Area Calculator
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Results
Area of the rectangle: 48 units², perimeter 28
How to use the area calculator
- Pick the shape. If you know a triangle's three sides but not its height, choose the Heron option.
- Enter the measurements in one unit. Heights are always measured at right angles to the base, not along a slanted side.
- Choose that unit from the list to label the answer. With feet, inches or yards you also see the area in square meters, and with metric units in square feet.
- For an irregular outline, split it into these shapes, calculate each one and add the areas.
Area formulas
Triangle: A = b × h ÷ 2 · Heron: A = √(s(s − a)(s − b)(s − c)), s = (a + b + c) ÷ 2
Trapezoid: A = (a + b) ÷ 2 × h · Parallelogram: A = b × h
Circle: A = πr² · Sector: A = πr² × θ ÷ 360°
Ellipse: A = πab · Regular polygon: A = n × s² ÷ (4 × tan(180° ÷ n))
Area measures the flat space inside a shape, in square units: a rectangle 8 cm long and 6 cm wide holds 48 squares of 1 cm × 1 cm. A triangle is half of a rectangle with the same base and height, and a parallelogram is a rectangle with one end sliced off and moved to the other, which is why both use the perpendicular height rather than the slanted side. A trapezoid's area is its average width, (a + b) ÷ 2, times its height. For circles, r is the radius; a sector is the fraction θ ÷ 360 of the whole circle. For an ellipse, a and b are the semi-axes, half the longest and shortest widths.
Formulas with worked examples
Each area below was calculated with this calculator's engine.
| Shape | Area formula | Example | Area |
|---|---|---|---|
| Rectangle | l × w | l = 8, w = 6 | 48 |
| Square | s² | s = 5 | 25 |
| Triangle | b × h ÷ 2 | b = 10, h = 4 | 20 |
| Triangle (Heron) | √(s(s − a)(s − b)(s − c)) | 13, 14, 15 | 84 |
| Trapezoid | (a + b) ÷ 2 × h | a = 6, b = 10, h = 4 | 32 |
| Parallelogram | b × h | b = 8, h = 3 | 24 |
| Circle | πr² | r = 3 | 28.274 |
| Sector | πr² × θ ÷ 360 | r = 6, θ = 90° | 28.274 |
| Ellipse | πab | a = 5, b = 3 | 47.124 |
| Regular hexagon | n·s² ÷ (4·tan(180° ÷ n)) | n = 6, s = 4 | 41.569 |
Worked example: an L-shaped room
An L-shaped room can be split into a 12 ft × 10 ft rectangle and a 6 ft × 4 ft rectangle. Their areas are 120 ft² and 24 ft², so the floor is 144 ft² in total, about 13.38 m² (1 ft² = 0.09290304 m² exactly). Measure each part along its own walls and make sure the pieces do not overlap. If a shape is easier to see as a big rectangle with a corner missing, subtracting works just as well: 12 × 14 minus the missing 6 × 4 corner gives the same answer.
Triangles without a height: Heron's formula
When you have three sides but no height, Heron's formula avoids trigonometry. For sides 13, 14 and 15 the semiperimeter is s = 21, and the area is √(21 × 8 × 7 × 6) = √7,056 = 84. The calculator uses a rearranged form of the formula that stays accurate for very long, thin triangles.
Perimeter of an ellipse
Unlike its area, an ellipse's perimeter has no exact formula in elementary terms. This calculator computes it exactly with the arithmetic–geometric mean (the method used for elliptic integrals). For semi-axes 5 and 3 it gives 25.526999; Ramanujan's well-known approximation, π(a + b)(1 + 3h ÷ (10 + √(4 − 3h))) with h = ((a − b) ÷ (a + b))², gives 25.526999, which is extremely close for everyday shapes.
Tips and common mistakes
- Use the perpendicular height. For triangles, parallelograms and trapezoids, the slanted side gives too large an answer.
- Diameter is not radius. Halve a measured diameter before using πr², or the area will be four times too big.
- Square the units, not just the number. 1 m² is 10,000 cm², not 100 cm². Convert lengths before calculating rather than converting the area by a length factor.
- Add waste for materials. For flooring, tile or turf, buy extra beyond the pure area: the flooring calculator and tile calculator handle that.
For a full triangle solution see the triangle calculator; for circles from any measurement, the circle calculator; and for 3D shapes, the surface area calculator.
Frequently asked questions
How do I calculate the area of a rectangle?
Multiply the length by the width. A rectangle 8 units long and 6 units wide has an area of 8 × 6 = 48 square units, and its perimeter is 2 × (8 + 6) = 28 units.
How do I find the area of a triangle?
Multiply the base by the perpendicular height and halve it: a base of 10 and height of 4 give 20 square units. If you only know the three sides, use Heron's formula: sides 13, 14 and 15 give an area of 84.
How do I find the area of a trapezoid?
Add the two parallel sides, halve the total and multiply by the height between them. With parallel sides of 6 and 10 and a height of 4, the area is (6 + 10) ÷ 2 × 4 = 32 square units.
How do I find the area of an L-shaped or irregular room?
Split it into rectangles, work out each area and add them. A room made of a 12 × 10 ft section and a 6 × 4 ft section is 120 + 24 = 144 square feet, about 13.38 square meters.
What is the area of a regular hexagon?
For side length s, the area is 3√3 ÷ 2 × s², from the general formula n × s² ÷ (4 × tan(180° ÷ n)). A hexagon with sides of 4 has an area of 24√3, about 41.57 square units.