Regular Polygon Notes

Published on: September 29, 2026

This Regular Polygon Calculator finds the perimeter and the area of a regular polygon from the number of sides and the length of one side, and shows the working line by line. Every side matches, so the perimeter counts that one length once per side. The polygon also splits into one identical triangle per side, and adding those triangles together gives the area. Following the same pattern every time is what makes a regular polygon easy to check and repeat.

Regular polygon with side labeled s

In the diagram, the side length is labeled \(s\).

Step-by-step method

  1. Set up the problem.
  2. Write the formulas.
  3. Substitute the number of sides and the side length into the perimeter formula.
  4. Work out the perimeter.
  5. Substitute the number of sides and the side length into the area formula.
  6. Work out the top of the area fraction.
  7. Work out the bottom of the area fraction.
  8. Divide the top by the bottom to get the area.

Formula:

\(P = n \times s\)
\(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\)
See Example 1 Hide Example 1

Problem

\(n = 7, s = 5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formulas.
  3. Substitute the number of sides and the side length into the perimeter formula.
  4. Work out the perimeter.
  5. Substitute the number of sides and the side length into the area formula.
  6. Work out the top of the area fraction.
  7. Work out the bottom of the area fraction.
  8. Divide the top by the bottom to get the area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the perimeter and the area of a regular polygon with \(n = 7\) sides, each of length \(s = 5\).

\(n = 7,\; s = 5\)

Step 2

Step 2 - Write the formulas.

In this problem: Every side of a regular polygon is the same length, so the perimeter counts that length once per side. The polygon also splits into one identical triangle per side, and adding those triangles together gives the area formula.

\(P = n \times s\)\(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\)

Step 3

Step 3 - Substitute the number of sides and the side length into the perimeter formula.

In this problem: Replacing \(n\) with \(7\) and \(s\) with \(5\) in \(P = n \times s\) gives \(P = 7 \times 5\).

\(P = 7 \times 5\)

Step 4

Step 4 - Work out the perimeter.

In this problem: Multiplying the number of sides by the side length gives \(7 \times 5 = 35\).

\(P = 7 \times 5 = 35\)

Step 5

Step 5 - Substitute the number of sides and the side length into the area formula.

In this problem: Replacing \(n\) with \(7\) and \(s\) with \(5\) in \(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\) gives \(A = \frac{7 \times 5^{2}}{4 \times \tan(\pi / 7)}\).

\(A = \frac{7 \times 5^{2}}{4 \times \tan(\pi / 7)}\)

Step 6

Step 6 - Work out the top of the area fraction.

In this problem: The top is still written as the number of sides multiplied by the square of the side length, so those are taken in turn.

\(A = \frac{7 \times 5^{2}}{4 \times \tan(\pi / 7)}\)

Step 6a

Step 6a - Work out the top of the area fraction.

In this problem: Squaring the side length gives \(5^{2} = 25\).

\(A = \frac{7 \times 25}{4 \times \tan(\pi / 7)}\)

Step 6b

Step 6b - Work out the top of the area fraction.

In this problem: Multiplying that by the number of sides gives \(7 \times 25 = 175\).

\(A = \frac{175}{4 \times \tan(\pi / 7)}\)

Step 7

Step 7 - Work out the bottom of the area fraction.

In this problem: The bottom is still written as four multiplied by a tangent, so those are taken in turn.

\(A = \frac{175}{4 \times \tan(\pi / 7)}\)

Step 7a

Step 7a - Work out the bottom of the area fraction.

In this problem: Splitting the polygon into 7 triangles makes each half-angle \(\pi / 7\), and \(\tan(\pi / 7) \approx 0.4816\).

\(A = \frac{175}{4 \times 0.4816}\)

Step 7b

Step 7b - Work out the bottom of the area fraction.

In this problem: Multiplying that by four gives \(4 \times 0.4816 \approx 1.9264\).

\(A = \frac{175}{1.9264}\)

Step 8

Step 8 - Divide the top by the bottom to get the area.

In this problem: Dividing the top by the bottom gives \(175 \div 1.9264 \approx 90.84\).

\(A = \frac{175}{1.9264} \approx 90.84\)

Final Answer

\(P = 35,\; A \approx 90.84\)
See Example 2 Hide Example 2

Problem

\(n = 4, s = 2.5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formulas.
  3. Substitute the number of sides and the side length into the perimeter formula.
  4. Work out the perimeter.
  5. Substitute the number of sides and the side length into the area formula.
  6. Work out the top of the area fraction.
  7. Work out the bottom of the area fraction.
  8. Divide the top by the bottom to get the area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the perimeter and the area of a regular polygon with \(n = 4\) sides, each of length \(s = 2.5\).

\(n = 4,\; s = 2.5\)

Step 2

Step 2 - Write the formulas.

