Area Calculator

How this area calculator works

Area is the amount of flat space a shape covers, measured in square units: square feet, square meters, square inches. This area calculator has the standard formula for each common shape built in. Pick the shape, type the dimensions in whichever unit you measured with, and the result comes out in that unit squared, with the formula filled in with your numbers and the same area converted to the two other units you are most likely to need.

The formulas, with worked examples

Rectangle: length × width. A 12 ft by 8 ft bedroom is 96 ft². Square: side². Triangle: ½ × base × height, where the height is the straight-up distance from the base to the opposite corner, not the slanted side. A triangle with base 10 and height 6 has area 30. If you know the three sides instead, Heron's formula does the job: for sides 3, 4 and 5, s = (3 + 4 + 5) / 2 = 6 and A = √(6 × 3 × 2 × 1) = √36 = 6.

Circle: π × radius². With a 5 ft radius that is π × 25 ≈ 78.54 ft². If you measured the diameter, halve it first. Ellipse: π × a × b using the two semi-axes (half the longest and half the shortest width). Trapezoid: average the two parallel sides, then multiply by the height: ½ × (10 + 6) × 4 = 32. Parallelogram: base × perpendicular height, just like a rectangle. Sector: a slice of a circle, so take the fraction of 360° that the angle covers: a 90° slice of a radius-10 circle is ¼ × π × 100 ≈ 78.54. Regular polygon: n × s² / (4 × tan(180° / n)); a hexagon with 4-unit sides is 6 × 16 / (4 × tan 30°) ≈ 41.57.

Converting between square units

This is where most homework points are lost. Because area is length × length, every conversion factor gets squared. There are 12 inches in a foot, so there are 12 × 12 = 144 square inches in a square foot, not 12. There are 3 feet in a yard, so 9 ft² in a yd². And since 1 ft = 0.3048 m, 1 ft² = 0.3048² ≈ 0.0929 m², which is why 1,000 ft² is only about 93 m². The calculator does these conversions for you, but it is worth understanding why the numbers look so different.

Common mistakes

• Using the slanted side as the height. For triangles and parallelograms the height must be perpendicular to the base. The slanted side is always longer and gives too big an answer.
• Using the diameter instead of the radius. πd² is four times too large. Halve the diameter first.
• Mixing units. A 3 ft by 20 in rectangle is not 60 of anything. Convert to one unit first (3 ft = 36 in, so 720 in² or 5 ft²).
• Entering three sides that cannot form a triangle. Each side must be shorter than the other two combined; 1, 2 and 3 flatten into a line and have zero area. The calculator will tell you.

Real-world uses

Flooring, paint, sod, tile and carpet are all sold by area, so measuring a room as a rectangle (or splitting an odd room into rectangles and adding them) tells you how much to buy, usually plus 10% for waste. A round patio or half-circle window is a job for the circle and sector formulas. For problems described by an equation rather than a shape, head to the quadratic formula calculator.