GCSE · Angles & polygons

Interior & exterior angles of polygonsExam-style questions

Why the exterior angles of any polygon add up to 360°, where (n − 2) × 180° comes from, how regular polygons follow instantly, and the reliable route from a given interior angle back to the number of sides.

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20 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
Here is a pattern made from $5$ regular octagons and $4$ squares.The octagons and the squares all have sides of length $a$ .

Find an expression, in terms of $a$ , for the total area of the pattern.

Give your answer in the form $p(7 + 5\sqrt{2})a^2$ where $p$ is an integer.

You must show all your working.

(5 marks)
Level 5 · 5 marksTry it
2
The diagram shows a quarter circle, centre $O$ , radius $r$ cm. $B$ and $C$ are points on the arc $AD$ so that $AB = BC = CD$

By considering the length of the arc $AD$ , show that

$\pi > 6\sqrt{2 - \sqrt{3}}$

(4 marks)
Level 5 · 4 marksTry it
3
$P$ is a regular polygon with 10 sides.

The interior angle of a regular polygon $Q$ is $12°$ greater than the interior angle of $P$ .

Work out the number of sides of $Q$ .
Level 5 · 4 marksTry it
4
The diagram shows a regular pentagon $ABCDE$ . The diagonals $AC$ and $BD$ cross at the point $F$ .Diagram NOT accurately drawn

Work out the size of the angle marked $x$ . You must give reasons for each stage of your working.
Level 5 · 4 marksTry it