Welcome to our Area of Equilateral Triangle page.

Here you will find support and worked examples to show you how to find the area of an equilateral triangle.

We also have some worksheets to help you practice this skill.

These sheets are aimed at children of a 6th grade and up.

Quicklinks to ...

- Area is the amount of space that is inside a shape.
- Because it is an amount of space, it has to be measured in squares.
- If the shape is measured in cm, then the area would be measured in square cm or cm
^{2} - If the shape is measured in inches, then the area would be measured in square inches or in
^{2}

An equilateral triangle is a triangle with all sides equal and all angles equal to 60°.

Formula for the Area of an Equilateral Triangle

\[A = { \sqrt 3 \over 4} s^2 \]

where A is the area, and s is the length of the side of the triangle.

To find the area of a any triangle, you simply need to multiply the base of the triangle by the perpendicular height and halve the answer.

In this case, we will call h the perpendicular height and s the length of one side of the triangle.

You will notice that the perpendicular height bisects the base of the triangle, splitting it in half and creating two 'half' right triangles.

Let's look closer at one of these right triangles.

As this is a right triangle, we can use Pythagoras' Theorem to work out the value of the height.

Pythagoras' Theorem states that the hypotenuse squared = base squared + height squared.

So substituting the values of the triangle into the equation gives us:

\[ s^2 = ({s \over 2})^2 + h^2 \]

\[ s^2 = {s^2 \over 4} + h^2 \]

\[ h^2 = s^2 - {s^2 \over 4} \]

\[ h^2 = {3 \over 4} s^2 \]

\[ h = \sqrt { {3 \over 4} s^2} \]

\[ h = { \sqrt 3 \over 2} s \]

Now we have found the height, we can use this to get the area of the triangle.

The area of a triangle is equal to half of the base multiplied by the perpendicular height.

\[ A = {1 \over 2} bh \]

where b is the length of the base, and h is the perpendicular height.

Substituting our values for the height and the base, this gives us:

\[ A = {1 \over 2} s ({ \sqrt 3 \over 2} s) \]

\[ A = { \sqrt 3 \over 4} s^2 \]

Example 1) Find the area of the equilateral triangle below. Give your answer to 1dp.

In this example the length of the side is 5 cm.

\[ So \; A = { \sqrt 3 \over 4} s^2 \]

\[ A = { \sqrt 3 \over 4} \times 5^2 \]

\[ A = 25 { \sqrt 3 \over 4} = 10.8 cm^2 \; to \; 1dp \]

Example 2) What is the area of the yellow triangle below. Give your answer to 2dp.

This is an equilateral triangle with side length 8 inches.

\[ So \; A = { \sqrt 3 \over 4} s^2 \]

\[ A = { \sqrt 3 \over 4} 8^2 \]

\[ A = 64 { \sqrt 3 \over 4} = 16 \sqrt 3 = 27.71 in^2 \; to \; 2dp \]

Example 3) Find the area shaded blue in the triangle below. Give your answer to 1dp.

This is an equilateral triangle with side length 3.5 m.

\[ So \; A = { \sqrt 3 \over 4} s^2 \]

\[ A = { \sqrt 3 \over 4} (3.5)^2 \]

\[ A = 12.25 { \sqrt 3 \over 4} = 5.3 m^2 \; to \; 1dp \]

Take a look at some more of our worksheets similar to these.

Our printable support page has the formulas for the area of common 2d shapes.

Here is our selection of free printable area worksheets for 3rd and 4th grade.

The sheets are all graded in order from easiest to hardest.

Using these sheets will help your child to:

- work out the areas of a range of rectangles;
- find the area of rectilinear shapes.

Here is our selection of free printable perimeter worksheets for 3rd and 4th grade.

The sheets are all graded in order from easiest to hardest.

Using these sheets will help your child to:

- work out the perimeter of a range of rectangles;
- find the perimeter of rectilinear shapes.

All the math practice worksheets in this section support Elementary Math Benchmarks.

Here is our range of volume worksheets.

Using these sheets will help your child to:

- know what volume is and how to find it;
- find the volume of shapes by counting cubes;
- find the volume of rectangular prisms;
- solving basic problems involving volume

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How to Print or Save these sheets

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