Area Calculator

Area formulas — jump to a shape

Circle · Square · Rectangle · Triangle · Parallelogram · Trapezoid · Ellipse

Area of a circle

A = π · r². Circumference = 2·π·r. Diameter = 2·r. Doubling the radius quadruples the area (it grows as r²).

Example: a pizza with radius 12 cm has area π·144 ≈ 452 cm².

Area of a square

A = side². Perimeter = 4·side. Diagonal = side · √2. All four sides equal; any face is both the base and the height.

Example: a 6m × 6m room has floor area 36 m² and a diagonal of 6√2 ≈ 8.49 m.

Area of a rectangle

A = width × height. Perimeter = 2·(w + h). Diagonal = √(w² + h²) (from the Pythagorean theorem).

Example: a 4 × 7 photo has area 28 sq units and a diagonal of √65 ≈ 8.06.

Area of a triangle

Two common routes depending on what you know:

• Base-height: A = ½ · base · height. Height is the perpendicular distance from the base to the opposite vertex — notthe slanted side.
• Three sides (Heron's formula): A = √(s·(s−a)·(s−b)·(s−c)) with s = (a + b + c) / 2.

Example: a 3-4-5 right triangle has area ½·3·4 = 6, and Heron's formula agrees: √(6·3·2·1) = √36 = 6.

Area of a parallelogram

A = base · height. Same as a rectangle once you realise that sliding one side over doesn't change the area — only the apparent shape. The height is the perpendicular distance between the two parallel sides, not the slanted side length.

Example: a parallelogram with base 8 and perpendicular height 3 has area 24.

Area of a trapezoid

A = ½ · (a + b) · h, where a and b are the two parallel sides and h is the perpendicular height between them. Intuitively: average the two parallel sides, then multiply by the height.

Example: a trapezoid with parallel sides 6 and 4 and height 3 has area ½·(6+4)·3 = 15.

Area of an ellipse

A = π · a · b, where a is the semi-major and b is the semi-minor axis. When a = b you get πr² — the circle is just a special ellipse.

Perimeter doesn't have a neat closed form; the calculator uses Ramanujan's 1914 approximation, accurate to better than 0.001% for normal ellipses.

Example: an ellipse with a=5, b=3 has area π·15 ≈ 47.12.

Which formula when?

• Got base and perpendicular height? Use the base-height formula. Works for triangles, parallelograms, and trapezoids.
• Got all three sides of a triangle? Use Heron's formula — no need to find the height.
• Got a radius? It's a circle or ellipse — use π.
• Got two perpendicular sides? It's rectangular — multiply.

Common mistakes to avoid

• Using the slant side as height. For triangles, parallelograms, and trapezoids, “height” is always the perpendicular distance between the base and the opposite side — not the length of the slanted side.
• Forgetting to square (or not squaring). Circle area uses r²; if you write r you get the radius times π, not the area.
• Mixing up π and 2π. Area uses π·r². Circumference uses 2·π·r. They're easy to swap when tired.
• Heron with invalid sides. Three lengths can only form a triangle if each pair sums to more than the third. The calculator checks the triangle inequality and errors out if it fails.