Area Calculator
Calculate the area and perimeter of rectangles, triangles, circles, ellipses, trapezoids and sectors, with the formula shown for each shape.
How to use this calculator
- 1Pick your shape — the form shows only the dimensions that shape needs.
- 2Enter every dimension in the same unit.
- 3For triangles, make sure the height is perpendicular to the base.
How the calculation works
Rectangle: A = lw Triangle: A = ½bh Circle: A = πr² Trapezoid: A = ½(a+b)h Ellipse: A = πab- l, w
- Length and width
- b, h
- Base and perpendicular height
- r
- Radius
- a, b
- The two parallel sides, or the two semi-axes of an ellipse
For a triangle given three sides, Heron’s formula applies: A = √(s(s−a)(s−b)(s−c)) where s is half the perimeter.
Ellipse perimeter has no exact closed form. This calculator uses Ramanujan’s second approximation, accurate to better than one part in a billion for ordinary shapes.
Worked example
A triangle with sides 5, 6 and 7
- 1.The semi-perimeter s = (5 + 6 + 7) ÷ 2 = 9.
- 2.Heron’s formula: A = √(9 × 4 × 3 × 2) = √216.
- 3.That gives 14.6969 m².
Result: 14.6969 m²
What area actually measures
Area is the amount of two-dimensional space a shape encloses — how much surface is inside its boundary. It is always expressed in squared units (square metres, square feet, square inches) because it is fundamentally different from length: a length tells you how far along a line to travel, while an area tells you how much flat space a boundary wraps around.
That squaring is what makes area behave so differently from perimeter. Double every dimension of a shape and its perimeter only doubles, but its area quadruples — a fact that trips up a lot of everyday estimates, from pizza sizes to paint coverage.
How each shape’s area is built up
Every formula this calculator uses reduces, eventually, to counting how many unit squares fit inside a boundary — the shapes just differ in how directly that count can be found.
- Rectangles and squares — the simplest case — area is just length times width, because a grid of unit squares fits the shape exactly with no leftover space to account for.
- Triangles — half of the rectangle that would enclose them, which is why the formula is always one-half of base times height — the height must be measured perpendicular to the base, not along a slanted side.
- Circles and sectors — built from π, the fixed ratio between a circle’s circumference and its diameter. A sector — a pie slice of a circle — takes the same idea and scales it down by the fraction of the full turn it covers.
- Trapezoids and parallelograms — both lean on the rectangle-and-triangle logic underneath them — a trapezoid averages its two parallel sides before multiplying by height, while a parallelogram is a rectangle that has been sheared sideways without changing its area.
- Ellipses — a circle stretched along one axis, so its area formula (π times both semi-axes) is a natural extension of a circle’s — though unlike a circle’s circumference, an ellipse’s perimeter has no exact formula at all.
Area versus perimeter, and why they’re often confused
Area and perimeter both describe a shape’s size, which is exactly why they get mixed up. Perimeter is a one-dimensional measurement — the distance you would walk tracing the outline — while area is two-dimensional, measuring the surface enclosed by that outline. Two shapes can have identical perimeters and very different areas: a long, thin rectangle and a square can be fenced with the same length of fencing yet enclose very different amounts of ground.
That is also why area and perimeter, taken alone, never fully describe a shape — knowing one doesn’t tell you the other unless you also know the shape’s proportions, which is exactly what this calculator’s separate inputs for each dimension are for.
Where area calculations show up
Area is one of the few pieces of maths almost everyone ends up using outside a classroom, usually to answer a very practical "how much do I need?".
- Flooring and carpet — sold by the square metre or square foot, so a room’s area (plus a margin for waste and offcuts) sets the order quantity directly.
- Paint and coatings — coverage is quoted per unit area on the tin, and wall area (minus doors and windows) tells you how many tins to buy.
- Land and real estate — property listings, land surveys and zoning rules are all built around plot area, often in acres or hectares rather than square metres.
- Agriculture — crop yields, fertiliser and seed rates are quoted per acre or hectare, so field area drives almost every input decision on a farm.
- Solar panels and roofing — the usable roof area limits how much solar capacity or how many shingles a given roof can carry.
Handling irregular and real-world shapes
Very few real surfaces are a perfect rectangle or circle. The standard approach for anything irregular is to break it apart rather than search for a more complicated formula.
- 1Split it into primitives — divide an L-shaped room, an odd-shaped plot or a curved garden bed into rectangles, triangles and circular pieces that this calculator already covers.
- 2Measure everything in one unit — mixing centimetres and metres in the same shape is the single most common source of a wrong answer — settle on one unit before measuring anything.
- 3Add the pieces back together — sum the area of each primitive shape to get the total. Overlapping pieces need to be subtracted once, not counted twice.
- 4Budget a margin for waste — trades commonly add 5–10% on top of the measured area for flooring, tiling or turf, to cover cuts, offcuts and mistakes.
What this assumes, and where it stops
Assumptions
- Shapes are flat and regular as described. Real-world surfaces may need to be broken into several shapes.
Limitations
- A triangle given only base and height has no determinable perimeter, since infinitely many triangles share those two measurements.
- Irregular shapes need to be divided into these primitives and the areas added.
Common questions
How do I calculate the square footage of a room?
Multiply length by width in feet. For an L-shaped room, split it into two rectangles, calculate each, and add them. Add about 10% when buying flooring to allow for cuts and waste.
Why does my triangle calculation say the sides are impossible?
Because of the triangle inequality: any two sides must add up to more than the third. Sides of 2, 3 and 9 cannot close into a triangle — the two short sides are not long enough to reach across.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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