Area is the measure of the two-dimensional space inside a flat shape, expressed in square units such as square meters (m^2) or square feet (ft^2). Whether you are buying flooring, painting a wall, or sizing a garden bed, you need to calculate area. This guide covers formulas and worked examples for the eight most common shapes.
Quick Answer
Multiply the relevant dimensions according to each shape's formula. For rectangles, multiply length by width. For circles, multiply pi by the radius squared. For triangles, multiply half the base by the height. A quick-reference table appears at the end of this article.
Use our area calculator to compute any shape instantly.
Rectangle
Formula: A = length x width
A rectangle has four right angles and opposite sides of equal length. Area equals the product of its two distinct side lengths.
Worked example: A patio measures 12 m by 8 m.
A = 12 x 8 = 96 m^2
This is the most frequently used area formula in everyday life -- room dimensions, land plots, and screens are all rectangular.
Square
Formula: A = side^2
A square is a special rectangle where all four sides are equal. The formula simplifies to the side length multiplied by itself.
Worked example: A floor tile has a side length of 5 m.
A = 5^2 = 25 m^2
Because a square is a rectangle, A = length x width also works (5 x 5 = 25), but side^2 is the conventional shorthand.
Triangle
Formula: A = (1/2) x base x height
The height must be perpendicular to the base -- not a slant side. For more methods (Heron's formula, coordinate geometry), see our triangle area guide.
Worked example: A triangular sail has a base of 10 m and a height of 6 m.
A = (1/2) x 10 x 6 = 30 m^2
Use the triangle calculator when you know three sides but not the height.
Circle
Formula: A = pi x r^2
Here r is the radius (half the diameter). The constant pi is approximately 3.14159.
Worked example: A circular garden has a radius of 7 m.
A = pi x 7^2 = pi x 49 = 153.94 m^2
If you know the diameter d instead, use A = pi x (d/2)^2 or equivalently A = (pi/4) x d^2. The circle calculator handles both inputs.
Trapezoid
Formula: A = (1/2)(a + b) x h
A trapezoid (trapezium in British English) has exactly one pair of parallel sides, called bases a and b. The height h is the perpendicular distance between them.
Worked example: A trapezoidal lot has parallel sides of 8 m and 12 m, with a height of 5 m.
A = (1/2)(8 + 12) x 5 = (1/2)(20) x 5 = 50 m^2
The formula works because a trapezoid's area equals the average of the two bases multiplied by the height. Use the trapezoid calculator for side lengths and diagonals.
Parallelogram
Formula: A = base x height
A parallelogram has two pairs of parallel sides. Unlike a rectangle, the height is not the slant side -- it is the perpendicular distance between the base and the opposite side.
Worked example: A parallelogram-shaped tile has a base of 9 m and a perpendicular height of 4 m.
A = 9 x 4 = 36 m^2
Note that the slant side length is irrelevant to the area calculation. Only the perpendicular height matters.
Ellipse
Formula: A = pi x a x b
An ellipse is a stretched circle with two radii: a semi-major axis (a) and a semi-minor axis (b). When a = b, the ellipse becomes a circle and the formula reduces to pi x r^2.
Worked example: An elliptical pond has a semi-major axis of 6 m and a semi-minor axis of 4 m.
A = pi x 6 x 4 = pi x 24 = 75.40 m^2
Ellipses appear in architecture (oval rooms), astronomy (planetary orbits), and landscape design.
Regular Hexagon
Formula: A = (3 x sqrt(3) / 2) x s^2
A regular hexagon has six equal sides of length s and six equal angles of 120 degrees. The coefficient 3 x sqrt(3) / 2 is approximately 2.5981.
Worked example: A hexagonal paving stone has a side length of 5 m.
A = (3 x sqrt(3) / 2) x 5^2 = 2.5981 x 25 = 64.95 m^2
A regular hexagon can be divided into six equilateral triangles, each with area (sqrt(3)/4) x s^2. Multiplying by six gives the full formula. The hexagon calculator computes area, perimeter, and diagonals.
