Area Formulas



In this lesson, we give

  • a table of area formulas and perimeter formulas used to calculate the area and perimeter of two-dimensional geometrical shapes: square, rectangle. parallelogram, trapezoid (trapezium), triangle, rhombus, kite, regular polygon, circle, and ellipse.
  • a more detailed explanation (in text and video) of each area formula.
Related Topics: More Geometry Lessons

Shape

Area

Perimeter

Square

s2
where s = length of side

4s
where s = length of side

Rectangle

lw
where l = length, w = width

2(l + w)
where l = length, w = width

Parallelogram

lh
where l = length, h = perpendicular height

2(l + w)
where l = length, w = width

Trapezoid

area trapezoid
where a and b are the lengths of the parallel sides, h = perpendicular height

a + b + c + d
where a, b, c, and d are the lengths of the sides

Triangle

halfbh
where b = base, h = height

area using sine ab sin C
where a and b are lengths of sides and C is the included angle

For equilateral triangles:
area equilateral triangle
where s = length of side

Heron’s formula:

heorns formula
a, b, and c are the lengths of the sides and s is half the perimeter.

a + b + c
where a, b, and c are the lengths of the sides

Rhombus

area rhombus
where a and b are the lengths of the diagonals

a2 sin c
where a is the length of the side and c is any interior angle

4s
where s = length of side

Kite

area rhombus
where a and b are the lengths of the diagonals

2(l + w)
where l = length of short side, w = length of long side

Regular Polygon

area polygon
where p is the perimeter and is the apothem

ns
where n = number of sides, s = length of side

Circle

πr2
where r is the radius

r
where r is the radius

Ellipse

πab
where 2a and 2b are the lengths of the major and minor axes

 



Area of a Square

The area of a square is equal to the length of one side squared.

square

Area of a square = s2

How to find the area of a square?
The formula for the area of a square is s × s = s2, where s is the length of a side of the square.




Area of Rectangle

A rectangle is a 4-sided polygon where all four of its angles are right angles. Normally, the longer side is called the length and the shorter side is called the width. If all the sides are of equal length then it will be called a square.

Area of rectangle = length × width

A = lw



Area of Parallelogram

A parallelogram is a 4-sided polygon that has two sets of parallel sides. The opposite sides of a parallelogram are of equal length and the opposites angles are equal.

Area of parallelogram = base × perpendicular height

A = bh


How to find the area of a rectangle, triangle and parallelogram, using base and height? Always use the height that is perpendicular to the base. Do not use the slant height.



Area of Trapezoid / Trapezium

A trapeziod or trapezium is a 4-sided polygon that has at least one pair of parallel side. It is called a trapezoid in North America and a trapezium in Britain and other countries.


Area of trapezium = × (sum of two parallel sides) × height

A = × (a + b) × h

How to find the area of a trapezoid?
Remember to use the height that is perpendicular to the base.






Area of Triangle (given base and height)

A triangle is a 3-sided polygon.


Area of triangle = × Base × Height

A = bh

The following video shows an example of using the formula of half the product of the base and height to calculate the area of a triangle.

 

Area of Triangle (given 2 sides and an included angle)

Area of triangle = ab sin C

How to find the area of a triangle given side-angle-side (SAS)?
We can find the area of the triangle using a formula that uses the sine function.

 

Area of Triangle (given 3 sides)

Area of triangle =

This is also called the Heron's Formula

How to find the area of a triangle given the 3 sides? The following video ahows how to use the Heron's Formula.





Area of an Equilateral Triangle

To find the area of an equilateral triangle, we can use the following formula:

The area of an equilateral triangle (with all sides congruent) is equal to

area equilateral triangle

where s is the length of any side of the triangle

The following video gives the formula for the area of an equilateral triangle given the length of its side.

 

Area of Rhombus (given the length of the diagonals)

A rhombus is a 4-sided polygon that has 4 equal sides. The diagonals of a rhombus bisects each other at right angles.


Area of rhombus = product of diagonals

 

Area of Rhombus (given length of side and an angle)


Area of rhombus = a2 sin c where a is the length of the side and c is any interior angle.

(You can use any interior angle because either they are equal or they are supplementary and supplementary angles have the same sine.)

How to find the area of a rhombus, given the diagonals?







Area of Kite (given the length of the diagonals)

A kite is a 4-sided polygon that has two distinct pairs of adjacent sides that are congruent.. The diagonals of a kite bisects each other at right angles.

Area of a kite uses the same formula as the area of a rhombus

Area of kite = product of diagonals

 

Area of Regular Polygon

A regular polygon is a polygon where all the sides are the same length and all the angles are equal.

The apothem of a regular polygon is a line segment from the centre of the polygon to the midpoint of one of its sides.

Area of regular polygon = where p is the perimeter and a is the apothem.

 

How to find the area of a regular polygon, given the apothem and the length of the side?

 




Area of Circle

A circles is a shape consisting of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center.

Area of circle = × (radius)2

A = r2

How to find the area of a circle given the radius?

 

Area of Sector

A sector ia the portion of circle that is enclosed by two radii and an arc. The following video shows how to calculate the area of a sector in degrees.


 

Area of Ellipse

An ellipse is a curved line forming a closed loop, where the sum of the distances from two points (foci) to every point on the line is constant. It looks like a circle that has bee squashed into an oval.

If 2a and 2b are the lengths of the major and minor axes of the ellipse, then the area of the ellipse is πab.

How to find the area of an ellipse given the lengths of the major axis and minor axis? If 2a and 2b are the lengths of the major and minor axes of the ellipse, then the area of the ellipse is πab.




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