Area Calculator

Area is the amount of flat surface a two-dimensional shape covers. It is measured in square units: a square meter is the area of a square one meter on each side, and an acre is a traditional land unit of 43,560 square feet. Area matters whenever you buy material by coverage, such as paint, flooring, turf, roofing or fabric, or compare the size of rooms, plots and screens.

Most area formulas grow out of the rectangle. A parallelogram can be rearranged into a rectangle, a triangle is half a parallelogram, and a trapezoid is the average of two rectangles. Curved shapes such as the circle need π, but they too can be understood by slicing them into thin pieces. Irregular regions are usually split into a handful of these basic shapes, each measured separately, and the pieces added together.

Rectangle

A rectangle is a four-sided shape whose corners are all right angles, so opposite sides are equal and parallel. Its area is length times width. You can picture it as rows of unit squares: a 30 by 20 foot room holds 20 rows of 30 one-foot squares, which is 600 square feet. A square is the special case where length and width are equal, so its area is the side length squared.

When estimating materials, measure to the nearest practical unit and add an allowance for waste; flooring installers commonly add 5 to 10 percent for cuts. If a room is L-shaped, divide it into two rectangles, find each area, and add them. Remember that converting units squares the conversion factor: one square yard is 3 × 3 = 9 square feet, and one square meter is about 10.764 square feet.

• A = l × w

Triangle

A triangle is half of a parallelogram with the same base and height, so the familiar formula is one half times base times height. The height must be measured at right angles to the chosen base, which for an obtuse triangle can fall outside the shape.

When you know the three side lengths but not the height, as in this calculator, Heron's formula gives the area directly. First compute the semi-perimeter s, half the sum of the sides. The area is then the square root of s(s − a)(s − b)(s − c). For sides of 30, 45 and 50 feet, s = 62.5 and the area is √(62.5 × 32.5 × 17.5 × 12.5) ≈ 666.6 square feet.

Three lengths make a triangle only if each is shorter than the other two combined; otherwise the sides cannot meet. If two sides and the angle between them are known instead, the area is ½ab sin C. For a right triangle, the two legs serve as base and height, so the area is half their product.

• A = ½ × b × h
• A = √( s(s − a)(s − b)(s − c) ), where s = (a + b + c) ÷ 2

Trapezoid

A trapezoid (called a trapezium in British English) is a four-sided shape with one pair of parallel sides, the bases. Its area is the average of the two bases multiplied by the height, the perpendicular distance between them. One way to see why: place a second, upside-down copy next to the first, and together they form a parallelogram whose base is b₁ + b₂. The trapezoid is half of that.

With bases of 30 and 45 feet and a height of 20 feet, the average base is 37.5 feet and the area is 750 square feet. The slanted sides do not appear in the formula, so any trapezoid with the same bases and height has the same area. Surveyors use the same averaging idea, the trapezoidal rule, to estimate areas under curves and the cross-sections of earthworks.

• A = (b₁ + b₂) ÷ 2 × h

Circle

A circle's area is π times the radius squared, where the radius is the distance from the center to the edge. If you have the diameter instead, halve it first, or use π d² ÷ 4. A circle with a 30-foot radius covers about 2,827 square feet.

The formula can be pictured by cutting a circle into many thin wedges and arranging them alternately point-up and point-down. The result approaches a rectangle whose height is the radius and whose length is half the circumference, πr. Multiplying gives πr². Among all shapes with the same perimeter, the circle encloses the greatest area, which is one reason pipes, tanks and many containers are round.

• A = π × r²

Sector

A sector is a slice of a circle bounded by two radii and the arc between them, the shape of a pizza slice or a fan. Its area is the same fraction of the whole circle as its angle is of a full turn. With the angle θ in degrees, the area is θ ÷ 360 × πr². With θ in radians the formula is simpler, ½ θ r², because a full turn is 2π radians.

A 90° sector of a 30-foot circle is a quarter of the circle, about 707 square feet. The arc length of the same sector is θ r in radians, or about 47.1 feet. A lawn sprinkler that sweeps a 120° arc to a distance of 8 meters waters one third of a circle, about 67 square meters.

• A = θ ÷ 360° × π × r² (degrees)
• A = ½ × θ × r² (radians)

Ellipse

An ellipse is a stretched circle. It has two axes of symmetry: the long one, the major axis, and the short one, the minor axis. Half of each is called the semi-major axis a and the semi-minor axis b. The area is π times a times b, which reduces to πr² when a and b are equal.

Unlike its area, an ellipse's perimeter has no simple exact formula. A popular approximation by Ramanujan is π[3(a + b) − √((3a + b)(a + 3b))], which is accurate to a tiny fraction of a percent for most shapes. Ellipses appear in planetary orbits, which Kepler showed are ellipses with the Sun at one focus, as well as in oval tables, running tracks and the cross-section of a cylinder cut at an angle. An ellipse with semi-axes of 30 and 20 feet covers about 1,885 square feet.

• A = π × a × b

Parallelogram

A parallelogram has two pairs of parallel sides. Rectangles, rhombuses and squares are all special parallelograms. Its area is base times height, where the height is the perpendicular distance between the base and the opposite side, not the length of the slanted side.

The reason is easy to see: cut a right triangle off one end and slide it to the other, and the parallelogram becomes a rectangle with the same base and height. If you know two adjacent sides a and b and the angle θ between them, the height is b sin θ, so the area is ab sin θ. A parallelogram with a 30-foot base and a 20-foot height covers 600 square feet, the same as a 30 by 20 rectangle.

• A = b × h

Common Area Units

Equivalent values for common units of area.

Frequently asked questions