Area formulas, and how to measure an irregular space
The area and perimeter formulas for a rectangle, circle, triangle, trapezoid, ellipse, sector and more, with worked examples, unit conversions and how to measure an irregular room or plot.
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In short
- Area is the surface inside a shape, in square units; perimeter is the length around it. A 20 ft by 12 ft room has an area of 240 sq ft and a perimeter of 64 ft.
- A circle's area is π × r², with r the radius, half the diameter. Mixing up radius and diameter makes the answer four times too big.
- A triangle with three known sides is solved by Heron's formula: sides 13, 14 and 15 give exactly 84 square units.
- One square metre is 10.7639 square feet, one square yard (a gaj) is 9 square feet, and one acre is 43,560 square feet.
- For an irregular space, split it into rectangles and triangles, find each area and add them.
Area answers one of the most common practical questions in a house, a shop or a field: how much surface is there? It is what decides how much paint, flooring, grass seed or tile to buy, and how a plot is priced. The calculation is easy for a rectangle and gets more interesting for a triangle, a circle or a sector. This page gives the formulas the calculator uses, works through examples, and shows how to handle a space that is not a neat shape.
Area and perimeter are different things
Area is the amount of flat surface inside a shape, measured in square units: square feet, square metres, square yards. Perimeter is the distance around the outside, measured in plain units: feet, metres, yards. A room 20 feet long and 12 feet wide has an area of 20 × 12 = 240 square feet and a perimeter of 2 × (20 + 12) = 64 feet.
You need the area for anything that covers the surface (flooring, paint, a lawn) and the perimeter for anything that goes around the edge (skirting, fencing, a border). Buying fencing by area, or flooring by perimeter, is a classic way to order the wrong amount.
The formulas
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | A = length × width | P = 2 × (length + width) |
| Square | A = side × side | P = 4 × side |
| Circle | A = π × r² | P = 2 × π × r |
| Triangle (base and height) | A = ½ × base × height | needs all three sides |
| Triangle (three sides) | A = √(s × (s − a) × (s − b) × (s − c)), where s = (a + b + c) ÷ 2 | P = a + b + c |
| Trapezoid | A = ½ × (a + b) × height | needs the slanted sides |
| Parallelogram | A = base × height | needs the slanted sides |
| Ellipse | A = π × a × b | about π × (a + b) × (1 + 3h ÷ (10 + √(4 − 3h))), with h = ((a − b) ÷ (a + b))² |
| Sector of a circle | A = π × r² × angle ÷ 360 | P = 2 × π × r × angle ÷ 360 + 2 × r |
| Ring (annulus) | A = π × (R² − r²) | P = 2 × π × (R + r) |
| Regular polygon | A = n × side² ÷ (4 × tan(180° ÷ n)) | P = n × side |
Two details trip people up. The height of a triangle, trapezoid or parallelogram is measured at a right angle to the base, not along a slanted side. And an ellipse has no simple exact perimeter formula; the calculator uses a close approximation due to Ramanujan, accurate far beyond what a measurement needs.
Worked examples
A rectangle. A room 20 ft by 12 ft is 240 sq ft. In metric that is 240 × 0.09290304 = 22.297 square metres, and in square yards it is 240 ÷ 9 = 26.67.
A circle. A round garden with a radius of 7 m has an area of π × 7² = 3.14159 × 49 = 153.94 m², and a boundary of 2 × π × 7 = 43.98 m. If you only know the diameter, halve it first. A garden 14 m across has a radius of 7 m. Using 14 instead of 7 in the formula gives 615.75 m², four times too large, which is the most common mistake with circles.
A triangle from three sides. A plot has sides of 13 m, 14 m and 15 m and no known height. Half the perimeter is s = (13 + 14 + 15) ÷ 2 = 21. Then the area is √(21 × 8 × 7 × 6) = √7056 = 84 m². This is Heron's formula, and it is the right tool whenever you have measured the three sides of a triangle on the ground and cannot easily measure a perpendicular height. The sides must satisfy the triangle rule: the two shorter sides together must be longer than the longest, or no triangle exists. Sides of 1, 2 and 10 do not make a triangle, and the calculator will say so.