Quick-Reference Table
| Shape | Formula | Key measurements |
|---|---|---|
| Rectangle | A = length x width | Length, width |
| Square | A = side^2 | Side length |
| Triangle | A = (1/2) x base x height | Base, perpendicular height |
| Circle | A = pi x r^2 | Radius |
| Trapezoid | A = (1/2)(a + b) x h | Two parallel sides, height |
| Parallelogram | A = base x height | Base, perpendicular height |
| Ellipse | A = pi x a x b | Semi-major axis, semi-minor axis |
| Regular hexagon | A = (3 sqrt(3) / 2) x s^2 | Side length |
Unit Conversions
Area units scale by the square of the linear conversion factor. Here are the most useful conversions:
- 1 m^2 = 10.764 ft^2
- 1 ft^2 = 0.0929 m^2
- 1 acre = 43,560 ft^2 = 4,046.86 m^2
- 1 hectare = 10,000 m^2 = 2.471 acres
- 1 km^2 = 1,000,000 m^2 = 247.105 acres
- 1 cm^2 = 0.155 in^2
- 1 in^2 = 6.452 cm^2
Source: NIST Guide for the Use of the International System of Units.
Real-World Applications
Flooring. Measure the room's length and width, multiply to get the area, then add 5-10% for waste and cuts. A room 4.5 m x 6 m needs at least 27 m^2 of material, so order about 29-30 m^2.
Painting walls. Each wall is a rectangle. Multiply width by height, subtract window and door areas, and divide the total by the paint's coverage rate (typically 8-12 m^2 per liter for one coat).
Landscaping. Calculate the area of garden beds, lawns, and patios to determine how much soil, seed, or mulch to buy. Mulch is typically sold by volume, so multiply the area by the desired depth.
Fabric cutting. Garment patterns require specific fabric areas. Knowing the area of each pattern piece (often approximated as rectangles and triangles) prevents waste and ensures you purchase enough material.
Irregular Shapes
Not every shape matches a standard formula. Two common strategies:
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Decomposition. Break the shape into rectangles, triangles, and other known shapes. Calculate each area and sum them. For example, an L-shaped room is two rectangles.
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Grid method. Overlay a square grid, count fully covered squares, estimate partial squares, and multiply by the grid square area. This works well for organic shapes like ponds or garden beds.
For precise irregular-area calculations using coordinates, the shoelace formula computes the area of any polygon given its vertices.
Frequently Asked Questions
How do I find the area of an irregular shape?
Break the irregular shape into smaller regular shapes (rectangles, triangles, etc.), calculate each area separately, and add them together. For curved irregular shapes, use numerical methods such as the trapezoidal rule or count grid squares.
What is the difference between area and perimeter?
Area measures the two-dimensional space inside a shape in square units. Perimeter measures the total length of the boundary around a shape in linear units. A rectangle with length 10 m and width 5 m has an area of 50 m^2 and a perimeter of 30 m.
How many square feet are in an acre?
One acre equals 43,560 square feet. This is equivalent to about 4,047 square meters or roughly 0.4047 hectares.
How do I calculate the area of a room?
Measure the length and width of the room in feet or meters, then multiply them together. For L-shaped rooms, divide the space into two rectangles, calculate each area, and add the results.
What are square units?
Square units express area measurement. They represent a square whose sides each measure one unit. Common square units include square meters (m^2), square feet (ft^2), square centimeters (cm^2), and square inches (in^2).
How do I convert between square meters and square feet?
Multiply square meters by 10.764 to get square feet. Multiply square feet by 0.0929 to get square meters. For example, 50 m^2 equals 538.2 ft^2.
Why is the area of a circle pi times r squared?
The formula A = pi x r^2 comes from dividing a circle into infinitely many thin triangular sectors. When rearranged, these sectors approximate a rectangle with height r and width pi x r, giving an area of pi x r^2. This result was first rigorously proved by Archimedes.
Can area be negative?
No. Area is always zero or positive. In signed-area calculations used in coordinate geometry (such as the shoelace formula), a negative result indicates clockwise vertex ordering, but the actual area is the absolute value.
How do I calculate the area of a shape with a hole in it?
Calculate the area of the outer shape, then subtract the area of the hole. For example, a circular ring (annulus) with outer radius 10 cm and inner radius 6 cm has area = pi x 10^2 - pi x 6^2 = 314.16 - 113.10 = 201.06 cm^2.
What is the easiest way to estimate area without exact measurements?
Use the grid method: overlay a grid of known square size on the shape, count all fully covered squares, then estimate partial squares as half. Multiply the count by the area of each grid square. This technique is taught in elementary geometry and used in fields like biology for estimating leaf area.