Area Calculator

Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. It can be visualized as the amount of paint that would be necessary to cover a surface, and is the two-dimensional counterpart of the one-dimensional length of a curve, and the three-dimensional volume of a solid. The standard unit of area in the International System of Units (SI) is the square meter, or m². Provided below are equations for some of the most common simple shapes, and examples of how the area of each is calculated.

Rectangle

A rectangle is a quadrilateral with four right angles. Calculating its area only requires that its length and width are known (or can be measured). When the length and width of a rectangle are equal, the shape is a special case of a rectangle, called a square. The equation for calculating the area of a rectangle is as follows:

area = length × width

For example, a perfectly rectangular plot of land with a length of 220 feet and a width of 99 feet has an area of:

area = 220 × 99 = 21,780 sq ft

That plot, at 21,780 square feet, equates to half an acre, where an acre is defined as 43,560 square feet.

Triangle

There are many equations for calculating the area of a triangle based on what information is available. Please use the Triangle Calculator for further details. Briefly, the equation used in the calculator above is known as Heron's formula, referring to the Hero of Alexandria, a Greek mathematician and engineer. The formula is as follows:

area = √s(s - x)(s - y)(s - z)

where s = x + y + z2, and x, y, z are the sides of the triangle

For example, an equilateral triangle with sides of 77 ft has s = (77 + 77 + 77) / 2 = 115.5, and:

area = √115.5 × (115.5 - 77)³ = 2,567.33 sq ft

Trapezoid

A trapezoid is a simple convex quadrilateral that has at least one pair of parallel sides. The parallel sides are referred to as the bases of the trapezoid, and the other two sides are called the legs. The area is calculated in the same manner regardless of the type of trapezoid, using the following equation:

area = b₁ + b₂2 × h

where b₁ and b₂ are the bases, and h is the height, or perpendicular distance between the bases. For example, a trapezoidal face with a height of 9 ft, a bottom base of 29.528 ft, and a top base of 9 ft has an area of (9 + 29.528) / 2 × 9 = 173.376 sq ft.

Circle

A circle is a simple closed shape formed by the set of all points in a plane that are a given distance from a given center point. This distance from the center to any point on the circle is called the radius. More detail can be found on the Circle Calculator page, but to calculate the area, it is only necessary to know the radius, and understand that values in a circle are related through the mathematical constant π. The equation is:

area = πr²

For example, a circle with a radius of 15 ft has an area of π × 15² = 706.858 sq ft.

Sector

A sector of a circle is essentially a proportion of the circle that is enclosed by two radii and an arc. Given a radius and an angle, the area of a sector can be calculated by multiplying the area of the entire circle by a ratio of the known angle to 360° or 2π radians:

area = θ360 × πr²  if θ is in degrees

area = θ2π × πr²  if θ is in radians

For example, a 60° slice of a pie with a radius of 16 inches has an area of 60/360 × π × 16² = 134.041 in².

Ellipse

An ellipse is the generalized form of a circle, and is a curve in a plane where the sum of the distances from any point on the curve to each of its two focal points is constant. When the two foci coincide, the resulting ellipse is a circle. The semi-major axis of an ellipse is the longest radius of the ellipse, while the semi-minor axis is the shortest. The equation for calculating the area of an ellipse is similar to that for a circle, with two radii rather than one:

area = πab

where a and b are the semi-major and semi-minor axes. For example, an ellipse with a = 18 ft and b = 20 ft has an area of π × 18 × 20 = 1,130.97 sq ft.

Parallelogram

A parallelogram is a simple quadrilateral which has two pairs of parallel sides, where the opposite sides and angles have equal lengths and angles. Rectangles, rhombuses, and squares are all special cases of parallelograms. A parallelogram uses base and height, where the height is the length of the perpendicular between a pair of bases:

area = b × h

For example, a face with a base of 20 ft and a height of 18 ft has an area of 20 × 18 = 360 sq ft.

Common Area Units