A trapezoid. A plot with parallel sides of 8 m and 14 m, 5 m apart, has an area of ½ × (8 + 14) × 5 = 55 m².
An ellipse. An oval pond with semi-axes of 10 m and 6 m (half of its long and short widths) covers π × 10 × 6 = 188.5 m², and its edge is about 51.05 m long.
A sector. A quarter of a circle of radius 10 m, such as a corner garden bed, has an area of π × 10² × 90 ÷ 360 = 78.54 m². Its boundary is the curved edge, 2 × π × 10 × ¼ = 15.71 m, plus the two straight sides of 10 m, which is 35.71 m in all.
A ring. A circular track between an outer radius of 10 m and an inner radius of 6 m covers π × (100 − 36) = 201.06 m².
A regular hexagon. A hexagonal paving stone with sides of 5 units has an area of 6 × 25 ÷ (4 × tan 30°) = 64.95 square units.
Converting between units
Areas multiply the conversion factor twice, which is why they do not convert like lengths. A foot is 0.3048 m, so a square foot is 0.3048 × 0.3048 = 0.09290304 square metres, and a square metre is 10.7639 square feet. The factors you will use most:
| Unit | In square feet | In square metres |
|---|---|---|
| 1 square foot | 1 | 0.0929 |
| 1 square yard (gaj) | 9 | 0.8361 |
| 1 square metre | 10.7639 | 1 |
| 1 acre | 43,560 | 4,046.86 |
| 1 hectare | 107,639 | 10,000 |
A hectare is therefore 2.471 acres. Land is also sold in regional units such as bigha, biswa, guntha and kanal, whose sizes differ from state to state; the land area converter holds those, and the unit converter handles the standard ones.
Measuring a space that is not a rectangle
Rooms and plots are rarely perfect. The reliable method is to cut the space into simple shapes, measure each, and add the areas.
An L-shaped room can be treated as two rectangles. If the main part is 5 m by 4 m and the extension is 3 m by 2 m, the area is 20 + 6 = 26 m². Subtracting works as well as adding: a rectangular room with a square alcove cut out of one corner is the full rectangle minus the alcove.
For a plot with an angled side, divide it into a rectangle and a triangle. For a plot with many sides, divide it into triangles and use Heron's formula on each, measuring the sides on the ground. Keep a sketch with every length written on it; most errors come from measuring in mixed units, so measure everything in feet or everything in metres.
Round features need their own formula. A circular bed in the corner of a lawn is subtracted from the lawn's rectangle, using the circle formula above.
From area to quantity
Area is the first step; the number to buy comes next.
- Tiles. Divide the floor area by the area of one tile. A 240 sq ft room with tiles of 2 ft by 2 ft (4 sq ft each) needs 240 ÷ 4 = 60 tiles. Add 10 percent for cuts and breakage, so buy 66.
- Paint. Work out the wall area, not the floor: perimeter × height, minus doors and windows, then divide by the coverage printed on the tin.
- Flooring in rolls or planks is sold by area, and the waste allowance is larger for diagonal layouts and patterned floors.
- Grass seed, fertiliser and weed control are all given per area on the packet.
The allowance for waste is not padding. Tiles break, cuts leave offcuts that cannot always be reused, and a second purchase months later may not match the first batch.
What this calculator assumes
- All lengths are in the one unit you choose, and the area comes out in that unit squared.
- Shapes are flat and exact: a "rectangle" has four right angles, a "circle" is perfectly round.
- The perimeter of a triangle from base and height, a trapezoid and a parallelogram is not shown, because it needs the slanted sides which the inputs do not include.
- The ellipse perimeter is an approximation, accurate to many decimal places for any ordinary shape.
- Areas of land are as measured on a flat plan, not on a slope, so a hillside's surface is larger than its plan area.