Area Formulas for Every Common Shape
Every area formula worth keeping, each one with the reasoning behind it rather than just the symbols. Pick a shape below to run the numbers and see the working, then read on for where each formula comes from and which one to reach for when the obvious measurement is the one you do not have.
The sketch is drawn to scale from the numbers you enter, so a shape that looks wrong on screen usually means a measurement went in wrong.
Four ideas generate nearly all of them
There are perhaps thirty area formulas in common use and about four reasons behind them. Once you can see which reason a formula belongs to, you stop memorising and start deriving, which matters because the formula you half remember is usually the one you get wrong.
Base times height covers the rectangle and the parallelogram, because leaning a shape over slides area sideways without creating any. Halving that covers every triangle. The mean of the parallel sides covers the trapezoid, and it quietly contains the other two as special cases: set the top side equal to the bottom and you have a parallelogram, shrink it to zero and you have a triangle. Half the perimeter times the apothem covers every regular polygon and, in the limit, the circle.
Every area formula on one page
All of these are checked against each other in the calculator above, and several are computed two independent ways so a disagreement would show up rather than hide.
| Triangle, given | Formula | Notes |
|---|---|---|
| Base and height | A = b h / 2 | The height is the perpendicular drop to the base, not a side. |
| Three sides | A = √(s(s−a)(s−b)(s−c)) | Heron, with s = (a + b + c) / 2. No height needed at all. |
| Two sides and the angle between | A = a b sin C / 2 | The angle has to sit between the two sides you measured. |
| Equilateral, one side | A = √3 a² / 4 | About 0.4330 a². Agrees with Heron to the last decimal. |
| Three corner coordinates | A = |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| / 2 | The shoelace, and it extends to any polygon. |
| Quadrilateral | Formula | Notes |
|---|---|---|
| Square | A = a² | The regular polygon formula with four sides gives the same thing. |
| Rectangle | A = l w | |
| Parallelogram | A = b h | Or a b sin θ if you have two sides and the angle instead. |
| Rhombus | A = p q / 2 | Diagonals, which cross at right angles. Also a² sin θ. |
| Kite | A = p q / 2 | Same rule as the rhombus, for the same reason. |
| Trapezoid | A = (a + b) h / 2 | The mean of the parallel sides times the distance between them. |
| Circle and curves | Formula | Notes |
|---|---|---|
| Circle | A = π r² | Radius, never diameter. Using the diameter quadruples the answer. |
| Semicircle | A = π r² / 2 | |
| Sector | A = π r² θ / 360 | Degrees. In radians it is r² θ / 2. |
| Circular segment | A = r² (θ − sin θ) / 2 | Radians only. The sector minus the triangle inside it. |
| Annulus | A = π (R² − r²) | The ring between two circles sharing a centre. |
| Ellipse | A = π a b | Semi axes. Set a equal to b and it collapses to the circle. |
| Regular polygon, given | Formula | Notes |
|---|---|---|
| Side length and side count | A = n s² / (4 tan(π / n)) | The general case. Every shape below is this one with n fixed. |
| Perimeter and apothem | A = P a / 2 | The apothem is the centre to edge distance, not centre to corner. |
| Circumradius | A = n R² sin(2π / n) / 2 | Useful when the shape is set out from a circle. |
| Hexagon | A = 3√3 a² / 2 | About 2.5981 a². It is six equilateral triangles. |
| Pentagon | A ≈ 1.7205 a² | The general formula with five sides. |
| Octagon | A = 2(1 + √2) a² | About 4.8284 a². |
A worked check you can repeat on paper
Take a trapezoid with parallel sides of 8 and 14 and a height of 5. The direct route: the mean of 8 and 14 is 11, and 11 times 5 is 55. Now check it twice more without using that formula at all.
Cut it along a diagonal and you get two triangles, one with base 8 and one with base 14, both with height 5. That is 8 times 5 over 2, which is 20, plus 14 times 5 over 2, which is 35. Total 55. Then put the corners on coordinates, at (0, 0), (14, 0), (11, 5) and (3, 5), and run the shoelace: the cross products come to 110, halved is 55. Three routes, one number, and none of them borrowed anything from the others.
When the height is the measurement you do not have
This is the situation that sends people looking for a formula in the first place. A surveyor has corner coordinates, not heights. Someone measuring a room has four wall lengths and no perpendicular anywhere. The right move is to change formula rather than to start estimating.
For a triangle with three sides, Heron needs no height. For two sides and the angle between them, half a b sin C needs no height. For anything with known corner coordinates, the shoelace handles any number of sides and does not care whether the shape is convex. The honest exception is the trapezoid known only by its four side lengths: the height can be recovered, but the algebra is unpleasant and the result is fragile when the two parallel sides are close in length. Measure the height instead if you possibly can.
Mistakes that catch people out
- Using a slant side as the height. In a parallelogram or a trapezoid the sloping side is always longer than the perpendicular height, so this inflates the area every time. The steeper the lean, the worse it gets.
- Feeding in the diameter instead of the radius. Area goes with the square, so this does not overstate the circle by a factor of two, it overstates it by four.
- Mixing units mid calculation. A length in centimetres multiplied by a length in metres gives nothing useful. Convert first, and remember that a factor of 100 in length is a factor of 10,000 in area.
- Running Heron on lengths that cannot close. If any side is longer than the other two combined, the product under the root goes negative. That is the formula telling you the triangle does not exist, not a rounding problem.
- Degrees where the formula wants radians. The sector formula exists in both forms, but the segment formula only works in radians because it subtracts sin θ from θ directly. Feeding 90 in where 1.5708 belongs produces a confident wrong answer.
Common questions
What is the general formula for area?
There is no single one, but most reduce to base times height or half of it. A rectangle is b h, a parallelogram is b h, a triangle is b h over 2, and a trapezoid is the mean of its parallel sides times h. Circles and regular polygons follow half the perimeter times the apothem instead.
How do you find area without knowing the height?
Use a formula built for what you do have. Three sides means Heron. Two sides and the angle between them means half a b sin C. Corner coordinates mean the shoelace, which works for any polygon regardless of shape.
What is the area formula for a trapezoid?
Add the two parallel sides, halve the total, and multiply by the perpendicular height. Sides of 8 and 14 with a height of 5 give a mean of 11 and an area of 55 square units.
Why is a triangle half of base times height?
Because any triangle is exactly half of a parallelogram. Copy the triangle, rotate the copy through half a turn and join the two along a common side, and you get a parallelogram with the same base and the same height. Two triangles, one parallelogram, so one triangle is half of b h.
Why does area use squared units?
Because it is a count of unit squares. Every area formula multiplies one length by another, so the units multiply too, and metres times metres is square metres. It is also why doubling every dimension of a shape multiplies its area by four rather than two.
How do you find the area of an irregular shape?
Two ways. Split it into triangles and rectangles, work each out and add them. Or take coordinates around the boundary and run the shoelace once, which handles any number of corners in a single pass and is what survey software does internally.
Every geometry lesson on the site
Each one takes a single question, works it through with real numbers and shows where it goes wrong in practice.
Related calculators and reference
- Area of 2D shapes calculator, the standalone version with Heron and the trapezoid built in.
- Triangle solver, for when you need the angles and the remaining sides as well as the area.
- Volume of 3D shapes calculator, the next step up when the shape has depth.
- Circle arc calculator, for chords, sagitta and radius on the curved cases above.
- Pythagoras and slope calculator, for recovering a height before an area formula can use it.
- NIST on SI units, the reference for how squared units are defined and written.