How to Find the Height of a Triangle

There are various methods to find the height of a triangle based on different given things and types of triangles. In one such case, to find the height of a triangle, use the formula for the area and rearrange the formula to solve for height i.e., Height = (2 × Area) ÷ base. If the area and base are known, divide twice the area by the base to get the height.

Methods to Find Height of Triangle

There are various methods to find height of triangle, for various different cases. Some of these methods are:

• Using Area of the Triangle
• Using the Trigonometry
• Using Pythagoras Theorem
• Height for Equilateral Triangles

Let's discuss these methods in detail.

Height Using Area of the Triangle

Area formula of triangle is the most straightforward way to find height when base and the area are known.

As we know, Area of triangle = 1/2 × base × height

Rearranging this formula, we can write

Height = (2 × Area)/Base

Example: If area of triangle is 50 square units and base is 10 units, then find its height.

Solution:

Given: Area = 50 Square Units
Base = 10 Units

⇒ Height = (2 × 50)/10 = 10 units

Thus, height of given triangle is 10 units.

Height Using the Trigonometry (Right-Angled Triangles)

If you have right triangle and know one of angles besides the 90° angle, you can use the trigonometric ratios to find height.

Formula:

sin (θ) = opposite/hypotenuse

Where:

• θ is known angle.
• Opposite side is height.
• Hypotenuse is longest side of the triangle.

Example: If hypotenuse is 15 units and angle is 30 degrees, then find it's height.

h = 15 × sin (30°) = 15 × 0.5 = 7.5 units [sin (30°) = 1/2 = 0.5)

Heigh of Right Angle Triangle (Using Pythagorean Theorem)

If you know two sides of the right triangle, you can find height using Pythagorean Theorem. This theorem states:

a² + b² = c²

Where a, b and c are lengths of sides of the triangle.

Example: If base is 6 units and hypotenuse is 10 units, height can be found by:

Solution:

6² + h² = 10²
⇒ 36 + h² = 100
⇒ h² = 64
⇒ h = 8 units

Thus, height of right angle triangle is 8 units.

Height for Equilateral Triangles

For equilateral triangles, height can be found using special formula derived from properties of equilateral triangles. If you know length of one side then the height h is:

h = √3/2 ✕ side

Example: If side of an equilateral triangle is 12 units then find its height.

Solution:

Given: side = 12 units
⇒ h = √3/2 × 12 = 6√3 = 10.39 units [As √3 ≈ 1.73]

Thus, height of an equilateral triangle with side 12 units is

Read More,

• Types of Triangle
• Equiangular Triangle
• Isosceles Triangle
• Scalene Triangle
• Properties of Triangles
• Acute Angled Triangle
• Right Angled Triangle
• Obtuse Angled Triangle