TheCalculatory
Math · Geometry Formula verified

Area Calculator

Pick a shape, enter its dimensions, and see the area (and perimeter, where that applies) update instantly, with the formula and a step-by-step proof underneath.

Shape
050100
050100
On-screen number pad
Tap to type into the highlighted field
Area
40
Rectangle · 8 × 5
Perimeter
26

Step-by-step proofCheck by hand

The formula for each shape

Every area formula below multiplies two lengths together (or a length by itself), which is why area is always measured in square units — square feet, square meters, and so on. Rectangle and parallelogram are the same idea: base times height, whether or not the sides are perpendicular. A triangle is exactly half of the parallelogram that shares its base and height. A trapezoid averages its two parallel sides before multiplying by height, since those two sides aren't the same length.

The formulas

rectangle = l × w  ·  circle = πr²  ·  triangle = ½bh  ·  trapezoid = ½(b₁+b₂)h

Worked example

An 8-by-5 rectangle: area = 8 × 5 = 40 square units, and perimeter = 2 × (8 + 5) = 26 units.

Frequently asked questions

Why doesn't the triangle calculator always show a perimeter?
It depends which two measurements you give it. Base and height alone tell you the area, but not the length of the two slanted sides, so there's nothing to add up for a perimeter. If you enter all three side lengths instead, the calculator can and does show the perimeter as well as the area.
Why does entering three side lengths for a triangle sometimes show an error?
Three lengths only form a real triangle if each one is shorter than the other two added together — this is called the triangle inequality. Three lengths like 1, 1 and 5 can't close into a triangle at all (the two short sides can never reach far enough to meet), so there's no area to compute.
What's the difference between area and perimeter?
Perimeter is the distance around the outside edge of a shape — walk the boundary and add up the distance. Area is how much surface the shape covers on the inside. They use different units too: perimeter is a length (feet, meters), while area is that length squared (square feet, square meters).
Sources and method
  • Standard geometric area and perimeter formulas — general published mathematics.
  • Triangle area from three sides — Heron's formula, general published mathematics.
Last reviewed: 19 Sep 2026 Sources last verified: 19 Sep 2026 Results use the assumptions explained on this page. Report an error How we check calculations

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