# Perimeter Calculator

Created by Hanna Pamuła, PhD
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Oct 19, 2022

With this perimeter calculator, you don't need to worry about perimeter calculations anymore. Below you'll find the perimeter formulas for twelve different shapes as well as a quick reminder about what a perimeter is and a perimeter definition. Read on, give it a try, or check this calculators twin brother - our comprehensive area calculator.

## What is perimeter? Perimeter definition

Perimeter is the boundary of a closed geometric figure. It may also be defined as the outer edge of an area, simply the longest continuous line that surrounds a shape. The name itself comes from Greek perimetros: peri meaning "around" + metron, understood as "measure". As it's the length of the shape's outline, it's expressed in distance units – e.g., meters, feet, inches, or miles.

## How to find perimeter – perimeter formulas

Usually the most simple and straightforward approach is to find the sum of all of the sides of a shape. However, there are cases where there are no sides (such as an ellipse, circle, etc), or one or more sides are unknown. In this paragraph, we'll list all of the equations used in this perimeter calculator. Scroll down to next sections if you're curious about a specific shape, and wish to see an explanation, derivation, and an image for each of the twelve shapes present in this calculator. We also have tools dedicated to each shape - just type the name of the shape in the search bar at the top of this webpage.

Here are the perimeter formulas for the twelve geometric shapes in this calculator:

• Square perimeter formula: $P = 4a$.
• Rectangle perimeter formula: $P = 2(a + b)$.
• Triangle perimeter formulas:
• $P = a + b + c$; or
• $P = a + b + \sqrt{a^2 + b^2 - 2ab \times \cos(\gamma))}$; or
• $P = a + (a / \sin(\beta + \gamma)) \times (\sin(\beta) + \sin(\gamma))$ .
• Circle perimeter formula: $P = 2\pi r$.
• Circle sector perimeter formula: $P = r(\alpha + 2)$ ($\alpha$ is in radians);
• Ellipse perimeter formula: $P = \pi(3(a + b) - \sqrt{(3a + b) \times (a + 3b)})$;
• Quadrilateral / Trapezoid perimeter formula: $P = a + b + c + d$.
• Parallelogram perimeter formulas:
• $P = 2(a + b)$;
• $P = 2a^2 + \sqrt{2e^2 + 2f^2 - 4a^2}$; or
• $P = 2(b + h/\sin(\alpha))$.
• Rhombus perimeter formulas:
• $P = 4a$; or
• $P = 2\sqrt{e^2 + f^2}$.
• Kite perimeter formula: $P = 2(a + b)$.
• Annulus perimeter formula: $P = 2\pi(R + r)$.
• Regular polygon area formula: $P = n \times a$.

## Perimeter of a square formula

A square has four sides of equal length. To calculate its perimeter, all you need to do is to multiply the side length by $4$:

$\begin{split} P_\mathrm{{square}} &= a+a+a+a\\ &=4a \end{split}$

Believe it or not, but we have a perimeter of a square calculator, too!

## Formula for the perimeter of a rectangle

The formula for the perimeter of a rectangle is almost as easy as the equation for the perimeter of a square. The only difference is that we have two pairs of equal length sides:

$\begin{split} P_\mathrm{{rectangle}}& = a+b+a+b\\ &=2a+2b=2(a+b) \end{split}$

## Perimeter of a triangle formula

The easiest formula to calculate the perimeter of a triangle is – as usual – by summing all sides:

$P_{\mathrm{triangle}} = a+b+c$

However, you aren't always given three sides. What can you do then? Well, instead of fretting, you can use the law of cosines calculator to find the missing side:

$c=\sqrt{a^2+b^2-2ab\times\cos(\gamma)}$

This can be incorporated into the perimeter formula:

$\begin{split} &P_{\mathrm{triangle}} = a^2+b^2\\ &+\sqrt{a^2+b^2-2ab\times\cos(\gamma)} \end{split}$

The other option is to use the law of sines if you have one side and the two angles that are formed by that side:

