What area means
Area is the amount of flat surface inside a shape. It is always measured in square units: square feet (ft²), square metres (m²), square inches and so on. One square foot is the space covered by a square 1 foot on each side. When you say a room is 120 ft², you mean 120 of those squares would tile the floor.
Because area comes from multiplying two lengths, both lengths must be in the same unit first. Multiply feet by feet and you get square feet. Mix feet and inches and the answer is meaningless. This calculator takes every measurement in one unit you choose, then converts the result to the other common area units for you.
Area formulas for common shapes
| Shape | Formula | What the letters mean |
|---|---|---|
| Square | A = a² | a = side |
| Rectangle | A = l × w | l = length, w = width |
| Triangle | A = ½ × b × h | b = base, h = perpendicular height |
| Triangle (3 sides) | A = √(s(s − a)(s − b)(s − c)) | s = (a + b + c) ÷ 2 |
| Circle | A = π × r² | r = radius (half the diameter) |
| Trapezoid | A = ½ × (a + b) × h | a, b = parallel sides, h = height |
| Ellipse | A = π × a × b | a, b = half the longest and shortest widths |
| Sector | A = (θ ÷ 360) × π × r² | θ = angle in degrees |
| Parallelogram | A = b × h | b = base, h = perpendicular height |
The key idea behind most of them is the rectangle. A parallelogram is a rectangle with one end sliced off and moved to the other side, so its area is still base × height. A triangle is half of a parallelogram, which is where the ½ comes from. A trapezoid uses the average of its two parallel sides as the “length”.
Worked examples
A bedroom floor. A room measures 12 ft by 10 ft.
- A = 12 × 10 = 120 ft².
- In square metres: 120 × 0.09290304 = 11.15 m².
A round patio. A circular patio has a diameter of 5 m, so the radius is 2.5 m.
- A = π × 2.5² = 3.14159 × 6.25 = 19.63 m².
- In square feet: 19.63 ÷ 0.09290304 = 211.3 ft².
A triangular garden bed with three known sides. The sides are 8 m, 11 m and 13 m, and you do not know the height.
- Semi-perimeter s = (8 + 11 + 13) ÷ 2 = 16.
- A = √(16 × (16 − 8) × (16 − 11) × (16 − 13)) = √(16 × 8 × 5 × 3) = √1,920 = 43.82 m².
A plot shaped like a trapezoid. The two parallel boundaries are 30 m and 50 m, and they are 40 m apart.
- A = ½ × (30 + 50) × 40 = ½ × 80 × 40 = 1,600 m².
- That is 0.16 hectares, or 1,600 ÷ 4,046.86 = 0.395 acres.
Converting between area units
Area conversions use the length factor squared. This is the most common mistake: 1 metre is 100 centimetres, but 1 square metre is 100 × 100 = 10,000 square centimetres. The same goes for imperial units: 1 yard is 3 feet, so 1 square yard is 9 square feet.
| From | To | Multiply by |
|---|---|---|
| ft² | m² | 0.09290304 |
| m² | ft² | 10.7639 |
| yd² | m² | 0.83613 |
| in² | cm² | 6.4516 |
| acres | ft² | 43,560 |
| acres | m² | 4,046.86 |
| hectares | m² | 10,000 |
| hectares | acres | 2.47105 |
| mi² | acres | 640 |
| km² | hectares | 100 |
The factors for feet, inches and yards are exact, because the inch has been defined as exactly 2.54 cm since 1959. For land units used in South Asia, such as marla and kanal, use the land area converter. To convert a single length before you start, the length converter handles mm to miles.
Shapes in more detail
Triangles. The height must be measured at a right angle to the base, not along a sloping side. If you only know the three sides, choose “Triangle (3 sides)” and the calculator uses Heron’s formula, named after Hero of Alexandria. It checks that the sides can actually form a triangle: each side has to be shorter than the other two added together.
Circles. π (pi) is the ratio of a circle’s circumference to its diameter, about 3.14159. Archimedes showed more than 2,000 years ago that a circle has the same area as a triangle whose base is the circumference and whose height is the radius, which gives ½ × 2πr × r = πr². If you measured the diameter, halve it first. The calculator also shows the circumference, 2πr.
Ellipses. An oval running track or a racetrack-shaped rug is close to an ellipse. Measure the longest width and the shortest width through the centre and halve both to get a and b. The area formula is exact. The perimeter of an ellipse has no simple formula, so the calculator uses Ramanujan’s approximation, which is accurate to a tiny fraction of a percent for everyday shapes.
Sectors. A sector is a slice of a circle, like a slice of pizza. A 90° sector is a quarter of the circle, so its area is a quarter of πr². The perimeter shown includes the two straight edges plus the curved arc.
Trapezoids and parallelograms. These are common in land plots and in building plans. For both, the height is the straight-across distance between the parallel sides.
Irregular shapes
Real rooms, gardens and plots are rarely a single neat shape. The usual method is to break them into pieces:
- Sketch the outline and split it into rectangles, triangles and part-circles.
- Measure each piece and find its area here.
- Add the pieces together, and subtract any cut-outs such as a kitchen island or a pond.
For a room shaped like an L, for example, split it into two rectangles. A 20 × 12 ft space with a 8 × 6 ft corner missing is 240 − 48 = 192 ft². The square footage calculator can add several areas at once and price materials per square foot.
Tips for accurate results
- Use one unit. Convert 10 ft 6 in to 10.5 ft before you enter it, or switch the unit to inches.
- Measure twice. Rooms are not always square. Measure the length at both ends and use the average, or the larger number if you are buying materials.
- Keep extra decimals until the end. Rounding early can add up, especially for circles.
- Add waste when buying. Flooring, tiles and turf usually need 5% to 15% extra for cuts and mistakes. The percentage calculator makes this quick.
- From area to volume. For concrete, soil or gravel, multiply the area by the depth. The concrete calculator does this and counts bags.
Common questions answered by the numbers
- A 10 × 10 ft room is 100 ft², or 9.29 m².
- A 4 × 3 m room is 12 m², or 129.2 ft².
- A circle with a 1 m radius is 3.1416 m².
- A typical US football field (360 × 160 ft, including end zones) is 57,600 ft², about 1.32 acres.
- A square plot 100 m on each side is exactly 1 hectare.