In this problem: Every side of a regular polygon is the same length, so the perimeter counts that length once per side. The polygon also splits into one identical triangle per side, and adding those triangles together gives the area formula.

\(P = n \times s\)\(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\)

Step 3

Step 3 - Substitute the number of sides and the side length into the perimeter formula.

In this problem: Replacing \(n\) with \(4\) and \(s\) with \(2.5\) in \(P = n \times s\) gives \(P = 4 \times 2.5\).

\(P = 4 \times 2.5\)

Step 4

Step 4 - Work out the perimeter.

In this problem: Multiplying the number of sides by the side length gives \(4 \times 2.5 = 10\).

\(P = 4 \times 2.5 = 10\)

Step 5

Step 5 - Substitute the number of sides and the side length into the area formula.

In this problem: Replacing \(n\) with \(4\) and \(s\) with \(2.5\) in \(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\) gives \(A = \frac{4 \times 2.5^{2}}{4 \times \tan(\pi / 4)}\).

\(A = \frac{4 \times 2.5^{2}}{4 \times \tan(\pi / 4)}\)

Step 6

Step 6 - Work out the top of the area fraction.

In this problem: The top is still written as the number of sides multiplied by the square of the side length, so those are taken in turn.

\(A = \frac{4 \times 2.5^{2}}{4 \times \tan(\pi / 4)}\)

Step 6a

Step 6a - Work out the top of the area fraction.

In this problem: Squaring the side length gives \(2.5^{2} = 6.25\).

\(A = \frac{4 \times 6.25}{4 \times \tan(\pi / 4)}\)

Step 6b

Step 6b - Work out the top of the area fraction.

In this problem: Multiplying that by the number of sides gives \(4 \times 6.25 = 25\).

\(A = \frac{25}{4 \times \tan(\pi / 4)}\)

Step 7

Step 7 - Work out the bottom of the area fraction.

In this problem: The bottom is still written as four multiplied by a tangent, so those are taken in turn.

\(A = \frac{25}{4 \times \tan(\pi / 4)}\)

Step 7a

Step 7a - Work out the bottom of the area fraction.

In this problem: Splitting the polygon into 4 triangles makes each half-angle \(\pi / 4\), and \(\tan(\pi / 4) \approx 1\).

\(A = \frac{25}{4 \times 1}\)

Step 7b

Step 7b - Work out the bottom of the area fraction.

In this problem: Multiplying that by four gives \(4 \times 1 \approx 4\).

\(A = \frac{25}{4}\)

Step 8

Step 8 - Divide the top by the bottom to get the area.

In this problem: Dividing the top by the bottom gives \(25 \div 4 \approx 6.25\).

\(A = \frac{25}{4} \approx 6.25\)

Final Answer

\(P = 10,\; A \approx 6.25\)
See Example 3 Hide Example 3

Problem

\(n = 6, s = 4\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formulas.
  3. Substitute the number of sides and the side length into the perimeter formula.
  4. Work out the perimeter.
  5. Substitute the number of sides and the side length into the area formula.
  6. Work out the top of the area fraction.
  7. Work out the bottom of the area fraction.
  8. Divide the top by the bottom to get the area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the perimeter and the area of a regular polygon with \(n = 6\) sides, each of length \(s = 4\).

\(n = 6,\; s = 4\)

Step 2

Step 2 - Write the formulas.

In this problem: Every side of a regular polygon is the same length, so the perimeter counts that length once per side. The polygon also splits into one identical triangle per side, and adding those triangles together gives the area formula.

\(P = n \times s\)\(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\)

Step 3

Step 3 - Substitute the number of sides and the side length into the perimeter formula.

In this problem: Replacing \(n\) with \(6\) and \(s\) with \(4\) in \(P = n \times s\) gives \(P = 6 \times 4\).

\(P = 6 \times 4\)

Step 4

Step 4 - Work out the perimeter.

In this problem: Multiplying the number of sides by the side length gives \(6 \times 4 = 24\).

\(P = 6 \times 4 = 24\)

Step 5

Step 5 - Substitute the number of sides and the side length into the area formula.

In this problem: Replacing \(n\) with \(6\) and \(s\) with \(4\) in \(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\) gives \(A = \frac{6 \times 4^{2}}{4 \times \tan(\pi / 6)}\).

\(A = \frac{6 \times 4^{2}}{4 \times \tan(\pi / 6)}\)

Step 6

Step 6 - Work out the top of the area fraction.

In this problem: The top is still written as the number of sides multiplied by the square of the side length, so those are taken in turn.

\(A = \frac{6 \times 4^{2}}{4 \times \tan(\pi / 6)}\)

Step 6a

Step 6a - Work out the top of the area fraction.

In this problem: Squaring the side length gives \(4^{2} = 16\).

