Calculate the area of common 2D shapes: rectangle, triangle, circle, trapezoid, ellipse, sector, polygon, and more.
| Shape | Formula |
|---|---|
| Square | A = a² |
| Rectangle | A = l × w |
| Triangle (Base & Height) | A = ½ × b × h |
| Triangle (3 Sides) | A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2 (Heron's) |
| Circle | A = πr² |
| Parallelogram | A = b × h |
| Trapezoid | A = ½(a + b) × h |
| Rhombus | A = (d₁ × d₂) / 2 |
| Ellipse | A = π × a × b |
| Sector | A = (θ/360°) × πr² |
| Regular Polygon | A = (n × s²) / (4 × tan(π/n)) |
How do I find the area of a triangle when I only know the three side lengths? That question comes up constantly, and it is the one case where the base-times-height habit stops working. This calculator handles eleven flat shapes, and for triangles it accepts either a base and height or all three sides.
Take sides of 13, 14 and 15. The calculator reports an area of 84 square units and a perimeter of 42. Here is where each figure comes from.
Step one is the semi-perimeter, s = (a + b + c) / 2 = (13 + 14 + 15) / 2 = 21. Step two subtracts each side from s: 21 − 13 = 8, 21 − 14 = 7 and 21 − 15 = 6. Step three multiplies all four numbers: 21 × 8 × 7 × 6 = 7,056. Step four takes the square root, and √7,056 = 84. Those four steps are Heron's formula, A = √(s(s − a)(s − b)(s − c)).
Before any of that runs, the page checks the triangle inequality, which says any two sides must add up to more than the third. If they do not, you get an error message in place of a number, since no such triangle exists.
The remaining shapes are one-line formulas. A regular hexagon with side 6 uses A = n × s² / (4 × tan(π/n)) = 6 × 36 / (4 × tan 30°), giving 93.53074361. A 90° sector of a circle with radius 10 uses (θ/360) × πr² = 0.25 × π × 100 = 78.53981634. Results are rounded to eight decimals.
Any two sides must add up to more than the third. Sides of 3, 4 and 8 fail, and so do 3, 4 and 7, because the triangle would be completely flat with zero area. Recheck the numbers before assuming the calculator is wrong.
Perimeter appears for the square, rectangle, three-sided triangle, circle, ellipse and regular polygon. Trapezoid, parallelogram, rhombus and sector would need extra lengths you did not enter. The ellipse area is exact, but its perimeter uses Ramanujan's approximation, since no simple exact formula exists. For semi-axes 5 and 3 the page gives 25.52698639.
No, height means the perpendicular distance between the parallel sides. A parallelogram with base 10, slanted side 6 and height 5 has area 50. Typing 6 would report 60, overstating the area by a fifth.
Convert the lengths before you calculate, or square the conversion factor afterwards. One square foot is 144 square inches, not 12, and one square metre is 10,000 square centimetres. Guessing a linear factor is the most common error here.