How to Calculate Area: Complete Geometry Guide

How to Calculate Area

Area measures how much surface a two-dimensional shape covers. Every formula below comes from one idea — counting unit squares — and most can be derived from the rectangle in a line or two.

Area is always in squared units: square metres, square feet, square centimetres. Reporting an area in metres is a units error, not a typo.

Rectangle and square

A = l × w \qquad A_square = s^2

This is the definition rather than a derivation. A 5 × 3 rectangle contains exactly 15 unit squares, which is what multiplication counts.

Every other formula on this page ultimately reduces to this one.

Parallelogram

A = b × h

Where b is the base and h the perpendicular height — not the slanted side.

The derivation is visual: cut a right triangle off one end of a parallelogram and slide it to the other end. You get a rectangle of the same base and height, and cutting-and-sliding does not change area.

This is why the slanted side never appears. A parallelogram that leans further has a longer side but the same area, provided base and perpendicular height are unchanged.

Triangle

A = (1)/(2)bh

Any triangle is exactly half a parallelogram: duplicate it, rotate the copy 180°, and the two fit together into a parallelogram with the same base and height.

Again h is the perpendicular height. For an obtuse triangle this falls outside the triangle itself, which is legitimate.

When you know three sides: Heron's formula

s = (a + b + c)/(2) \qquad A = √(s(s-a)(s-b)(s-c))

Where s is the semi-perimeter.

Example. Sides 5, 6, 7.

s = (5 + 6 + 7)/(2) = 9 A = √(9(9-5)(9-6)(9-7)) = √(9 × 4 × 3 × 2) = √(216) = 14.697

Heron's formula is invaluable in surveying and construction, where measuring three sides is easy and measuring perpendicular height is not.

When you know two sides and the angle between them

A = (1)/(2)ab\sin C

Example. Sides 8 and 11 with a 40° angle between them:

A = (1)/(2)(8)(11)\sin 40° = 44 × 0.6428 = 28.28

Trapezoid

A = ((a + b))/(2) × h

Where a and b are the parallel sides.

The derivation is worth seeing, because the formula looks arbitrary until you do. Take two identical trapezoids, rotate one 180°, and join them. The result is a parallelogram with base (a + b) and height h. That has area (a+b)h, and your trapezoid is half of it.

Read another way: (a+b)/2 is the average of the two parallel sides. A trapezoid has the area of a rectangle whose width is the average of its two ends. That reading makes the formula memorable rather than arbitrary.

Circle

A = π r^2

Here is why. Slice a circle into many thin wedges and interleave them alternately point-up and point-down. As the wedges get thinner the arrangement approaches a rectangle whose height is r and whose width is half the circumference, π r. So the area approaches π r × r = π r^2.

From the diameter, A = π d^2/4 — a form worth remembering, since pipes and circular ducts are always specified by diameter.

Sector and segment

A sector is a pizza slice; a segment is the region cut off by a straight chord.

A_sector = (θ)/(360°) × π r^2 \qquad (θ \text in degrees)

A_sector = (1)/(2)r^2θ \qquad (θ \text in radians)

A_segment = (1)/(2)r^2(θ - \sinθ) \qquad (θ \text in radians)

A sector is simply a fraction of the whole circle. A segment is that sector minus the triangle formed by the two radii and the chord.

Ellipse

A = π ab

Where a and b are the semi-major and semi-minor axes. When a = b = r this collapses to π r^2, as it must.

Regular polygons

A = (1)/(2) × \textperimeter × \textapothem

The apothem is the perpendicular distance from the centre to the midpoint of a side.

The derivation: split the polygon into n identical triangles from the centre. Each has base equal to a side and height equal to the apothem, so each has area (1)/(2)(\textside)(\textapothem). Summing over n sides gives (1)/(2)(\textperimeter)(\textapothem).

Notice this is the circle argument again in discrete form — as n grows, the perimeter approaches 2π r and the apothem approaches r, giving (1)/(2)(2π r)(r) = π r^2.

From the side length alone:

A = (n s^2)/(4\tan(π/n))

Example. A regular hexagon with 4 cm sides:

A = (6 × 16)/(4\tan(30°)) = (96)/(4 × 0.57735) = (96)/(2.3094) = 41.57 \text cm^2

Any polygon from coordinates: the shoelace formula

For a polygon with vertices (x_1,y_1), (x_2,y_2), \ldots, (x_n,y_n) listed in order:

A = (1)/(2)|Σ_i=1^n(x_i y_i+1 - x_i+1 y_i)|

Where the last vertex wraps back to the first.

