Exterior Angles in Geometry: Meaning, Formulas & Examples

The concept of Exteror Angle is a crucial part of geometry, especially when studying polygons and triangles. Understanding exterior angles is fundamental for solving geometry problems in school, competitive exams like JEE and NEET, and for recognizing shapes and patterns in everyday life.

Understanding Exterior Angles

An exterior angle is formed when one side of a polygon or triangle is extended outwards. The angle between this extension and the adjacent side is known as the exterior angle. In any polygon, at each vertex, you can draw an exterior angle. Understanding the relationship between exterior and interior angles helps students confidently solve various geometry problems.

For triangles, the exterior angle is always equal to the sum of the two non-adjacent interior angles. For polygons, the sum of all exterior angles, one at each vertex, is always 360°, no matter how many sides the polygon has.

For example, in a triangle ABC, extend side BC to D. The angle CBD is the exterior angle at vertex B. This concept is widely used in geometry, construction, and even in designing everyday objects.

Formulas and Properties of Exterior Angles

Here are the key formulas associated with exterior angles:

• Exterior Angle Sum Formula (Polygons): The sum of the exterior angles of any polygon (one angle at each vertex) is always 360°.
• Exterior Angle of a Regular Polygon: Each exterior angle = \( \frac{360^\circ}{n} \), where n = number of sides.
• Exterior Angle Theorem (Triangles): The measure of an exterior angle of a triangle equals the sum of the measures of the two opposite (remote) interior angles.

These formulas make it easier to calculate missing angles, solve geometry questions, and check your work during exams.

Worked Examples

Example 1: Exterior Angle of a Triangle

In triangle ABC, side BC is extended to point D. If angle A = 40° and angle C = 60°, find the exterior angle at vertex B (angle CBD).

1. Apply the Exterior Angle Theorem:
   Exterior angle at B = angle A + angle C = 40° + 60° = 100°

Example 2: Exterior Angle of a Regular Hexagon

Find each exterior angle of a regular hexagon.

1. There are 6 sides, so n = 6.
2. Each exterior angle = \( \frac{360^\circ}{6} = 60^\circ \)

Example 3: Sum of All Exterior Angles in a Pentagon

No matter how many sides, sum of all exterior angles = 360°.

Example 4: Solving for Missing Angle

In a triangle, two interior angles are 55° and 35°. Find the exterior angle at the vertex of the remaining angle.

1. Sum the two remote interior angles: 55° + 35° = 90°
2. The exterior angle at the third vertex is 90°.

Practice Problems

• Find the exterior angle of a regular octagon.
• If two interior angles of a triangle are 50° and 45°, what is the exterior angle at the third vertex?
• Calculate the sum of exterior angles in a 12-sided polygon.
• If the exterior angle of a regular polygon is 30°, how many sides does the polygon have?
• In quadrilateral ABCD, find an exterior angle at A if the adjacent interior angle is 110°.
• Draw a triangle and extend one side. Measure the exterior angle and compare it to the sum of the two non-adjacent interior angles.

Common Mistakes to Avoid

• Confusing interior and exterior angles—remember, the exterior angle forms outside the polygon at the extended side.
• Forgetting that the sum of all exterior angles, one at each vertex, is always 360°, no matter the number of sides.
• Mixing up the formula for regular polygons—always divide 360° by the number of sides for one exterior angle.
• Ignoring the fact that in a triangle, the exterior angle equals the sum of the two non-adjacent interior angles, not just the adjacent one.

Real-World Applications

Exterior angles play a practical role in construction, navigation, and design. For instance, architects use the sum of exterior angles when designing polygons in floor layouts, engineers calculate angles when creating road turns or bridges, and even artists use exterior and interior angles to ensure accurate patterns in designs. By understanding these concepts, students can see the connection of geometry to real life.

At Vedantu, we simplify complex topics like exterior angles to help students build confidence and solve questions accurately. To study related concepts, explore our pages on Interior Angles of a Polygon and Angles and its Types.

In this topic, we have explored the meaning, properties, and formulas related to exterior angles, with detailed examples and practice problems. This knowledge is essential for tackling geometry questions in both school exams and competitive exams, building a strong foundation in mathematics.