Mathematical Calculators

Area Calculator

Our intuitive tool lets you choose from different shapes and calculates their area in the blink of an eye.

area calculator


Table of contents

What is area in math? Definition of area in mathematics
How cam you calculate area?
Square area formula
Rectangle area formula
Triangle area formula
Circle area formula
Sector area formula
Ellipse area formula
Trapezoid area formula
Parallelogram area formula
Rhombus area formula
Kite area formula
Pentagon area formula
Hexagon area formula
Octagon area formula
Annulus area formula
Quadrilateral area formula
Regular polygon area formula
Which quadrilateral has the largest area?
What shape has the largest area given perimeter?
How can I calculate the area of an irregular shape?
How can I calculate the area under a curve?

What is area in math? Definition of area in mathematics

Area is the size of a surface. In other words, it may be defined as the space occupied by a flat shape. To understand the concept, it's usually helpful to think about the area as the amount of paint necessary to cover the surface. This makes sense because area is the amount of substance or material that is occupied by a figure or object.
There are a number of useful formulas for calculating the area of simple shapes. In this section, you'll find not only the well-known formulas for triangles, rectangles, and circles but also other shapes, such as parallelograms, kites, or annuli. By the end of the section, you'll have a comprehensive understanding of how to calculate the area of any shape.

How cam you calculate area?

Formulaic content can be tricky to write, but we've got you covered. In this section, you'll learn all about the formulas for the sixteen shapes featured in our area calculator. We'll list the equations only - their images, explanations and derivations may be found in the separate paragraphs below (and also in tools dedicated to each specific shape). So whether you need to know the volume of a cone or the surface area of a trapezoid, we've got you covered!

Square area formula

ₛqᵤₐᵣₑ ₐᵣₑₐ ₌ ₐ × ₐ ₌ ₐ²
a: square side

Rectangle area formula

ᵣₑ꜀ₜₐₙ₉ₗₑ ₐᵣₑₐ ₌ ₐ × ₆
a and b: being the sides of the rectangle

Triangle area formula

When base and height are given
ₜᵣᵢₐₙ₉ₗₑ ₐᵣₑₐ ₌ ₆ × ₕ / ₂
When two sides and the angle between them are
ₜᵣᵢₐₙ₉ₗₑ ₐᵣₑₐ ₌ ₀.₅ × ₐ × ₆ × ₛᵢₙ₍γ₎
When two angles and the side between them is given
ₜᵣᵢₐₙ₉ₗₑ ₐᵣₑₐ ₌ ₀.₂₅ × √₍ ₍ₐ ₊ ₆ ₊ ꜀₎ × ₍₋ₐ ₊ ₆ ₊ ꜀₎ × ₍ₐ ₋ ₆ ₊ ꜀₎ × ₍ₐ ₊ ₆ ₋ ꜀₎ ₎
When three sides are given
ₜᵣᵢₐₙ₉ₗₑ ₐᵣₑₐ ₌ ₐ² × ₛᵢₙ₍β₎ × ₛᵢₙ₍γ₎ / ₍₂ × ₛᵢₙ₍β ₊ γ₎₎

Circle area formula

Cᵢᵣ꜀ₗₑ ₐᵣₑₐ ₌ πᵣ²
r: it is the radius of the circle
Cᵢᵣ꜀ₗₑ ₐᵣₑₐ ₌ πᵣ² ₌ π × ₍ₔ / ₂₎²
Cᵢᵣ꜀ₗₑ ₐᵣₑₐ ₌ ꜀² / ₄π

Sector area formula

α / ₃₆₀° ₌ ₛₑ꜀ₜₒᵣ ₐᵣₑₐ / Cᵢᵣ꜀ₗₑ ₐᵣₑₐ
₃₆₀° ₌ ₂π
α / ₂π ₌ ₛₑ꜀ₜₒᵣ ₐᵣₑₐ / πᵣ²
ₛₑ꜀ₜₒᵣ ₐᵣₑₐ ₌ ᵣ² × α / ₂

Ellipse area formula

ₑₗₗᵢₚₛᵢₛ ₐᵣₑₐ ₌ π × ₐ × ₆

Trapezoid area formula

ₜᵣₐₚₑ₂ₒᵢₔ ₐᵣₑₐ ₌ ₍ₐ ₊ ₆₎ × ₕ / ₂
a and b: being the lengths of the parallel sides
h: being the height
ₜᵣₐₚₑ₂ₒᵢₔ ₐᵣₑₐ ₌ ₘ × ₕ
m: being the arithmetic mean of the lengths of the two parallel sides of the trapezoid.

