Regular Polygon Calculator

Published on: September 29, 2024
Final Answer: Free • Full Steps: Plus

This Regular Polygon Calculator helps you find the perimeter and area of a regular polygon when the number of sides and side length are known. It uses the number of sides n and side length s to calculate both measurements, with answers written in units and square units. First find the perimeter by multiplying the number of sides by the side length, then use the area formula for a regular polygon. It is a simple way to check answers, understand regular polygon formulas, and practise basic geometry step by step.

Regular polygon with side labeled s

In the diagram, the side length is labeled s.

Step-by-step method

  1. Identify what is given.
  2. Write the formulas.
  3. Substitute the values to calculate the perimeter.
  4. Substitute the values into the area formula.
  5. Calculate the area.

Formula:

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

Example 1:

\(n = 6, s = 4\)

Step 1 - Identify what is given.

In this problem: The given values are \(n = 6\) and \(s = 4\).

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

Step 2 - Write the formulas.

In this problem: Use the formulas: \(P = n \times s\) and \(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\).

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

Step 3 - Substitute the values to calculate the perimeter.

In this problem: Substitute into \(P = n \times s\): \(P = 6 \times 4 = 24\).

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

Step 4 - Substitute the values into the area formula.

In this problem: Substitute into the area formula: \(A = \frac{6 \times 4^{2}}{4 \times \tan(\pi / 6)}\).

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

Step 5 - Calculate the area.

In this problem: Compute: \(n \times s^{2} = 6 \times 16 = 96\) and \(4 \times \tan(\pi / 6) \approx 2.31\). So \(A = \frac{96}{2.31} \approx 41.57\).

\(A = \frac{96}{2.31} \approx 41.57\)

Final answer:

\(P = 24,\; A \approx 41.57\)

Example 2:

\(n = 5, s = 3.5\)

Step 1 - Identify what is given.

In this problem: The given values are \(n = 5\) and \(s = 3.5\).

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

Step 2 - Write the formulas.

In this problem: Use the formulas: \(P = n \times s\) and \(A = \frac{n \times s^{2}}{4 \times \tan(\pi / n)}\).

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

Step 3 - Substitute the values to calculate the perimeter.

In this problem: Substitute into \(P = n \times s\): \(P = 5 \times 3.5 = 17.5\).

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

Step 4 - Substitute the values into the area formula.

In this problem: Substitute into the area formula: \(A = \frac{5 \times 3.5^{2}}{4 \times \tan(\pi / 5)}\).

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

Step 5 - Calculate the area.

In this problem: Compute: \(n \times s^{2} = 5 \times 12.25 = 61.25\) and \(4 \times \tan(\pi / 5) \approx 2.91\). So \(A = \frac{61.25}{2.91} \approx 21.08\).

\(A = \frac{61.25}{2.91} \approx 21.08\)

Final answer:

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