$b = \sin(\beta)\times\frac{a}{\sin(\beta+\gamma)}$

And:

$b = \sin(\gamma)\times\frac{a}{\sin(\beta+\gamma)}$

so the triangle perimeter may be expressed as:

$\begin{split} P_{\mathrm{triangle}} &= a+\frac{a}{\sin(\beta+\gamma}\\ &\times(\sin(\beta)+\sin(\gamma)) \end{split}$

## Perimeter of a circle formula (circumference formula)

A perimeter of a circle has a special name - it's also known as the circumference. The most well-known perimeter of a circle formula uses only one variable - circle radius:

$\mathrm{C} = 2\pi\times r$

Have you ever wondered how many times your bike wheel will rotate on a ten-mile trip? Well, that's one of the cases where you'll need to use the circumference formula. Input the radius of your wheel (half of the wheel's diameter), and divide 10 miles by the obtained circumference (but don't forget about the [conversion of the units of length!). If you want to be even more accurate, you can include the size of the bike tire. If you're a keen cyclist, don't forget to have a look at our bike cadence and speed calculator and the bike gear calculator.

## Perimeter of a circle sector formula

Calculating the perimeter of a circle sector may sound tricky – is it only the arc length, or is it the arc length plus two radii? Just keep in mind the perimeter definition! The sector perimeter is the sum of the lengths of all its boundaries, so it's the latter:

P_{\mathrm{circle\ sector} = r\times(\alpha+2)

Where $\alpha$ is in radians.

## Perimeter of an ellipse formula (ellipse circumference formula)

Although the formula for the area of an ellipse is really simple and easy to remember, the perimeter of an ellipse formula is the most troublesome of all the equations listed here. We've chosen to implement one of the Ramanujan approximations in this perimeter calculator:

$\begin{split} &P_\mathrm{ellipse} = \pi\times\bigl(3(a+b)\\ &-\sqrt{(3a+b)\times(a+3b)}\bigr) \end{split}$

Where $a$ is the shortest possible radius and $b$ in the longest possible radius of an ellipse. The other, more accurate Ramanujan approximation is:

$\begin{split} &P_\mathrm{ellipse} = \pi\times(a+b)\\ &\times\frac{1+3\frac{(a-b)^2}{(a+b)^2}}{10+\sqrt{4-3\frac{(a-b)^2}{(a+b)^2}}} \end{split}$

There is also a simpler form, using an additional variable $h$:

$h = \frac{(a-b)^2}{(a+b)^2}$

That is:

$\begin{split} &P_\mathrm{ellipse} = \pi\times(a+b)\\ &\times\frac{1+3h}{10+\sqrt{4-3h}} \end{split}$

Or you could just use our calculator!

## Perimeter of a trapezoid formula

If you want to calculate the perimeter of an irregular trapezoid, there's no special formula - just add all four sides:

$P_{|mathrm{trapezoid}} = a+b+c+d$

Maybe you've noticed, but it's the formula for any quadrilateral perimeter.

There's also an option that presents itself with certain special trapezoids – like an isosceles trapezoid, where you need $a$, $b$, and $c$ sides. Another example is a right trapezoid, where the length of the bases and one leg are enough to find the shape's perimeter (to find the last leg, we calculate Pythagoras' Theorem).

## Perimeter of a parallelogram formula

In this perimeter calculator, you'll find three formulas to calculate the perimeter of a parallelogram:

1. The most straightforward one, adding all sides together
$\begin{split} &P_{\mathrm{parallelogram}} = a+b+a+b\\ &=2(a+b) \end{split}$
1. The perimeter of a parallelogram formula that requires one side and diagonals
$\begin{split} &P_{\mathrm{parallelogram}} = 2a^2\\ &+\sqrt{2e^2+2f^2-4a^2} \end{split}$
1. The perimeter is given in terms of base, height, and any parallelogram angle.
$P_{\mathrm{parallelogram}} = 2\left(b+\frac{h}{\sin(\alpha)}\right)$