Area Calculator

MathLast updated: August 17, 2026

Area is the measure of the surface a plane figure covers, and every shape has its own formula: the product of the sides for a rectangle, half the base times the height for a triangle, and π times the radius squared for a circle. This calculator works out the area of the five most common basic shapes on a single screen.

Two methods are offered for triangles: the classic formula when the base and height are known, and Heron's formula when only the three sides are known. Wherever it can be computed, the perimeter is reported alongside the area. Whatever unit you enter the lengths in (cm, m, km), the area comes out in the square of that unit, which makes the tool a practical helper for carpet, tile, paint, and land calculations.

Formula

Square: Area = a², Perimeter = 4a
Rectangle: Area = a × b, Perimeter = 2(a + b)
Triangle: Area = (base × height) / 2
Heron: s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c))
Circle: Area = πr², Circumference = 2πr
Trapezoid: Area = ((a + b) / 2) × h

Forgetting to divide by 2 in the triangle formula and using the diameter instead of the radius in the circle formula are the two most frequent mistakes. If you are given the diameter, halve it first to obtain the radius; in a triangle the height is the length PERPENDICULAR to the base, not a slanted side.

How to Calculate

  1. Select the shape whose area you want to calculate.
  2. Enter the lengths of that shape in the same unit (all in meters, for instance).
  3. Read the area and, where available, the perimeter; the unit is the square of the one you entered (m² if you entered meters).
  4. The steps written out in the interpretation line let you check the result by hand as well.

Worked Examples

Rectangular rug: 5 × 8

A room with sides of 5 and 8 meters has a floor area of 5 × 8 = 40 m²; the perimeter needed for the baseboard is 2 × (5 + 8) = 26 meters.

Area: 40 unit² · Perimeter: 26 units

Round tablecloth: r = 3

A circle with a radius of 3 units has an area of π × 3² ≈ 28.27 unit² and a circumference of 2π × 3 ≈ 18.85 units. Its diameter is 6 units; if the diameter is what you are given, halve it before entering it.

Area: 28.27 unit² · Circumference: 18.85 units · Diameter: 6 units

The 3-4-5 triangle (Heron)

In the famous 3-4-5 right triangle the semiperimeter is s = 6 units; Heron's formula √(s(s−a)(s−b)(s−c)) gives an area of 6 unit² and a perimeter of 12 units.

Area (Heron): 6 unit² · Perimeter: 12 units · Semiperimeter (s): 6 units

Frequently Asked Questions

How is the unit of area determined?
Area is expressed in the square of the length unit: if you enter the sides in meters the result is in m², and if you enter them in centimeters it is in cm². All lengths MUST be entered in the same unit; mixing meters with centimeters ruins the result completely.
Why is the area of a triangle half the base times the height?
Any triangle, joined with a copy of itself, forms a parallelogram whose dimensions are base by height. Since the triangle is exactly half of that parallelogram, its area is (base × height) / 2.
When is Heron's formula used?
It is used when the height of the triangle is unknown and only the three side lengths are available. First the semiperimeter s = (a+b+c)/2 is found, then the area is computed as √(s(s−a)(s−b)(s−c)). If the sides do not satisfy the triangle inequality the formula yields no result, and the calculator says so.
What is the difference between diameter and radius?
The radius (r) is the distance from the center to the circle, while the diameter (d) runs from one side of the circle to the other through the center: d = 2r. Circle formulas are written with the radius, so if you have the diameter you must halve it first; this is the most common mistake of all.
How do you convert between m² and cm²?
Since 1 m = 100 cm, 1 m² = 100 × 100 = 10,000 cm². In other words, area conversion uses the SQUARE of the length factor: you multiply by 10,000, not by 100. In the same way, 1 km² = 1,000,000 m².
Do shapes with the same perimeter have the same area?
No. A 6×6 square with a perimeter of 24 units has an area of 36, while a 2×10 rectangle with the same perimeter has an area of 20. For a fixed perimeter the circle gives the largest area, so the perimeter alone never determines the area.