Angles in Polygons (WJEC GCSE Maths & Numeracy (Double Award): Foundation): Revision Note

Angles in polygons

What is a polygon?

• A polygon is a 2D shape with $n$ straight sides

  • A triangle is a polygon with 3 sides

  • A quadrilateral is a polygon with 4 sides

  • A pentagon is a polygon with 5 sides

• In a regular polygon all the sides are the same length and all the angles are the same size

  • A regular polygon with 3 sides is an equilateral triangle

  • A regular polygon with 4 sides is a square

What are the interior angles and the exterior angles of a polygon?

• Interior angles are the angles inside a polygon at the corners

• The exterior angle at a corner is the angle needed to make a straight line with the interior angles

  • It is not the angle that forms a full turn at the corner

• The interior angle and exterior angle add up to 180° at each corner

What is the sum of the interior angles in a polygon?

• To find the sum of the interior angles in a polygon of $n$ sides, use the rule

  • Sum of interior angles = $180°\times (n–2)$

    • This formula comes from the fact that $n$-sided polygons can be split into $n-2$ triangles

• Remember the sums for these polygons

  • The interior angles of a triangle add up to 180°

  • The interior angles of a quadrilateral add up to 360°

  • The interior angles of a pentagon add up to 540°

What is the sum of the exterior angles in a polygon?

• The exterior angles in any polygon always sum to 360°

How do I find the size of an interior or exterior angle in a regular polygon?

• To find the size of an interior angle in a regular polygon:

• Method 1
  Find the sum of the interior angles

  • For a pentagon: $180°\times \left(5-2\right)=540°$

  • Divide by the number of sides ($n$)

    • For a pentagon: $540°÷5=108°$

• Method 2
  Use the formula $\frac{180\left(n-2\right)}{n}$ which calculates the interior angle directly

• To find the size of an exterior angle in a regular polygon:

  • either divide 360° by the number of sides ($n$)

    • For a pentagon: $360°÷5=72°$

  • or use the formula $\frac{360}{n}$

• The interior angle and exterior angle add to 180°

  • Subtract the exterior angle from 180° to find the interior angle

  • Subtract the interior angle from 180° to find the exterior angle

How do I find a missing angle in a polygon?

• To find a missing angle in a polygon:

  • Use the formula $180°\times \left(n-2\right)$ to work out the sum of the interior angles

  • Subtract the other interior angles in the polygon

How do I find the number of sides in a regular polygon?

• If you are given the interior angle of a regular polygon

  • set the angle equal to $\frac{180\left(n-2\right)}{n}$

  • then solve the equation to find $n$