Area of equilateral triangle

Here you will learn about how to find the area of equilateral triangles, including what formula to use and why it works.

Students will first learn about the area of equilateral triangles as part of geometry in 6th grade.

What is the area of an equilateral triangle?

The area of an equilateral triangle is the amount of the space inside an equilateral triangle.

An equilateral triangle is a triangle where all angles and sides are equal – making it a regular polygon.

Since the angles in a triangle add up to 180^{\circ}, each of the interior angles (internal angles) in an equilateral triangle is 60^{\circ}.

In order to find the area of an equilateral triangle, start with the area of a rectangle.

The area of a rectangle is calculated by multiplying the \text {base } \times \text { height }.

Within certain rectangles, there is an equilateral triangle. It can be shown by drawing lines from a midpoint of the rectangle to the corners.

Creating the equilateral triangle (in pink) also creates two congruent right triangles (in green).

If you rotate the two right triangles and place them back to back, they form an equilateral triangle that is the same area as the blue equilateral triangle.

The area of the rectangle is the same as two congruent equilateral triangles. This means the area of each equilateral triangle is exactly half the area of the rectangle. 

The general formula to find the area of any triangle is:

\text { Area of a triangle }=\cfrac{\text{ base } \times \text{ height }}{2}

This can also be written as the following formula:

A=\cfrac{1}{2} \, b h

where b is the base length and h is the height of the triangle.

The space inside of shapes is measured in square units, so the final answer must be given in \text{units}^2. For example, \mathrm{cm}^2, \mathrm{~m}^2, or \mathrm{~mm}^2.

For example,

What is the area of the equilateral triangle?

A=\cfrac{1}{2} \, b h

The base is 13 and the height is 11.3.

A few things to note:

• The height of an equilateral triangle is always a perpendicular line that starts at a vertex and intersects with the opposite side of the triangle at a 90 degree angle.
• The other side lengths of the triangle may be given, but they are extra information.

\begin{aligned} & A=\frac{1}{2} \times 13 \times 11.3 \\\\ & A=6.5 \times 11.3 \\\\ & A=73.45 \end{aligned}

The area of the equilateral triangle is 73.45 \, cm^2.

What is the area of an equilateral triangle?

Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – Geometry (6.G.A.1)
  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

How to find the area of equilateral triangles

In order to find the area of equilateral triangles:

1. Identify the base and perpendicular height of the triangle.
2. Write the area formula.
3. Substitute known values into the area formula.
4. Solve the equation.
5. Write the answer, including the units.

Area of equilateral triangles examples

Example 1: given base length and height

Find the area of the triangle below:

1. Identify the base and perpendicular height of the triangle.

b = 3 \, cm

h = 2.6 \, cm

2Write the area formula.

A=\cfrac{1}{2} \, b h

3Substitute known values into the area formula.

\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (3)(2.6) \end{aligned}

4Solve the equation.

\begin{aligned} A &=\cfrac{1}{2} \, b h \\\\ &=\cfrac{1}{2} \, (3)(2.6)\\\\ &=3.9 \, cm^2 \end{aligned}

5Write the answer, including the units.

A=3.9 \mathrm{~cm}^2

Remember: Your final answer must be in units squared.

Example 2: given base length and height

Find the area of the triangle below:

Identify the base and perpendicular height of the triangle.

Rotating the triangle makes it easier to identify the base and height.

b = 5 \, m

h = 430 \, cm

Notice, there are 2 different units here. You must convert them to a common unit:

430 \, cm = 4.3 \, m

b = 5 \, m

h = 4.3 \, m

Example 3: given base length and height – fractions

Find the area of the triangle below:

Identify the base and perpendicular height of the triangle.

Even though the base length is not labeled, since all sides of the equilateral triangle are the same the base will also be 7 \, \cfrac{3}{4} \mathrm{~mm}.

\mathrm{b}=7 \, \cfrac{3}{4} \mathrm{~mm}

\mathrm{h}=6 \, \cfrac{2}{3} \mathrm{~mm}

Example 4: word problem

Shown below is a triangular shaped field. Each chicken needs 5 \mathrm{~m}^2 to graze. How many chickens can fit into this field?

