How to Find the Angle of a Triangle?

A triangle is a shape with three sides and three angles. One important rule in math is that the sum of all angles in a triangle is always 180°. This rule is called the triangle angle formula and it helps us quickly find the angle when other angles are known.

Moreover, the angle formula is used to calculate missing angles in all types of triangles like right-angle triangles, isosceles triangles, equilateral triangles, and scalene triangles. To further understand how to calculate angles using triangle angle formulas, keep reading.

Angle Formula Overview

As we know, an angle is formed when two lines or rays meet at a common point. This common point is called the vertex, and the two lines are called the arms of the angle. The angle formula helps us to find the angle in different triangles. In simple words, an angle shows how much one line has turned to meet another line. We use the angle calculation formula to measure this turn.

In geometry, learning how to calculate angles is very useful because angles tell us the shape and size of objects. The angle of triangle formula is one of the most common formulas that students use to solve many geometrical questions.

How to Calculate Angle in a Triangle

As we learned above, in every triangle, the sum of all three angles is always 180°. This is called the triangle angle formula. We can use this rule to find the angle that is missing in the triangle. Here are the steps on how to calculate angle in a triangle:

• Step 1: Add the given angles.

• Step 2: Subtract the sum from 180° to find the missing angle.

For example, if one angle of a triangle is 50° and the other angle is 60°, let’s find the third angle.

• Step 1: Add the two known angles: 50° + 60° = 110°

• Step 2: Subtract this sum from 180°: 180° – 110° = 70°

• So, the missing angle is 70°.

This is one of the easiest ways to calculate an angle in a triangle using the angle of triangle formula.

Triangle Angle Formula

In every triangle, the total of all three angles is always 180°. This is a basic rule in geometry and is known as the triangle angle formula. For a triangle with three angles, the triangle angle formula is:

• Triangle Angle Formula: ∠a + ∠b + ∠c = 180°

This is also called the Triangle Angle Sum Theorem. We can use this angle formula to find the angle that is missing when the other two angles are given.

Read More: Composite Numbers

Angle Calculation Formulas

When we learn about triangles, we use different angle calculation formulas to find the angle when some sides or angles are already given. These formulas help us easily calculate angles in a triangle. Let's look at them one by one:

1. Triangle Angle Sum Formula

In every triangle, the sum of all angles is always 180°. So, if we know two angles, we can easily find the third angle.

• Angle Formula: Angle 1 + Angle 2 + Angle 3 = 180°

This is the most common angle of triangle formula used to solve simple triangle-related questions in mathematics.

2. Trigonometric Ratios (For Right-Angled Triangles)

In a right-angled triangle, we can use the trigonometric formulas to find the angle.

Formula:

• sin θ = Opposite side ÷ Hypotenuse

• cos θ = Adjacent side ÷ Hypotenuse

• tan θ = Opposite side ÷ Adjacent side

We use these angle calculation formulas when we know the lengths of the sides.

3. Law of Sines (For Any Triangle)

The Law of Sines helps to find the angle when we know:

• Two sides and a non-included angle, or

• Two angles and a non-included side.

Angle Formula: a / sin A = b / sin B = c / sin C

Here, a, b, and c are the sides and A, B, and C are the opposite angles.

4. Law of Cosines (For Any Triangle)

The Law of Cosines helps to find the angle when we know:

• All three sides, or

• Two sides and the included angle.

Formula:

• a² = b² + c² – 2bc × cos A

• b² = c² + a² – 2ca × cos B

• c² = a² + b² – 2ab × cos C

This angle formula is useful for tough triangle questions.

So, these are the main angle calculation formulas that help us understand how to calculate angles in a triangle step by step.

Read More: Acute Angle

Finding the Angle of a Triangle: Solved Examples

Example 1: Two angles of a triangle are 40° and 70°. Find the third angle.

Solution: We know from the triangle angle formula that the sum of all angles in a triangle is 180°.

• So, the sum of the given angles = 40° + 70° = 110°.

• Third angle = 180° – 110° = 70°.

Example 2: One angle of a triangle is 90° and another is 45°. Find the missing angle.

Solution: In a triangle, the total of all angles is always 180°.

• Sum of the given angles = 90° + 45° = 135°

• So, the missing third angle = 180° – 135° = 45°.

Example 3: In a right-angled triangle, the base is 9 cm and the hypotenuse is 18 cm. Find the base angle.

Solution: We can find the angle using the cosine angle formula.

Given: Base = 9 cm and Hypotenuse = 18 cm.

We use the angle calculation formula: cos θ = base ÷ hypotenuse

• cos θ = 9 ÷ 18 = 1 ÷ 2

• Now, find θ: θ = cos⁻¹(1 ÷ 2) = 60°

So, the base angle is 60°.

Read More: Measurement

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