Area Calculator: Area of Common Shapes
Pick a shape, enter its measurements and get the area with the formula worked out. Includes a triangle from three sides (Heron formula), regular polygons, ellipses and circle sectors, plus perimeter where it follows.
Area formulas used on this page
| Shape | Formula |
|---|---|
| Square | s² |
| Rectangle | length × width |
| Triangle | ½ × base × height |
| Triangle, three sides | √(s(s−a)(s−b)(s−c)), with s = (a+b+c) ÷ 2 |
| Circle | π r² |
| Ellipse | π a b |
| Trapezoid | ½ (a + b) × height |
| Parallelogram | base × height |
| Rhombus | ½ × d1 × d2 |
| Regular polygon | n s² ÷ (4 tan(π ÷ n)) |
| Sector | (angle ÷ 360) × π r² |
All of these measure how many unit squares fit inside the shape, so the result is in squared units. If your sides are in metres the area is in square metres.
Triangles when you only know the sides
The base-and-height formula needs a perpendicular height, which is often hard to measure. Heron formula avoids that by using just the three side lengths. First find the semi-perimeter s = (a + b + c) ÷ 2, then take √(s(s − a)(s − b)(s − c)). For the 13, 14, 15 triangle, s = 21, and the product 21 × 8 × 7 × 6 equals 7,056, whose square root is 84. The sides must satisfy the triangle inequality: the two shorter sides together must be longer than the longest. If they are not, the pieces cannot meet and the calculator reports an error rather than an area.
Common mistakes with area
- Using a slanted side as the height. The height of a triangle, parallelogram or trapezoid is the perpendicular distance between the parallel sides, not the length of a slanted edge.
- Confusing radius and diameter. A circle with diameter 10 has radius 5 and area about 78.54, not 314.16.
- Forgetting to square the unit. Converting 1 m² is 10,000 cm², not 100 cm². Convert lengths before multiplying, or use the squared factor.
- Mixing units. Multiplying feet by inches gives a meaningless number; use one unit throughout.
- Treating a rhombus like a rectangle. Its area comes from the diagonals or from base times perpendicular height, not from side times side, unless it is a square.
Perimeter and limits
Perimeter is shown whenever the entered values determine it. A triangle given by base and height, a trapezoid and a parallelogram do not fix their slanted sides, so no perimeter is given for them. Ellipse perimeter uses Ramanujan first approximation, which is very close for most shapes but is still an approximation. Irregular outlines can be split into the shapes above, calculated separately and added, or subtracted where a hole is cut out. Results are rounded for display to at most four decimal places, while the calculation itself uses full precision.
Worked example
A triangular plate has sides of 13 cm, 14 cm and 15 cm and no known height. Heron formula still gives its area.
Inputs
| Shape | Triangle (three sides, Heron) |
|---|---|
| Length unit | cm |
| Side a | 13 |
| Side b | 14 |
| Side c | 15 |
Result
Area is always in square units of the length unit you chose.
| Working | s = 21, then √(s(s−a)(s−b)(s−c)) = √(7,056) |
|---|---|
| Perimeter | 42 cm |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I find the area of a triangle with three known sides?
Use Heron formula. Find half the perimeter, subtract each side from it, multiply those four numbers and take the square root. A 3, 4, 5 triangle gives 6.
Why is the area of a circle pi times radius squared?
Cutting a circle into thin slices and rearranging them approximates a rectangle whose width is half the circumference and whose height is the radius, giving π r × r.
What is the difference between area and perimeter?
Perimeter is the distance around the outside of a shape, measured in length units. Area is the surface enclosed, measured in square units.
How do I find the area of an irregular shape?
Divide it into rectangles, triangles and other simple pieces, find each area and add them. For a hole or cut-out, subtract that part.
Which unit should I use for the answer?
The area comes out in the square of the length unit you enter. Choose the unit that matches your measurements and convert afterwards if you need another.
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References and sources
- Standard Euclidean geometry area formulas; Heron of Alexandria formula for triangles; Ramanujan perimeter approximation for ellipses.