Shape Areas with Explanations
Pick a shape, enter the dimensions — you get the area formula, your numbers substituted, and the answer. Circles also get the circumference, polygons the perimeter.
Solution
Where the formulas come from. All area formulas grow out of the rectangle's a · b. A parallelogram rearranges into a rectangle, a triangle is half a parallelogram, a trapezoid is its “average width” times the height. The circle's πr² comes from slicing the circle into thin sectors. The calculator shows the formula and the substitution for every shape — ready to copy into a notebook.
FAQ
How do you find a triangle's area from the sides alone?
With Heron's formula: compute the semi-perimeter p = (a + b + c) / 2, then the area is the square root of p(p−a)(p−b)(p−c). No height needed. The calculator has a switch for this method.
What value of π should I use?
π ≈ 3.14159… School problems usually take 3.14; the calculator uses higher precision. If the problem says “take π = 3.14”, the answer may differ slightly from the calculator's — that is normal.
What units does the area come in?
Square units: dimensions in centimeters give cm², in meters — m². Just make sure all dimensions of one shape use the same unit.
Why isn't a parallelogram's area the product of its sides?
Because it leans: the height is shorter than the slanted side. The product of the sides would be the area of a rectangle with those sides, and a parallelogram with the same sides is always “lower”. Hence the height in the formula: S = a · h.
Free online area calculator for geometric shapes with formulas and explanations: circle, triangle, square, rectangle, trapezoid, and parallelogram. Every shape gets its formula, your dimensions substituted, and the answer.
The triangle is computed two ways: via base and height or from three sides with Heron's formula, including the semi-perimeter step. The circle additionally gets its circumference, the square and rectangle their perimeter.
Everything is computed in the browser. Handy for geometry classes, for home projects — the area of a room or material — and anywhere you need a shape's area with the working shown.