It is called the shoelace formula because writing the coordinates in two columns and multiplying diagonally looks like lacing a shoe.

Example. A quadrilateral at (0,0), (4,0), (5,3), (1,4).

Cross products:

• (0)(0) - (4)(0) = 0
• (4)(3) - (5)(0) = 12
• (5)(4) - (1)(3) = 17
• (1)(0) - (0)(4) = 0

Sum = 29, so A = (1)/(2)|29| = 14.5

This handles any simple polygon, convex or concave, and is what GIS and CAD software uses for land parcels and floor plans. The vertices must be in order around the perimeter — a jumbled order gives a wrong answer silently.

Awkward shapes: decompose

Most real-world shapes are not textbook figures. Two reliable approaches:

Addition. Split into rectangles, triangles and circle parts, compute each, and sum. An L-shaped room is two rectangles.

Subtraction. Compute the bounding shape, then subtract the holes. A rectangular plate with a circular hole is lw - π r^2. A running track infield is a rectangle plus two semicircles — which together make one full circle.

Example: a room 6 m × 4 m with a 1.5 m × 1 m alcove and a circular pillar of radius 0.3 m.

A = (6 × 4) + (1.5 × 1) - π(0.3)^2 = 24 + 1.5 - 0.283 = 25.22 \text m^2

Why area scales with the square of length

Double every dimension of a shape and its area quadruples. Triple them and area grows nine-fold. Area scales with the square of the linear scale factor.

This has consequences well beyond geometry:

• A 16-inch pizza has not twice the food of a 12-inch. It has (16/12)^2 = 1.78 times as much — but it is often priced as if the relationship were linear, which is why larger pizzas are almost always better value.
• Doubling a room's dimensions quadruples the flooring needed.
• A 20-inch monitor and a 40-inch monitor differ by four times the screen area, not two.

The same principle in three dimensions gives volume scaling with the cube, which is why small animals lose body heat disproportionately fast — surface area grows as the square while mass grows as the cube.

Unit conversions

Because area is squared, conversion factors must be squared too. This is a frequent error.

One metre is 3.281 feet, so one square metre is 3.281^2 = 10.764 square feet — not 3.281. Likewise a square metre is 10,000 square centimetres, not 100.

Common mistakes

Using the slanted side instead of the perpendicular height. The most common error in parallelogram and triangle problems.

Forgetting to square the unit conversion. Converting m² to ft² by multiplying by 3.281 instead of 10.764 gives an answer three times too small.

Confusing radius and diameter. Using diameter in π r^2 gives four times the true area.

Mixing units within one calculation. Convert everything to a single unit first.

Listing shoelace vertices out of order. The formula assumes sequential vertices around the perimeter and fails silently otherwise.

Reporting area in linear units. If your answer is in metres rather than square metres, something went wrong.

Frequently asked questions

What is the difference between area and perimeter? Area is the surface covered; perimeter is the distance around the boundary. Shapes with equal perimeters can have very different areas — of all shapes with a given perimeter, the circle encloses the most.

Can area be negative? No. The shoelace formula can produce a negative intermediate value if vertices are listed clockwise, which is why the formula takes an absolute value. The sign indicates orientation, not negative area.

How do I find the area of an irregular plot of land? Get the corner coordinates and apply the shoelace formula, or decompose into triangles and use Heron's formula on each. Surveyors do both.

Why is a circle's area πr² rather than something simpler? Because π is defined as the ratio of circumference to diameter, and the wedge-rearrangement argument above shows the area works out to half the circumference times the radius. π is irrational, so no simpler closed form exists.

How do I calculate area under a curve? That requires integration. For a function f(x) between a and b, the area is ∫_a^b f(x)\,dx. Numerical approximations such as the trapezoidal rule work when no closed form exists.

What is surface area? The total area of all faces of a three-dimensional object — a cube of side s has surface area 6s^2. Distinct from volume, which measures the space enclosed.

Summary

Every area formula descends from counting unit squares in a rectangle. Triangles are half a parallelogram, parallelograms are rearranged rectangles, circles are the limit of many thin wedges, and regular polygons are fans of triangles.

For shapes that fit no formula, decompose into ones that do, or use the shoelace formula on coordinates. And remember that area scales with the square of length — the fact behind everything from pizza pricing to why elephants have thick legs.

Check your working with our Area Calculator, which covers every shape above.

Topics: area, geometry, mathematics, shapes, measurement, shoelace formula