# Equilateral Triangle

Here we will learn about equilateral triangles, including what an equilateral triangle is and how to solve problems involving the sides and angles of equilateral triangles.

There are also equilateral triangles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is an equilateral triangle?

An equilateral triangle is a type of triangle that has the following properties,

• All sides are equal length.

• All angles are equal size (60^{\circ}) .

An equilateral triangle is the simplest regular polygon.

All equilateral triangles are similar to (an enlargement of) each other as they have the same angles.

Equilateral triangles are only congruent if the lengths of the sides are the same.

An equilateral triangle is a special case of an acute triangle, where all the angles are acute angles.

Properties of equilateral triangles and other types of triangles are very important within geometry.

They are one of the most common shapes to recognise for angles in parallel lines, circle theorems, interior angles, trigonometry, Pythagoras’ theorem and many more. You therefore must be familiar with their individual properties.

### Symmetry

An equilateral triangle has three lines of symmetry. The lines go from one vertex to the middle of the opposite side.

The lines of symmetry are angle bisectors of the vertices.

The lines of symmetry are also perpendicular bisectors of the sides of the equilateral triangle.

An equilateral triangle has an order of rotational symmetry of 3 as it looks the same three times in a full turn.

In other words, if you rotate an equilateral triangle about its centre, it looks like the original every 120^{\circ} rotation clockwise.

### Area of an equilateral triangle

The area of an equilateral triangle can be found by using the formula A=\cfrac{1}{2} \, bh, where b is the base length and h is the perpendicular height of an equilateral triangle. Sometimes these values need to be calculated.

Remember that an area uses square units.

For example, calculate the area of the equilateral triangle ABC below, where M is the midpoint of the line AC , and angle AMB = 90^{\circ} .

Here, the base AC is double the length of AM and so AC = 8{~cm} .

Substituting the base length b = 8 and the perpendicular height h = 6.92 into the formula for the area of a triangle, we have

A=\cfrac{1}{2}\times{8}\times{6.92}= 27.68\text{~cm}^{2}.

The area of the triangle is 27.68 \, cm^2.

Step-by-step guide: Area of an equilateral triangle

### 3D shapes

A regular tetrahedron can be made by using 4 equilateral triangles.

An octahedron can be made by using 8 equilateral triangles.

An icosahedron can be made by using 20 equilateral triangles.

Step-by-step guide: 3D shape names

## How to solve problems involving equilateral triangles

In order to solve problems involving equilateral triangles:

1. Locate known angles and calculate any necessary unknown angles.
2. Locate known sides and calculate any necessary unknown side lengths.
3. Solve the problem using any necessary values.

### Related lessons on triangles

Equilateral triangle is part of our series of lessons to support revision on polygons and triangles. You may find it helpful to start with the main triangles lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

## Equilateral triangle examples

### Example 1: perimeter of an equilateral triangle

Find the perimeter of an equilateral triangle with side 7{~cm} .

1. Locate known angles and calculate any necessary unknown angles.

We do not need to calculate any angles as the perimeter is a measure of length and we can determine the side lengths of the triangle from the information provided.

2Locate known sides and calculate any necessary unknown side lengths.

As the triangle is an equilateral triangle, all three side lengths are the same and are equal to 7{~cm} .

3Solve the problem using any necessary values.

We can calculate the perimeter by multiplying one of the side lengths by 3 (or adding the three side lengths), 3\times 7=21 .

The perimeter of the equilateral triangle is 21{~cm} .

### Example 2: perimeter of a 2D compound shape

Find the perimeter of this trapezium made from congruent equilateral triangles with side 20{~cm} .

Locate known angles and calculate any necessary unknown angles.

Locate known sides and calculate any necessary unknown side lengths.

Solve the problem using any necessary values.

### Example 3: external angles of an equilateral triangle

Calculate the size of angle x .

Locate known angles and calculate any necessary unknown angles.

Locate known sides and calculate any necessary unknown side lengths.

Solve the problem using any necessary values.

### Example 4: similar equilateral triangles problem solving

An equilateral triangle ABC has a perimeter of 30{~cm} .