Parallelogram area formula

base and height
ₚₐᵣₐₗₗₑₗₒ₉ᵣₐₘ ₐᵣₑₐ ₌ ₐ × ₕ
sides and angle between them
ₚₐᵣₐₗₗₑₗₒ₉ᵣₐₘ ₐᵣₑₐ ₌ ₐ × ₆ × ₛᵢₙ₍α₎
diagonals and an angle between them
ₚₐᵣₐₗₗₑₗₒ₉ᵣₐₘ ₐᵣₑₐ ₌ ₑ × բ × ₛᵢₙ₍θ₎

Rhombus area formula

side and height
ᵣₕₒₘ₆ᵤₛ ₐᵣₑₐ ₌ ₐ × ₕ
ᵣₕₒₘ₆ᵤₛ ₐᵣₑₐ ₌ ₍ₑ × բ₎ / ₂
side and any angle
ᵣₕₒₘ₆ᵤₛ ₐᵣₑₐ ₌ ₐ² × ₛᵢₙ₍α₎

Kite area formula

when the kite diagonals are given
ₖᵢₜₑ ₐᵣₑₐ ₌ ₍ₑ × բ₎ / ₂
when two non-congruent side lengths and the angle between those two sides are given
ₖᵢₜₑ ₐᵣₑₐ ₌ ₐ × ₆ × ₛᵢₙ₍α₎

Pentagon area formula

ₚₑₙₜₐ₉ₒₙ ₐᵣₑₐ ₌ ₐ² × √₍₂₅ ₊ ₁₀√₅₎ / ₄
a is the side of a regular pentagon

Hexagon area formula

ₕₑₓₐ₉ₒₙ ₐᵣₑₐ ₌ ₃/₂ × √₃ × ₐ²

Octagon area formula

ₒ꜀ₜₐ₉ₒₙ ₐᵣₑₐ ₌ ₂ × ₍₁ ₊ √₂₎ * ₐ²
Octagon Area = perimeter × apothem / 2
h = (1 + √2) × a / 4
Octagon Area = perimeter * apothem / 2 = (8 × a × (1 + √2) × a / 4) / 2 = 2 × (1 + √2) × a²

Annulus area formula

Annulus area = πᵣ² ₋ πᵣ² ₌ π₍ᵣ² ₋ ᵣ²₎

Quadrilateral area formula

Quadrilateral Area ₌ ₑ × բ × ₛᵢₙ₍α₎
e and f are the diagonals of the quadrilateral

Regular polygon area formula

Regular Polygon Area ₌ ₙ × ₐ² × ꜀ₒₜ₍π/ₙ₎ / ₄
n is the number of sides the polygon has

Which quadrilateral has the largest area?

For a given perimeter, the quadrilateral with the maximum area is always a square. This follows from geometry - a perfect square has four equal side lengths, and a quadrilateral with four equal sides has the maximum area possible.

What shape has the largest area given perimeter?

Given a given perimeter, the closed figure with the maximum area is a circle.

How can I calculate the area of an irregular shape?

Before you can calculate the area of an irregular shape, you need to break it down into smaller shapes that you can calculate the area easily. This can be done by dividing the shape into triangles, rectangles, trapezoids, etc. Then, you can calculate the area of each of these subshapes. Finally, you can sum up the areas of all subshapes to get the final result.

How can I calculate the area under a curve?

To find the area under a curve, you need to compute the definite integral of the function describing the curve between the two points that correspond to the endpoints of the interval in question. This can be done by finding the height of the curve between those points or by using another method if you know the specific function that you're approximating.

Parmis Kazemi
Article author
Parmis Kazemi
Parmis is a content creator who has a passion for writing and creating new things. She is also highly interested in tech and enjoys learning new things.

Area Calculator English
Published: Tue Aug 30 2022
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