How to calculate perimeter, step by step
To calculate a perimeter, add up the lengths of every side. That is the whole method, and every shape below is just a faster way of doing that one addition. A rectangle 8 by 5 has a perimeter of 26, because 8 + 5 + 8 + 5 is 26. A square of side 4 has a perimeter of 16, because 4 + 4 + 4 + 4 is 16. And a triangle with sides 2, 3 and 4 has a perimeter of 9, because 2 + 3 + 4 is 9.
The worked example: a rectangle 8 by 5
The worked example. Step 1 — list the four sides: 8, 5, 8 and 5, because the opposite sides of a rectangle match. Step 2 — add them: 8 + 5 + 8 + 5 is 26. Step 3 — check it with the rectangle formula: 2 × (8 + 5) is 2 × 13, which is 26 — the same answer, two ways. Step 4 — write the answer with its units: the perimeter is 26, in whatever units the sides were measured in. Step 5 — check it against the fence test: walking around the rectangle covers 8, then 5, then 8, then 5 — 26 units in total.
Rectangles: twice the length plus the width
A rectangle has two pairs of equal sides, so you never need to add all four one by one: add the length and the width, then double it. Take a rectangle 3 long and 4 wide. Step 1 — add the length and the width: 3 + 4 is 7. Step 2 — double it: 2 × 7 is 14. The perimeter is 14. The formula is P = 2 × (length + width). The perimeter-calculator shows the side-by-side breakdown for any rectangle you type.
Squares: four times one side
A square has four equal sides, so the perimeter is one side multiplied by four. Take a square of side 4. Step 1 — take one side: 4. Step 2 — multiply by four: 4 × 4 is 16. The perimeter is 16. The formula is P = 4 × side. If you know the perimeter and need the side, divide by four instead: a square with a perimeter of 16 has sides of 4, because 16 ÷ 4 is 4.
Triangles: add the three sides
A triangle has three sides and no shortcut — add all three. Take a triangle with sides 2, 3 and 4. Step 1 — list the sides: 2, 3 and 4. Step 2 — add them: 2 + 3 + 4 is 9. The perimeter is 9. Before you trust it, check that the triangle can close: the two shorter sides have to add up to more than the longest — 2 + 3 is 5, more than 4, so this one closes. The formula is P = side 1 + side 2 + side 3.
Circles: twice π times the radius — and why π appears
A circle has no straight sides, so there is nothing to add — instead, the perimeter, called the circumference, is 2π times the radius. Take a circle of radius 2. Step 1 — take the radius: 2. Step 2 — multiply by 2π: 2 × π × 2 is 12.566371 to six places. The circumference is 12.566371. Why π? Because π is the ratio of a circle’s circumference to its diameter, and the diameter is twice the radius, so the circumference is 2π times the radius. The decimal never terminates, so the answer is rounded to six places — carry the rounded value into further working only if that precision is enough.
Regular polygons: the number of sides times one side
A regular polygon has all its sides equal and all its angles equal, so the perimeter is the number of sides times one side. Take a regular pentagon of side 4. Step 1 — count the sides: a pentagon has five. Step 2 — multiply: 5 × 4 is 20. The perimeter is 20. The formula is P = number of sides × side length. If the sides are not all equal, the shape is not regular, and you have to add the sides one by one instead — that is the triangle’s job.
Checking by measuring
Whatever method you used, check it by measuring: walk a tape around the shape, or add the sides a second time in a different order. For a rectangle, add all four sides, then check with 2 × (length + width) — the two ways must agree, and if they do not, one of the additions is wrong. For a circle, measure the diameter and multiply by π — the diameter is twice the radius, so the check is C = π × diameter. Run your numbers through the perimeter-calculator and compare them with the worked example above.
Common mistakes
The first is adding the area instead of the perimeter: a rectangle 8 by 5 has a perimeter of 26 and a different area, and the two numbers answer different questions — fencing measures the perimeter, carpet measures the area. The second is counting a side twice or missing one: write all four sides down before you add. The third is using the diameter where the formula wants the radius: the circumference is 2π times the radius, which is π times the diameter — mixing them up doubles or halves the answer. And the fourth is forgetting that a triangle has to close: if the two shorter sides do not add up to more than the longest, there is no triangle.
A quick check you can do in your head
You can sanity-check any perimeter without a calculator. First, re-add the sides in a different order: 8 + 5 + 8 + 5 is 26, and so is 8 + 8 + 5 + 5 — if the two orders disagree, one addition is wrong. Second, use the rectangle formula as a cross-check: 2 × (8 + 5) is 26, the same as the long addition. If a calculator says the perimeter of a rectangle 8 by 5 is far from 26, the numbers are wrong.