Perimeter Formula for Every Shape, with Circumference

What perimeter means and the general perimeter formula

Imagine walking once around the edge of a park and measuring every step. The total distance is its perimeter. Because it is a length, its unit is cm, m or km, never square units. The word comes from Greek: "peri" means around and "metron" means measure.

For a polygon, the general rule is to add up all the side lengths. Each standard shape just gives you a shortcut, because some sides repeat. The perimeter formula in maths is written as

Class 6 and 7 books treat perimeter together with area. Class 10 returns to it through the circumference of circles and the length of arcs.

Perimeter of triangle formula

A triangle has three sides $a$, $b$ and $c$, so its perimeter is $a + b + c$. Special triangles give shorter forms:

• Equilateral (all sides $a$): $P = 3a$.
• Isosceles (two equal sides $a$, base $b$): $P = 2a + b$.
• Right angle triangle with legs $p$ and $q$: find the hypotenuse first from Pythagoras, $h = \sqrt{p^2 + q^2}$, and then $P = p + q + h$.

Half perimeter of triangle

Half the perimeter, written $s$, is called the semi-perimeter. It is the starting point of Heron's formula which finds a triangle's area from its three sides.

Perimeter of square and rectangle formula

Square

All four sides of a square are equal, so the perimeter of square formula is $P = 4a$. A tile of side 12.5 cm has a boundary of 50 cm. If the diagonal $d$ is given, one side is $\tfrac{d}{\sqrt{2}}$, so $P = 2\sqrt{2}\,d$; a diagonal of 10 cm gives about 28.28 cm.

Rectangle

A rectangle has two lengths and two breadths, so the formula of perimeter of rectangle is $P = 2(l + b)$. Add length and breadth first, then double. Writing $2l + b$ by mistake is a very common slip.

Rhombus and parallelogram follow the same logic: a rhombus has four equal sides, so $4a$; a parallelogram has two pairs, so $2(a + b)$. A trapezium has no shortcut: add its four sides.

Circumference of circle formula

The circumference is the perimeter of a circle. For every circle, the ratio of circumference to diameter is the same number, $\pi$. That is the definition of pi in maths, and it gives

Use the diameter form when the diameter is given, and the radius form when the radius is given; both are the same formula because $d = 2r$.

Proof of circumference of a circle using calculus

On a circle of fixed radius r, turning through a small angle $d\theta$ (in radians) traces an arc of length $r\,d\theta$. Going once round, $\theta$ runs from 0 to $2\pi$:

Read more on the integration formulas page.

Area of a circle using circumference

Many questions give the boundary and ask for the inside. Get the radius from $r = \tfrac{C}{2\pi}$, then square it. In one line,

Radius, diameter, circumference and area are linked: know any one and you can find the other three. The area side of that link, and the reason it is pi r squared, are explained on the area formulas page.

Perimeter of semicircle, sector and polygon

Semicircle

The boundary of a semicircle is half the circumference plus the straight diameter. Students often forget the diameter part.

For radius 7 cm, with $\pi = \tfrac{22}{7}$: $22 + 14 = 36$ cm.

Arc and sector

An arc of angle $\theta$ is $\tfrac{\theta}{360}$ of the full circumference. A sector is bounded by that arc and two radii.

Regular polygon

A regular polygon of $n$ equal sides of length $a$ has $P = na$. A regular hexagon of side 5 cm has a perimeter of 30 cm; a regular octagon of side 3 m has 24 m.

Perimeter of cube, cuboid and sphere

Solids do not have a perimeter in the strict sense. What books sometimes call the perimeter of a cube is the total length of its 12 edges: $12a$. For a cuboid it is $4(l + b + h)$, the wire needed to make a frame. A sphere has no edges at all; the closest idea is the circumference of its largest circle, $2\pi r$. For the space and surface of solids, use the volume and surface area page.

When the perimeter of a triangle is given, find the area

A perimeter alone does not fix the area, so the question must give something more: the ratio of sides, or the fact that the triangle is equilateral.

For an equilateral triangle with perimeter 36 cm, each side is 12 cm and the area is $\tfrac{\sqrt{3}}{4} \times 144 = 36\sqrt{3} \approx 62.35$ square cm.

Scale factor, area and perimeter

Enlarge any figure by a scale factor $k$ and every length, including the perimeter, is multiplied by $k$, but the area is multiplied by $k^2$. Double the sides of a square, and its perimeter doubles while its area becomes four times as large. In percentage terms, see the percentage formula page.

Prove that AQ is half the perimeter of triangle ABC

This is a well known class 10 problem on tangents to a circle. A circle touches side BC of triangle ABC at P, and touches AB and AC extended beyond B and C at Q and R. Prove that $AQ = \tfrac{1}{2}(\text{perimeter of } \triangle ABC)$.

The key fact: the two tangents drawn to a circle from an outside point are equal. So $AQ = AR$, $BQ = BP$ and $CP = CR$.

Hence $AQ$ is half the perimeter. The same idea appears in exams in the form "prove that AR is half the perimeter" with the letters changed.

Mistakes that cost marks in perimeter questions

• Writing square units for a perimeter. Perimeter is a length: cm, not $\mathrm{cm^2}$.
• Mixing units. Convert 2 m 40 cm to 240 cm before adding it to other sides in cm.
• Using the radius in $\pi d$ or the diameter in $2\pi r$. Check which one the question gives.
• Leaving out the straight edge of a semicircle or the two radii of a sector.
• In combined figures, counting an inside edge that is not part of the outer boundary. Trace the outline with your finger first.

Other geometry and algebra shortcuts are listed in the full maths formula list.