\(A = \frac{6 \times 16}{4 \times \tan(\pi / 6)}\)

Step 6b

Step 6b - Work out the top of the area fraction.

In this problem: Multiplying that by the number of sides gives \(6 \times 16 = 96\).

\(A = \frac{96}{4 \times \tan(\pi / 6)}\)

Step 7

Step 7 - Work out the bottom of the area fraction.

In this problem: The bottom is still written as four multiplied by a tangent, so those are taken in turn.

\(A = \frac{96}{4 \times \tan(\pi / 6)}\)

Step 7a

Step 7a - Work out the bottom of the area fraction.

In this problem: Splitting the polygon into 6 triangles makes each half-angle \(\pi / 6\), and \(\tan(\pi / 6) \approx 0.5774\).

\(A = \frac{96}{4 \times 0.5774}\)

Step 7b

Step 7b - Work out the bottom of the area fraction.

In this problem: Multiplying that by four gives \(4 \times 0.5774 \approx 2.3096\).

\(A = \frac{96}{2.3096}\)

Step 8

Step 8 - Divide the top by the bottom to get the area.

In this problem: Dividing the top by the bottom gives \(96 \div 2.3096 \approx 41.57\).

\(A = \frac{96}{2.3096} \approx 41.57\)

Final Answer

\(P = 24,\; A \approx 41.57\)
See Example 4 Hide Example 4

Problem

\(n = 5, s = 3.5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the formulas.
  3. Substitute the number of sides and the side length into the perimeter formula.
  4. Work out the perimeter.
  5. Substitute the number of sides and the side length into the area formula.
  6. Work out the top of the area fraction.
  7. Work out the bottom of the area fraction.
  8. Divide the top by the bottom to get the area.

Step 1

Step 1 - Set up the problem.

In this problem: We are finding the perimeter and the area of a regular polygon with \(n = 5\) sides, each of length \(s = 3.5\).

\(n = 5,\; s = 3.5\)

Step 2

Step 2 - Write the formulas.

In this problem: Every side of a regular polygon is the same length, so the perimeter counts that length once per side. The polygon also splits into one identical triangle per side, and adding those triangles together gives the area formula.

\(P = n \times s\)\(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\)

Step 3

Step 3 - Substitute the number of sides and the side length into the perimeter formula.

In this problem: Replacing \(n\) with \(5\) and \(s\) with \(3.5\) in \(P = n \times s\) gives \(P = 5 \times 3.5\).

\(P = 5 \times 3.5\)

Step 4

Step 4 - Work out the perimeter.

In this problem: Multiplying the number of sides by the side length gives \(5 \times 3.5 = 17.5\).

\(P = 5 \times 3.5 = 17.5\)

Step 5

Step 5 - Substitute the number of sides and the side length into the area formula.

In this problem: Replacing \(n\) with \(5\) and \(s\) with \(3.5\) in \(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\) gives \(A = \frac{5 \times 3.5^{2}}{4 \times \tan(\pi / 5)}\).

\(A = \frac{5 \times 3.5^{2}}{4 \times \tan(\pi / 5)}\)

Step 6

Step 6 - Work out the top of the area fraction.

In this problem: The top is still written as the number of sides multiplied by the square of the side length, so those are taken in turn.

\(A = \frac{5 \times 3.5^{2}}{4 \times \tan(\pi / 5)}\)

Step 6a

Step 6a - Work out the top of the area fraction.

In this problem: Squaring the side length gives \(3.5^{2} = 12.25\).

\(A = \frac{5 \times 12.25}{4 \times \tan(\pi / 5)}\)

Step 6b

Step 6b - Work out the top of the area fraction.

In this problem: Multiplying that by the number of sides gives \(5 \times 12.25 = 61.25\).

\(A = \frac{61.25}{4 \times \tan(\pi / 5)}\)

Step 7

Step 7 - Work out the bottom of the area fraction.

In this problem: The bottom is still written as four multiplied by a tangent, so those are taken in turn.

\(A = \frac{61.25}{4 \times \tan(\pi / 5)}\)

Step 7a

Step 7a - Work out the bottom of the area fraction.

In this problem: Splitting the polygon into 5 triangles makes each half-angle \(\pi / 5\), and \(\tan(\pi / 5) \approx 0.7265\).

\(A = \frac{61.25}{4 \times 0.7265}\)

Step 7b

Step 7b - Work out the bottom of the area fraction.

In this problem: Multiplying that by four gives \(4 \times 0.7265 \approx 2.906\).

\(A = \frac{61.25}{2.906}\)

Step 8

Step 8 - Divide the top by the bottom to get the area.

In this problem: Dividing the top by the bottom gives \(61.25 \div 2.906 \approx 21.08\).

\(A = \frac{61.25}{2.906} \approx 21.08\)

Final Answer

\(P = 17.5,\; A \approx 21.08\)