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Geometry Two-dimensional measurement

Area of regular polygons (using apothem)

20 practice questions 0 video lessons Theory + worked examples

Area of Regular Polygons

Common Core Geometry • Standard G-MG.1 • Two-Dimensional Measurement

Area of Regular Polygons is a topic in Two-Dimensional Measurement in the Common Core State Standards. It is aligned to Standard G-MG.1, which requires students to use geometric shapes and their measures to model and solve problems, including the area of regular polygons.

A regular polygon's area is \(\dfrac12 aP\), one half the apothem times the perimeter.

Common Core Geometry › Two-Dimensional Measurement › Area of Regular Polygons  —  Standard G-MG.1

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Theory

A regular polygon has equal sides and equal angles. Its area uses the apothem \(a\) — the perpendicular distance from the center to the midpoint of a side:

\[A=\dfrac12 aP,\]

where \(P\) is the perimeter (number of sides times the side length).

Why it works: the polygon splits into triangles of height \(a\); their total area is \(\dfrac12 a\) times the total base, the perimeter.
Regular polygon with apothem The apothem is the perpendicular distance from the center to a side; the area is one half the apothem times the perimeter. apothem A = ½ × apothem × perimeter
The apothem is the center-to-side distance; \(A=\dfrac12 aP\).
Regular polygon area Regular polygon area Regular polygon area A = ½ a P a = apothem, P = perimeter P = number of sides × side length
The regular-polygon area formula.

The area and perimeter:

\[A=\dfrac12 aP,\qquad P=n\times(\text{side length})\]
area is one half the apothem times the perimeter; the perimeter is the number of sides times the side length
Find the perimeter first if you are given the side length and number of sides.

How to find a regular polygon's area

  1. Find the perimeter \(P=n\times\) side.
  2. Identify the apothem \(a\).
  3. Apply \(A=\dfrac12 aP\).
  4. Rearrange to solve for \(a\) or \(P\) if needed.
Example 1 — Area from apothem and perimeter
A regular polygon has apothem \(6\) and perimeter \(40\). Find its area.
Solution

Use \(A=\dfrac12 aP\).

\(A\)\(=\)\(\dfrac12(6)(40)=120\)
the area is 120 square units
Example 2 — Regular hexagon
A regular hexagon has side \(8\) and apothem \(4\sqrt3\). Find its area.
Solution

Perimeter \(=6\times 8=48\), then \(A=\dfrac12 aP\).

\(A\)\(=\)\(\dfrac12(4\sqrt3)(48)\)
\(=\)\(96\sqrt3\)
the area is 96 root 3
Example 3 — Find the apothem
A regular pentagon has area \(100\) and perimeter \(40\). Find its apothem.
Solution

Solve \(\dfrac12 aP=100\).

\(\dfrac12 a(40)\)\(=\)\(100\)
\(20a\)\(=\)\(100\)
\(a\)\(=\)\(5\)
the apothem is 5
Example 4 — What the apothem is
What is the apothem of a regular polygon?
Solution

The perpendicular distance from the center to the midpoint of a side.

the perpendicular distance from the center to a side

Common pitfalls

The apothem goes to the midpoint of a side, not to a vertex (that's the radius).
Use the whole perimeter, not a single side.
Include the \(\dfrac12\) — the formula is half the apothem times perimeter.

Frequently asked questions

What is the apothem?

The perpendicular distance from the center of a regular polygon to the midpoint of a side.

What is the area formula for a regular polygon?

\(A=\dfrac12 aP\): one half the apothem times the perimeter.

How do you find the perimeter?

Multiply the number of sides by the side length.

How is the apothem different from the radius?

The apothem reaches the midpoint of a side; the radius reaches a vertex.

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