Area and perimeter, explained

Two different questions

Perimeter is the distance around the edge of a shape. It is a length — feet, metres — and it answers questions like how much baseboard, fence, edging or trim to buy.

Area is the amount of surface inside the edge. It is measured in square units — square feet, square metres — and answers how much flooring, paint, sod or roofing to buy. Multiply an area by a depth and you get a volume, which is how concrete and mulch are ordered.

Two shapes can share a perimeter and have very different areas: a 10 × 10 square and a 19 × 1 strip both have a 40-foot perimeter, but 100 and 19 square feet inside.

Square units, and converting them

A square foot is a square one foot on each side. Because area is length × length, unit conversions are squared too: 1 foot = 12 inches, so 1 square foot = 12 × 12 = 144 square inches; 1 yard = 3 feet, so 1 square yard = 9 square feet; 1 metre ≈ 3.281 feet, so 1 square metre ≈ 10.764 square feet. Always measure every side in the same unit before multiplying.

Rectangles and squares — the starting point

Area      A = length × width
Perimeter P = 2 × (length + width)
Square:   A = a²,  P = 4a

A rectangle 12 by 8 can be tiled with 12 × 8 = 96 one-unit squares — that is all area means. Its perimeter is two lengths and two widths. Almost every other formula is this one in disguise.

Parallelograms

Slice the triangle off one end of a parallelogram and slide it to the other: you get a rectangle with the same base and the same height. So A = base × height — where the height is measured square to the base, not along the slanted side. The perimeter still uses the slanted sides: P = 2(base + side).

Triangles

Two copies of any triangle make a parallelogram, so a triangle is half of one: A = base × height ÷ 2. The perimeter is just the three sides added.

• Right triangle — the two legs are base and height, so A = a × b ÷ 2; the long side comes from Pythagoras, c = √(a² + b²). 3-4-5 is the builder's square-corner check.
• Three sides known — Heron's formula needs no height: with s = (a + b + c) ÷ 2, A = √(s(s−a)(s−b)(s−c)). Any two sides must add to more than the third, or there is no triangle.
• Two sides and the angle between — A = ½ a b sin C, and the third side from the law of cosines, c² = a² + b² − 2ab cos C.
• Isosceles and equilateral — the height drops through the middle, splitting it into two right triangles; an equilateral triangle's height is side × √3 ÷ 2, so A = (√3 ÷ 4) a².

Trapezoids, rhombuses and kites

A trapezoid (trapezium in the UK) has two parallel sides. Its area is the average of the parallel sides times the height between them: A = (a + b) ÷ 2 × h — the same as a rectangle of the average width.

A rhombus and a kite both have diagonals that cross at right angles. Each is made of four right triangles whose legs are halves of the diagonals, which adds up to A = p × q ÷ 2. Their sides come from Pythagoras on those half-diagonals.

Regular polygons

A regular polygon with n equal sides splits into n identical triangles meeting at the centre. Each has the side as its base and the apothem (centre to mid-side) as its height, so A = ½ × perimeter × apothem. Writing the apothem in terms of the side gives A = n s² ÷ (4 tan(180° ÷ n)). Each interior angle is 180° × (n − 2) ÷ n — 120° for a hexagon, 135° for an octagon — and the mitre for each joint is half the outside turn, 22.5° for an octagon.

Circles and the curved family

Circle         A = π r²                 C = 2 π r   (π ≈ 3.14159)
Semicircle     A = π r² ÷ 2             P = π r + 2r
Quarter circle A = π r² ÷ 4             P = π r ÷ 2 + 2r
Sector (θ°)    A = (θ ÷ 360) π r²       P = (θ ÷ 360) 2 π r + 2r
Ring           A = π (R² − r²)          P = 2 π (R + r)
Ellipse        A = π a b                P ≈ π [3(a+b) − √((3a+b)(a+3b))]

π is the ratio of any circle's circumference to its diameter. Cut a circle into thin pie slices and lay them alternately point-up and point-down and they form a near-rectangle half the circumference long and one radius high — π r × r = π r². A sector is the same fraction of the circle's area as its angle is of 360°. A circular segment — the arch above a chord — is the sector minus the triangle under the chord: A = ½ r² (θ − sin θ) with θ in radians. The ellipse has no exact perimeter formula in elementary terms; Ramanujan's approximation above is accurate to a few parts in a million for normal ovals.

Shapes that are not in any book

Add and subtract. An L-shaped room is the full rectangle minus the missing corner. A room with a bay is a rectangle plus a trapezoid. Split, work out each piece, then add or subtract.

Perimeter does not split the same way. Only the outside edges count. For an L-shape the perimeter equals the surrounding rectangle's, because the notch moves two walls without changing their total length.

Any straight-sided outline — an irregular lot, a room with angled walls — can be done from its corner points. Pick a corner as 0,0, measure every other corner's x and y from it, list them in order around the edge and use the shoelace formula:

A = ½ × | (x₁y₂ − x₂y₁) + (x₂y₃ − x₃y₂) + … + (xₙy₁ − x₁yₙ) |
P = sum of the distances between consecutive corners

It works for any polygon whose edges do not cross, however many corners it has. The polygon calculator does it for you.

Practical tips

• Measure to the nearest ⅛″ (or few millimetres) and round only the final answer.
• For materials, add waste: typically 5–10% for flooring, more for diagonal layouts and patterned tile.
• Check a “rectangle” is square by measuring both diagonals — if they differ, it is a parallelogram or a trapezoid and length × width will be slightly wrong.

Open the area & perimeter calculators · How the stair calculations work