š Area Calculator
Calculate the area of nine common shapes. Enter the dimensions in any unit and the result comes back in that unit squared, with conversions to the other common units underneath.
Area formulas for nine shapes, and how to handle the awkward ones
Area is the amount of surface a two-dimensional shape covers, measured in square units. Every formula below reduces to the same idea: find an equivalent rectangle. A triangle is half a rectangle, a parallelogram is a rectangle with its top slid sideways, and a circle is the limit of a shape cut into ever thinner wedges and laid alternately head to tail.
The nine formulas
| Shape | Formula | You need |
|---|---|---|
| Rectangle | l × w | Length and width |
| Square | s² | One side |
| Triangle | ½ × b × h | Base and perpendicular height |
| Triangle (three sides) | √[s(s−a)(s−b)(s−c)] | All three sides; s is half the perimeter |
| Circle | πr² | Radius, which is half the diameter |
| Ellipse | πab | Both semi-axes |
| Trapezoid | ½(a + b) × h | The two parallel sides and the height |
| Parallelogram | b × h | Base and perpendicular height |
| Circle sector | πr² × angle/360 | Radius and the angle in degrees |
Perpendicular height, not slant height
The commonest error in triangle and parallelogram problems is using the sloping side instead of the vertical height. The height must be measured at right angles to the base. For a leaning parallelogram with a base of 8 and a slanted side of 5 at an angle, the area is 8 times the perpendicular height, not 8 times 5. If you only have the slant and the angle, the perpendicular height is slant × sin(angle).
Irregular shapes
Nothing in the real world is a clean rectangle. The practical method is decomposition: split the shape into rectangles, triangles and part-circles, calculate each and add them. An L-shaped room is two rectangles. A room with a bay window is a rectangle plus a sector. Subtract rather than add for holes: a patio with a circular fire pit is the rectangle minus πr².
Measure twice, and measure the diagonals of any room you believe is rectangular. If the two diagonals differ, it is not square, and treating it as one will leave you short of material.
Why area scales as the square
Doubling every linear dimension multiplies the area by four, not two, because both dimensions grow. A 3 metre by 4 metre room is 12 m²; a 6 by 8 room is 48 m². This catches people out when scaling plans, pricing by the square metre, or estimating paint. The same relationship in three dimensions makes volume scale as the cube, which is why a pizza twice the diameter is four times the pizza.
Unit conversions
| From | To | Multiply by |
|---|---|---|
| Square feet | Square metres | 0.092903 |
| Square metres | Square feet | 10.7639 |
| Square yards | Square metres | 0.836127 |
| Acres | Square metres | 4,046.86 |
| Hectares | Acres | 2.47105 |
Note that square conversions are the linear factor squared. A foot is 0.3048 m, and 0.3048² = 0.092903. Using the linear factor by mistake is a classic error that understates area by about a factor of three.
A worked example
An L-shaped patio: a main rectangle 6 m by 4 m, plus a wing 2.5 m by 3 m, with a circular planter of radius 0.8 m cut out. Main 24 m², wing 7.5 m², planter π × 0.64 = 2.01 m². Total area to pave: 24 + 7.5 − 2.01 = 29.49 m², which is 317.4 square feet.
Frequently Asked Questions (FAQs)
How do I find the area of an irregular shape?
Split it into rectangles, triangles and part-circles, work out each piece and add them. Subtract any holes. For genuinely curved boundaries, overlay a grid and count squares, or use the surveyor's formula if you have coordinates for the corners.
What is the difference between area and perimeter?
Area is the surface inside a shape, measured in square units; perimeter is the distance around its edge, in linear units. Two shapes can share a perimeter and have very different areas: a circle encloses the most area for a given perimeter, and a long thin rectangle the least.
Why do I need perpendicular height, not the slanted side?
Because the area formula measures how far the shape extends at right angles to the base. Using the slant overstates the area. If you only have the slant length and the angle, the perpendicular height is slant × sin(angle).
How many square feet are in a square metre?
10.764. Going the other way, a square foot is 0.0929 square metres. Note these are the linear conversions squared, which is where most conversion errors come from.
What is Heron's formula?
A way to find a triangle's area from its three sides alone, with no angle or height needed. Take half the perimeter, call it s, then the area is the square root of s(s−a)(s−b)(s−c). If the expression under the root is negative, those sides cannot form a triangle.
Does doubling a room's size double the material I need?
Only if you doubled the area. Doubling both the length and the width quadruples the area and therefore the material. This is the single most expensive misunderstanding in DIY estimating.