Angles in polygons Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Angles in polygons questions. See exactly how to solve problems on interior angle sum, polygon rule, naming polygons, exterior angles.

interior angle sumpolygon rulenaming polygonsexterior anglessum of exterior angles is 360regular polygon
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
Work out the sum of the interior angles of a polygon with 55 sides.

Worked solution

  1. Write down the rule for the sum of the interior angles.

    S=(n−2)×180∘S = (n - 2) \times 180^\circ

    A polygon with nn sides can be split into n−2n - 2 triangles from one vertex, and each triangle contributes 180∘180^\circ.

  2. Substitute the number of sides into the rule.

    S=(5−2)×180∘S = (5 - 2) \times 180^\circ

    The polygon has 55 sides, so n=5n = 5.

  3. State the sum of the interior angles.

    S=540∘S = 540^\circ

    The interior angles add up to 540∘540^\circ.

Answer
S=540∘S = 540^\circ
Question 2
2 markseasy
Work out the sum of the interior angles of a polygon with 77 sides.

Worked solution

  1. Write down the rule for the sum of the interior angles.

    S=(n−2)×180∘S = (n - 2) \times 180^\circ

    A polygon with nn sides can be split into n−2n - 2 triangles from one vertex, and each triangle contributes 180∘180^\circ.

  2. Substitute the number of sides into the rule.

    S=(7−2)×180∘S = (7 - 2) \times 180^\circ

    The polygon has 77 sides, so n=7n = 7.

  3. State the sum of the interior angles.

    S=900∘S = 900^\circ

    The interior angles add up to 900∘900^\circ.

Answer
S=900∘S = 900^\circ
Question 3
1 markeasy
Work out the sum of the interior angles of a hexagon.

Worked solution

  1. Write down the rule for the sum of the interior angles.

    S=(n−2)×180∘S = (n - 2) \times 180^\circ

    A polygon with nn sides can be split into n−2n - 2 triangles from one vertex, and each triangle contributes 180∘180^\circ.

  2. Substitute the number of sides into the rule.

    S=(6−2)×180∘S = (6 - 2) \times 180^\circ

    The polygon has 66 sides, so n=6n = 6.

  3. State the sum of the interior angles.

    S=720∘S = 720^\circ

    The interior angles add up to 720∘720^\circ.

Answer
S=720∘S = 720^\circ
Question 4
2 markseasy
Work out the sum of the interior angles of an octagon.

Worked solution

  1. Write down the rule for the sum of the interior angles.

    S=(n−2)×180∘S = (n - 2) \times 180^\circ

    A polygon with nn sides can be split into n−2n - 2 triangles from one vertex, and each triangle contributes 180∘180^\circ.

  2. Substitute the number of sides into the rule.

    S=(8−2)×180∘S = (8 - 2) \times 180^\circ

    The polygon has 88 sides, so n=8n = 8.

  3. State the sum of the interior angles.

    S=1080∘S = 1080^\circ

    The interior angles add up to 1080∘1080^\circ.

Answer
S=1080∘S = 1080^\circ
Question 5
1 markeasy
A regular polygon has 88 sides. Work out the size of one exterior angle of the polygon.

Worked solution

  1. Recall that the exterior angles of any polygon add up to 360 degrees.

    sum of exterior angles=360∘\text{sum of exterior angles} = 360^\circ

    Going once round the outside of a polygon turns you through one complete turn, so the exterior angles always total 360∘360^\circ, whatever the number of sides.

  2. Divide 360 degrees by the number of sides.

    e=360∘8e = \frac{360^\circ}{8}

    The polygon is regular, so its 88 exterior angles are all equal.

  3. State the size of one exterior angle.

    e=45∘e = 45^\circ

    Each exterior angle is 45∘45^\circ.

Answer
e=45∘e = 45^\circ

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