Identify the base and perpendicular height of the triangle.

b = 45 \, m

h = 39 \, m

Example 5: compound shapes

Below is the layout for a new garden that needs to be filled with soil. Each bag of soil costs \$4.50 and covers an area of 2.5 \, m^2. How much would it cost to cover the entire garden with soil?

Identify the base and perpendicular height of the triangle.

Split the garden into 2 shapes; a rectangle and an equilateral triangle.

For the triangle:

b = 7.5 \, m

h = 6.5 \, m

Example 6: calculating base length

Triangle ABC is an equilateral triangle with an area of 173 \, cm^2. The height of the triangle is 17.3 \, cm. Find the length of the base of the triangle.

Solve the equation.

173=8.65 \times b

To solve, find what times 8.65 is equal to 173. You can use the inverse operation of division to solve. Divide both sides by 8.65.

\begin{aligned} &173 \quad \quad = \quad \; 8.65 \times b \\ & \div 8.65 \quad \quad \quad \div 8.65 \end{aligned}

20=b

Solve the original equation to check your work.

Teaching tips for area of equilateral triangles

• Give students the opportunity to measure the length of the sides and the height of equilateral triangles. Once they have found one height measurement, encourage them to rotate the triangle to find the other two to see that the height of the equilateral triangle is the same no matter which side is used as a base. Understanding these attributes of an equilateral triangle helps students think flexibly when different measurements are given.

• Introduce how to find the area of an equilateral triangle by connecting the base of the triangle to the rectangle formed around it. This helps students understand why the area of a triangle is half the area of a rectangle, giving meaning to the formula and making it easier to remember.

• Worksheets are a useful tool, but be sure to choose ones that include a variety of types of triangles and scenarios. Triangles where all measurements are given, triangles where no measurements are given (either on grid paper or give students tools to measure), word problems that include a picture of the triangle, word problems that do not, etc.

Easy mistakes to make

• Thinking the height of an equilateral triangle is the same as the base
  Even though the length of all sides are equal, the height will not be the same as the sides – which means the base and the height will never be equivalent.

• Forgetting to write the units
  It is common to forget the units for area in the final answer. When calculating area, your answer must always include the square units.

Related area lessons

• Area of a triangle
• Area of isosceles triangle
• Area of obtuse triangle
• Area of a right triangle
• Area of composite shapes
• Area of a hexagon
• Area of a pentagon
• Area of irregular shapes

How to find the area of an equilateral triangle in later grades

You will not always be given the height of the equilateral triangle, but if you know the side length there are ways to calculate the height or area using other methods.

To find the area of an equilateral triangle when only the side length has been given, we will need to calculate the height of the triangle using Pythagoras’ Theorem or Trigonometry (SOHCAHTOA).

Finding the height using Pythagoras’ Theorem

Let the side length of the equilateral triangle represent the base, b.

Drawing a perpendicular line from the top vertex to the base forms the height, h, and also forms a right angled triangle with hypotenuse b and short sides h and \cfrac{1}{2} \, b.

\begin{aligned} h &=\sqrt{{{b}^{2}}-{{\left( \frac{1}{2}b \right)}^{2}}} \\\\ & =\sqrt{\frac{3}{4}{{b}^{2}}} \\\\ & =\frac{\sqrt{3}}{2}b \end{aligned}

Finding the height using SOHCAHTOA

Let the side length of the equilateral triangle represent the base, b.

Drawing a perpendicular line from the top vertex to the base forms the height, h, and also forms a right angled triangle with hypotenuse b and angle 60^{\circ}.

\sin \left( 60 \right)=\cfrac{h}{b}

So, h=b\sin \left( 60 \right)

We can also find the area of the equilateral triangle using the area of a triangle formula

\text{Area }=\cfrac{1}{2} \, ab\sin C

The formula would give the area as

\text{Area }=\cfrac{1}{2} \, {{b}^{2}}\sin \left( 60 \right)

Using the exact value of \sin \left( 60 \right)=\cfrac{\sqrt{3}}{2}, this gives us

Area of an equilateral triangle =\cfrac{\sqrt{3}}{4}{{b}^{2}}

The next lessons are

• Area of a quadrilateral
• Perimeter
• 3D shapes
• Volume

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