Points M and N are midpoints of the lines AB and AC respectively where MN is parallel to BC . Show that the line MN = 5{~cm} .

Locate known angles and calculate any necessary unknown angles.

Locate known sides and calculate any necessary unknown side lengths.

Solve the problem using any necessary values.

### Example 5: equilateral triangles and regular hexagons problem solving

Six congruent equilateral triangles are placed next to each other to form a hexagon ABCDEF . The side length of a triangle is 3{~cm} . Show that the hexagon is a regular hexagon.

Locate known angles and calculate any necessary unknown angles.

Locate known sides and calculate any necessary unknown side lengths.

Solve the problem using any necessary values.

### Example 6: bearings problem with equilateral triangles problem solving

Three planes set off from an airfield at the same height.

Plane A travels at a bearing of 020^{\circ} , Plane B travels at a bearing of 140^{\circ} .

Plane C travels at a bearing of 260^{\circ} . Each plane travels at the same speed.

After 30 minutes, each plane is exactly D{~km} from the airport.

A diagram of this is shown below.

Without calculating the exact value, show that the planes are all the same distance apart from each other

Locate known angles and calculate any necessary unknown angles.

Locate known sides and calculate any necessary unknown side lengths.

Solve the problem using any necessary values.

### Common misconceptions

• Angles in polygons
Make sure you know your angle properties. Getting these confused causes quite a few misconceptions.
• Angles in a triangle total 180^{\circ} .
• Angles in a quadrilateral total 360^{\circ} .

• Angle facts
Make sure you know your angle facts.
• Adjacent angles on a straight line add up to 180^{\circ} .
• Vertically opposite angles are equal.
• Alternate angles are equal.
• Corresponding angles are equal.

### Practice equilateral triangle questions

1) Which of these diagrams shows an equilateral triangle?

The marks on the sides of the triangle indicate which lengths are equal. We need the triangle which has the same mark on all three sides.

2) How many lines of symmetry does an equilateral triangle have?

1

2

3

6

Equilateral triangles have 3 lines of symmetry that bisect an angle and the opposite side to the vertex.

3) What is the interior angle of an equilateral triangle?

360^{\circ}

50^{\circ}

180^{\circ}

60^{\circ}

The sum of angles in any triangle is 180^{\circ} . To find one interior angle of this regular shape, we can simply divide 180 by 3 to get 180\div 3=60^{\circ}.

4) Calculate the perimeter of an equilateral triangle with sides 3.5{~cm} .

7{~cm}

10.5{~cm}

6.5{~cm}

14{~cm}

The perimeter of an equilateral triangle can be found by multiplying the length of one side of the equilateral triangle by 3 .

3.5\times 3=10.5

5) This parallelogram is made from 2 congruent equilateral triangles. Find angle BCD .

120^{\circ}

60^{\circ}

90^{\circ}

180^{\circ}

The angle can be found by multiplying the size of one angle in an equilateral triangle by 2 .

2\times 60= 120^{\circ}

6) This 2D shape has been made with 4 congruent triangles with side 12{~cm} . Find the perimeter of the shape.

144{~cm}

120{~cm}

72{~cm}

84{~cm}

The perimeter of the shape can be found by multiplying the length of one side of the equilateral triangle by 6 .

12\times 6=72

### Equilateral triangle GCSE questions

1. Shape A is a triangle.

Circle the type of triangle it is.

Isosceles Scalene Equilateral Right-angled

(1 mark)

Equilateral

(1)

2. This parallelogram has been made using 4 congruent equilateral triangles with side length 5{~cm} .

Calculate the perimeter of the shape.

(3 marks)

6 \times 5

(1)

30

(1)

cm

(1)

3. ABC is a straight line.

CBD = 2x+50^{\circ}

Calculate the value of x .

(4 marks)

ABD= 60^{\circ}

(1)

CBD= 120 ^{\circ}

(1)

2x+50=120 or 2x=70

(1)

x=35

(1)

## Learning checklist

You have now learned how to:

• Recognise an equilateral triangle
• Use the properties of an equilateral triangle to solve problems

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