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Practice Problems
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