Height of Equilateral Triangle - Definition, Formula, Examples

What is the Height of an Equilateral Triangle?

• The height of an equilateral triangle, also called the altitude, is the perpendicular line drawn from the top vertex to the opposite side. It always makes a 90° angle with the base.

• The height of equilateral triangle also divides the triangle into two equal right-angled triangles. This property helps in deriving the formula and solving different types of questions.

Height of Equilateral Triangle Formula

To calculate the height of equilateral triangle, we use a standard formula. If the side of the triangle is represented by a, then the height of equilateral triangle formula is:

h = (√3/2) × a

Here:

• h = height of the triangle

• a = side length of the equilateral triangle

• √3 = square root of 3

You can also write it in decimal form as:

Height ≈ 0.866 × Side

This is the most commonly used height of equilateral triangle formula in geometry.

Read More - Right Angle Triangle: Definition, Properties, Formula & Examples

Formula to calculate Height of Equilateral Triangle 

We use a certain math method to find the height without a ruler. If we call the side of the triangle "a," the height (h) is provided by the formula h = \frac{\sqrt{3}}{2} \times a

Breaking Down the Formula:

• 'a': This is the length of the side.

• \sqrt{3}: This is the "square root of 3," which is approximately 1.732.

• 2: We divide the result by 2.

Fun Fact: If you don't like square roots yet, you can use the decimal version!

Height ≈ 0.866 × Side

How to Find the Height of Equilateral Triangle?

If you are wondering how to find the height of an equilateral triangle, follow these simple steps:

Step 1: Identify the side length

First, note the side length of the equilateral triangle. Suppose the side is 10 cm.

Step 2: Use the formula

Substitute the side length into the formula:

h = (√3/2) × 10

Step 3: Solve the equation

Using the decimal value:

h = 0.866 × 10 = 8.66 cm

So, the height of the triangle is 8.66 cm.

This is the easiest method for how to find the height of an equilateral triangle when the side is given.

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Why is the Height of Equilateral Triangle Important?

The height of equilateral triangle is important because it helps in calculating the area of the triangle. The area formula for any triangle is:

Area = (1/2) × Base × Height

So, once you know the height, you can easily calculate the area and solve many geometry questions.

Height of Equilateral Triangle if Side is Given

When the side length is given, finding the height of equilateral triangle is very simple. You just need to apply the height of equilateral triangle formula:

h = (√3/2) × a

Since all sides of an equilateral triangle are equal, knowing one side is enough to calculate the height.

For example, if the side is 12 units:

h = (√3/2) × 12 = 6√3 ≈ 10.39 units

This is one of the most direct methods used in questions on how to find the height of an equilateral triangle.

Height of Equilateral Triangle when Area is Given

Sometimes, the area of the triangle is given instead of the side. In that case, first use the area formula of an equilateral triangle:

Area = (√3/4) × a²

From this formula, find the side length a. Once you know the side, substitute it into the height of equilateral triangle formula:

h = (√3/2) × a

This method is useful when the side is not directly given but you still need to calculate the height of equilateral triangle.

For example, if the area is known, first calculate the side using the area formula, then use that side to find the height.

Height of Equilateral Triangle when Perimeter is Given

If the perimeter of the equilateral triangle is given, first find the side length. Since all three sides are equal:

Perimeter = 3a

So,

a = Perimeter ÷ 3

After finding the side, substitute it into the height of equilateral triangle formula:

h = (√3/2) × a

For example, if the perimeter is 21 units:

a = 21 ÷ 3 = 7 units

Now substitute:

h = (√3/2) × 7 ≈ 6.06 units

So, the height is approximately 6.06 units.

Read More - 30-60-90 Triangle - Sides, Formula, Examples

Height of an Equilateral Triangle Examples

Here are some height of an equilateral triangle examples to understand the concept better.

Example 1

An equilateral triangle has a side of 4 cm. Find its height.

Side = 4 cm
Formula = 0.866 × 4
Height = 3.464 cm

So, the height is 3.46 cm.

Example 2

An equilateral triangle has a side of 20 inches. Find its height.

Side = 20 inches
Formula = (√3/2) × 20
Height = 17.32 inches

So, the height is 17.32 inches.

These height of an equilateral triangle examples show that once you know the formula, solving such questions becomes easy.

How is the Height of Equilateral Triangle Formula Derived?

The height of equilateral triangle formula is derived using the Pythagorean Theorem.

When the height is drawn in an equilateral triangle, it divides the triangle into two equal right-angled triangles. In one of those right triangles:

• Hypotenuse = a

• Base = a/2

• Height = h

Using the Pythagorean Theorem:

a² = h² + (a/2)²

a² = h² + a²/4

h² = a² - a²/4

h² = 3a²/4

h = √(3a²/4)

h = (√3/2)a

This is how the height of equilateral triangle formula is obtained.

Difference between Side and Height of Equilateral Triangle

 Although both measurements belong to the same triangle, they are not the same.

This table shows that the height is always smaller than the side length.

Practice Questions for Equilateral Triangle

1. Find the height of equilateral triangle whose side is 8 cm.

2. Find the height of equilateral triangle whose perimeter is 18 units.

3. An equilateral triangle has an area of 16√3 square units. Find its side and height.

4. If the side of an equilateral triangle is 14 cm, what is its height?

Practising these questions will help you better understand what is the height of an equilateral triangle and how to find the height of an equilateral triangle.

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