Area Calculator

Area is a measure of the size or extent of a two-dimensional shape or surface in a plane. It can be imagined as the amount of paint needed to completely cover the surface. Just as length measures one-dimensional distance and volume measures three-dimensional space, area represents the two-dimensional equivalent. The standard unit of area in the International System of Units (SI) is the square meter (m²).

Below are formulas for calculating the area of some of the most common simple shapes, along with examples.

Rectangle

A rectangle is a four-sided polygon (quadrilateral) with four right angles. It is one of the most basic geometric shapes. To calculate its area, you need to know its length and width.

In a rectangle:

• Length usually refers to the longer sides.

• Width refers to the shorter sides.

• If all sides are equal, the rectangle becomes a special type called a square.

Area formula:

This simple multiplication gives you the total surface area inside the rectangle.

Area = Length × Width

The Farmer and His Daughter – Unsold Land

Imagine a farmer trying to sell a piece of perfectly rectangular land. To keep his cows from roaming freely, he fenced off the area and knew the exact length and width of the land. Since the farmer lives in the United States, he measured the land using feet instead of the metric system.

Interestingly, the foot was officially defined as exactly 0.3048 meters in 1959. Before that, measurements varied greatly over time and region, as units were often based on the human body, which led to inconsistencies.

Now, back to the farmer: his land measures 220 feet in length and 99 feet in width.

Using the area formula for a rectangle:

Area = 220 × 99 = 21,780 square feet

This gives the farmer a plot of land covering 21,780 square feet. For reference, this is exactly half an acre. (One acre is traditionally defined as the area of one chain by one furlong—an outdated system which illustrates why the metric SI system was introduced.)

Unfortunately, the farmer lives in a region dominated by foreign investors who are used to metric measurements and believe they’re getting fewer square feet for their money. Because of this misunderstanding, the land remains unsold to this day.

Triangle

There are several formulas to calculate the area of a triangle depending on what measurements are available. For detailed calculations and methods, it’s best to use a Triangle Calculator. One of the most commonly used methods, especially when all three sides of the triangle are known, is Heron’s formula.

Heron’s formula (also known as Hero’s formula) is named after Hero of Alexandria, a Greek mathematician and engineer who was regarded as one of the greatest experimenters of ancient times.

The formula is as follows:

If a triangle has sides of length a, b, and c, then:

The Farmer and His Daughter – Triangle Daze

After much effort and persistence, the farmer has finally managed to sell his 21,780 square foot plot of land. With some of the earnings, he decides to build a swimming pool for his family. However, he overlooks the fact that the annual maintenance cost of a private pool could easily fund countless visits to public pools or water parks for years.

To add to his challenge, the farmer’s 7-year-old daughter—newly fascinated by triangles after an imaginary trip to Egypt via Dora the Explorer—insists that the pool must be triangular. Not only that, but she demands that all measurements of the triangle include the number 7, as a tribute to her age and a way to “immortalize” this stage of her life through the design of the pool.

Being a loving and supportive father, the farmer agrees to her wish and begins planning the triangular pool.

Still uncomfortable with metric units, the farmer sticks with feet. After considering several options, he concludes that the only suitable shape is an equilateral triangle with each side measuring 77 feet. Any other dimensions would make the pool either too large or too small for the space available in his backyard.

To determine whether this triangle-shaped pool will fit, he calculates the area using the formula for the area of an equilateral triangle:

Trapezoid

A trapezoid has at least one pair of parallel sides (bases).
Formula:
Area = (b₁ + b₂) / 2 × height

The Farmer and His Daughter – Ramping Endeavors

At age 9, the daughter is into BMX biking. They build a ramp with a trapezoidal side: top base = 9 ft, bottom base = 29.528 ft, height = 9 ft.
Area = (9 + 29.528) / 2 × 9 = 173.376 sq ft

Circle

A circle is defined by all points equidistant from a center. The radius (r) is key to finding area.
Formula:
Area = πr²

The Farmer and His Daughter – Circle of Li(f)es

Now 15, the daughter pranks her father by making a crop circle (outer radius = 15 ft).
Area = π × 15² = 706.86 sq ft

“Circle investigators” overrun the farm, mistaking it for alien activity.

Sector

A sector is a slice of a circle, defined by a central angle (θ).
Formula (degrees):
Area = (θ / 360) × πr²

The Farmer and His Daughter – Sectioning Family

On her 16th birthday, her pie is partially eaten by their raccoon. Only 180° remains, split among 3 people. Each gets a 60° slice with a 16 in radius.
Area = (60 / 360) × π × 16² = 134.04 in² per person

Ellipse

An ellipse is a stretched circle with two radii: the semi-major axis (a) and semi-minor axis (b).
Formula:
Area = πab

The Farmer and His Daughter – Falling out of Orbit

At 18, her dreams of studying astrophysics fall short. She builds a model of Earth’s orbit in her room, using an ellipse with a = 20 ft and b = 18 ft.
Area = π × 18 × 20 ≈ 1,130.97 sq ft

Parallelogram

A parallelogram has opposite sides that are both parallel and equal in length.
Formula:
Area = base × height

The Farmer and His Daughter – Diamond in the Sky

At 20, she finds an octahedral diamond-shaped asteroid on the farm. One rhomboid face has dimensions 20 ft by 18 ft.
Area = 20 × 18 = 360 sq ft

She eventually sells it, giving up her ideals for a life of comfort.

Common Area Units