Free Perimeter Calculator

Understanding Perimeter and Its Use

The perimeter of any closed geometric figure is the total length of its outer boundary, measured in linear units such as meters, feet, or inches. It is a fundamental measurement that appears in countless real‑world tasks – from fencing a property to cutting a frame for a picture. Whether you need to calculate perimeter online for a quick homework check or a detailed construction plan, this geometric perimeter calculator provides immediate results for 12 common shape families, simplifying the process and minimizing errors.

Rather than memorizing formulas for every shape, you can use this shape perimeter calculator to handle the calculations. Below we present the essential perimeter formulas for each supported shape, along with helpful variations and guidance for cases when not all side lengths are known.

Perimeter Formulas at a Glance

Shape	Basic Perimeter Formula
Square	$P = 4a$
Rectangle	$P = 2(a + b)$
Triangle	$P = a + b + c$ (base formula)
Circle (circumference)	$P = 2\pi r$
Circle sector	$P = r(\alpha + 2)$ (with $\alpha$ in radians)
Ellipse (approximation)	$P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]$
Trapezoid / Quadrilateral	$P = a + b + c + d$
Parallelogram	$P = 2(a + b)$
Rhombus	$P = 4a$
Kite	$P = 2(a + b)$
Annulus	$P = 2\pi(R + r)$
Regular polygon	$P = n \times a$

For any polygon – regular or irregular – the perimeter is simply the sum of its side lengths: $P = \sum_{i=1}^{n} a_i$. If vertex coordinates are known, the distance formula can be applied: $P = \sum_{i=1}^{n} \sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2}$, with the indices wrapping around.

Detailed Notes on Selected Shapes

Triangle

The simplest formula requires all three sides: $P = a + b + c$. When only two sides and the included angle $\gamma$ are known, use the law of cosines to find the missing side: $c = \sqrt{a^2 + b^2 - 2ab\cos\gamma}$, and then add the three sides. Alternatively, if one side $a$ and the two adjacent angles $\beta$ and $\gamma$ are known, the law of sines gives:

$$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$

which leads to $P = a + a\frac{\sin\beta + \sin\gamma}{\sin(\beta + \gamma)}$ (since $\alpha = 180^\circ - \beta - \gamma$).

Circle & Sector

The circumference is simply $2\pi r$. For a sector of angle $\alpha$ (radians), the perimeter includes the two radii plus the arc: $P = r\alpha + 2r = r(\alpha + 2)$.

Ellipse

No exact closed‑form exists; the tool uses the well‑known Ramanujan approximation:

$$P \approx \pi\left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]$$

where $a$ is the semi‑minor axis and $b$ the semi‑major axis. A more accurate approximation is $P \approx \pi(a+b)\left(1 + \frac{3h}{10+\sqrt{4-3h}}\right)$ with $h = \frac{(a-b)^2}{(a+b)^2}$.

Parallelogram

The standard formula is $P = 2(a+b)$. If the diagonals $e,f$ and one side are given, the perimeter can be expressed as $P = 2a + \sqrt{2e^2 + 2f^2 - 4a^2}$. Alternatively, using base $b$, height $h$, and the angle $\alpha$ between sides: $P = 2\left(b + \frac{h}{\sin\alpha}\right)$.

Rhombus

A rhombus has four equal sides, so $P = 4a$. When the diagonals $e,f$ are known, the side length can be derived via Pythagoras (the diagonals bisect each other at right angles): $a = \frac{\sqrt{e^2+f^2}}{2}$, thus $P = 2\sqrt{e^2+f^2}$.

Regular Polygon

For an $n$-sided regular polygon with side length $a$, $P = n \times a$. The calculator automatically recognizes polygon names up to 12 sides (e.g., pentagon, hexagon, octagon).

Annulus

The perimeter (often called the circumference) of an annulus is the sum of the inner and outer circle circumferences: $P = 2\pi r_{\text{inner}} + 2\pi r_{\text{outer}} = 2\pi(R+r)$.

Additional Considerations

• Units: Because perimeter is a linear measure, the result will be in the same unit as the inputs (e.g., cm, m, ft).
• Irregular Shapes: For any irregular polygon, simply enter the length of each side into the tool, and it will sum them.
• Complex Curves: Shapes containing circular arcs require the arc length (radius × central angle in radians) plus the straight‑edge lengths.

For example, a triangle with sides of 5 cm, 6 cm, and 7 cm yields a perimeter of 18 cm. Selecting “Triangle” in the calculator and entering these three lengths returns the total immediately. Similarly, a rectangular plot measuring 8 m by 5 m gives a perimeter of 26 m ($P = 2(8+5)$).

This free geometric perimeter calculator eliminates manual arithmetic and error‑prone conversions. By selecting the shape that matches your problem and providing the required dimensions, you instantly obtain the perimeter. Whether you're studying geometry, planning a garden fence, or checking material needs, the shape perimeter calculator makes it easy to find perimeter of shapes online.