Area Calculator
Find the area (and perimeter) of 11 common 2D shapes. Pick a shape, enter its dimensions, and get the result with a step-by-step worked solution. Toggle on the cost estimator to price flooring, turf, paint, or any other material by the square unit.
Formula
Worked example
Trapezoid with a = 8, b = 12, h = 5: A = 0.5 x (8 + 12) x 5 = 50 sq units. Circle of radius 5: A = pi x 25 = 78.54 sq units. Triangle by Heron with sides 6, 8, 10: s = 12; A = sqrt(12 x 6 x 4 x 2) = sqrt(576) = 24 sq units. Regular hexagon, side 5: A = (6 x 25) / (4 x tan(30 deg)) = 150 / (4 x 0.5774) = 64.95 sq units.
How each area formula works
Area is the amount of flat surface a shape covers, always expressed in squared units. A rectangle tiles neatly into unit squares, so its area is simply length times width. A square is the special case where both sides are equal. A triangle is exactly half the rectangle that encloses it, giving the well-known half-base-times-height rule. A parallelogram shears that rectangle sideways without changing the base or the perpendicular height, so its area is identical. A trapezoid has two parallel sides of different lengths, so it averages them before multiplying by the height. A rhombus is most easily solved with its two diagonals: they bisect each other at right angles, so the area is half their product. A circle of radius r has area pi times r squared. Cutting it in half gives the semicircle. A sector is a pie slice: its fraction of the full circle equals the central angle divided by 360, so multiply that fraction by pi r squared. An ellipse stretches the circle across two axes, giving pi times the semi-major axis times the semi-minor axis. A regular polygon with n equal sides of length s uses A = (n x s squared) divided by (4 x tan(pi/n)), which handles everything from an equilateral triangle to a 100-sided figure.
Multiple ways to solve a triangle
The calculator offers three triangle modes. Base-and-height is the quickest when you can measure straight down from the apex to the base. Side-angle-side (SAS) works when you know two sides and the angle between them: A = 0.5 x a x b x sin(C). This is especially useful when the perpendicular height is hard to measure but a protractor reading is easy. Three-sides (SSS) uses Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then A = sqrt(s x (s - a) x (s - b) x (s - c)). Check the triangle inequality first: each side must be shorter than the sum of the other two, otherwise the shape cannot be closed and the result is not a real number.
Using the perimeter and cost outputs
Most shapes also return a perimeter alongside the area. For a circle that is the circumference (2 pi r); for a semicircle it is the curved arc plus the diameter. The trapezoid perimeter is exact when you supply a leg length, otherwise an isosceles approximation is used. For the parallelogram, supply the slanted side to unlock the perimeter. The optional cost estimator multiplies the computed area by your price per square unit and is a quick way to budget flooring, turf, roof sheathing, wall paint, or any other material priced by the square. Enter the price in whatever currency and unit combination suits your project, then read the estimated total straight off the result card.
Choosing the right height and keeping units consistent
For triangles, parallelograms, and trapezoids the height must be the perpendicular distance between the relevant sides, not the length of a slanted edge. Using a sloped edge instead of the true vertical height is the most common error and always overstates the area. Keep every measurement in the same unit before calculating: if one side is in feet and another in inches, convert one first. The area result carries squared units, so a shape measured in metres yields an answer in square metres. If you then want square feet, multiply by 10.764.
Area formulas by shape
| Shape | Formula | Inputs needed |
|---|---|---|
| Rectangle | A = l x w | Length, width |
| Square | A = s^2 | Side |
| Triangle (b+h) | A = (1/2) x b x h | Base, height |
| Triangle (SAS) | A = (1/2) x a x b x sin(C) | Two sides, included angle |
| Triangle (SSS) | A = sqrt(s(s-a)(s-b)(s-c)) | Three sides (Heron) |
| Circle | A = pi x r^2 | Radius |
| Semicircle | A = pi x r^2 / 2 | Radius |
| Sector | A = (theta/360) x pi x r^2 | Radius, central angle |
| Ellipse | A = pi x a x b | Semi-major & semi-minor axes |
| Trapezoid | A = (1/2)(a + b) x h | Two parallel sides, height |
| Parallelogram | A = b x h | Base, perpendicular height |
| Rhombus | A = (d1 x d2) / 2 | Both diagonals |
| Regular polygon | A = (n x s^2) / (4 tan(pi/n)) | Number of sides, side length |
Variables: b = base, h = perpendicular height, r = radius, a/b = parallel sides or semi-axes, d1/d2 = diagonals, n = number of sides, s = side length.
Frequently asked questions
What units does the area come out in?
The area is in squared units matching whatever length unit you enter. If you measure in centimetres, the area is in square centimetres; feet gives square feet. The calculator labels the result with generic "units^2" so you can apply your own unit. To convert, for example, square metres to square feet, multiply by 10.764.
How do I find the area of a triangle when I only know the three side lengths?
Switch the triangle mode to "Three sides (Heron's formula)" and enter all three sides. The calculator computes the semi-perimeter s = (a + b + c) / 2, then evaluates sqrt(s x (s - a) x (s - b) x (s - c)). The three sides must satisfy the triangle inequality: each side must be less than the sum of the other two, otherwise no valid triangle exists.
How do I find the area of a circle from its diameter?
Divide the diameter by two to get the radius, then enter that radius. The calculator applies A = pi x r^2. For example, a 10-unit diameter gives a 5-unit radius and an area of about 78.54 square units.
What is the difference between a sector and a semicircle?
A semicircle is exactly half a circle (central angle 180 degrees). A sector is any pie-slice portion of a circle defined by a central angle between 0 and 360 degrees. Set the sector angle to 180 and the sector area equals the semicircle area. The sector formula is A = (angle / 360) x pi x r^2.
How does the cost estimator work?
Toggle "Estimate material cost" on and enter your price per square unit. The calculator multiplies the computed area by that price. Use it to budget flooring (price per square foot), artificial turf (price per square metre), paint coverage (price per square yard), or any other material sold by area. Make sure your price unit matches the length unit you entered for the shape dimensions.
Why does the parallelogram need a side length for the perimeter but not for the area?
Area of a parallelogram only needs the base and the perpendicular height, which is independent of the slant of the sides. Perimeter, however, is the total boundary length, so it needs the actual length of the slanted side as well as the base. Leave the side field blank if you only need the area.