Angles in Polygons (Cambridge International IGCSE Maths)

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 = $180^{\circ}\times(n--2)$

    • 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: $180^{\circ} \times ( 5 - 2 ) = 540^{\circ}$

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

    • For a pentagon: $540^{\circ}\div5=108^{\circ}$

• Method 2 Use the formula $\frac{180 ( n - 2 )}{n}$ 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: $360^{\circ}\div5=72^{\circ}$

  • 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 $180^{\circ} \times ( n - 2 )$ 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 $\frac{180 ( n - 2 )}{n}$
  • then solve the equation to find $n$

Exam Hint Make sure you identify whether you are dealing with a regular or irregular polygon before you start a question.

Finding the sum of the interior angles using $180 \times ( n - 2 )$ can often be a good starting point for finding missing angles.

Worked Example The exterior angle of a regular polygon is 45°.

Write down the name of the polygon.

Answer:

The formula for the exterior angle of a regular polygon is $\mathrm{Exterior} \mathrm{Angle}=\frac{360^{\circ}}{n}$* *

Substitute the 45 for the exterior angle

$45^{\circ}=\frac{360^{\circ}}{n}$

Solve by rearranging

$\begin{gathered} n = \frac{360}{45} \\ n = 8 \end{gathered}$

Write down the name of a shape with 8 sides

Final answer: Regular octagon