Polygons and quadrilaterals · Angles, triangles & polygons

A polygon is a flat shape with straight sides. From one corner you can draw lines to every other corner and cut an $n$ -sided polygon into $n - 2$ triangles. Each triangle has $180^\circ$ , so the interior angles of any $n$ -sided polygon add up to

(n - 2) \times 180^\circ

At each corner, producing a side makes an exterior angle, the turn you make there as you walk round. Walking all the way round turns you through one full turn, so the exterior angles of any polygon add up to $360^\circ$ . At every corner, interior angle + exterior angle $= 180^\circ$ .

Interior anglesA pentagon is 3 triangles: (5 − 2) × 180° = 540°

Exterior anglesThey add up to 360° for every polygon

Try it

Angles of a polygonChoose the number of sides

3triangles, n − 23 × 180° = 540°interior angles add to540° ÷ 5 = 108°each angle (regular)

sides, n = 5

A pentagon (5 sides) splits into 3 triangles from one corner, each with 180°. So its interior angles add to (5 − 2) × 180° = 540°. It's regular, so each angle is 540° ÷ 5 = 108°.

Change the number of sides and count the triangles: always two fewer than the sides. Switch on “Irregular”: the corners move and the angles change, but their sums don’t.

Regular polygons

In a regular polygon all the sides and all the angles are equal. So each exterior angle is $\dfrac{360^\circ}{n}$ , and each interior angle is $180^\circ$ minus that. Turned round, this finds the number of sides:

n = \frac{360^\circ}{\text{exterior angle}}

The quickest route is nearly always through the exterior angle.

Worked example · WAEC 2023

WAEC 2023 · Paper 2 · Q11 (a)

Two regular polygons $P$ and $Q$ are such that the number of sides of $P$ is twice the number of sides of $Q$ . The difference between the exterior angles of $Q$ and $P$ is $45^\circ$ . Find the number of sides of $P$ .

Name the sides
Let $Q$ have $n$ sides; then $P$ has $2n$ .
Think first. If Q has n sides, how many has P?
The exterior angles
Exterior angle of $Q = \dfrac{360}{n}$ ; exterior angle of $P = \dfrac{360}{2n} = \dfrac{180}{n}$ .
Think first. Write each exterior angle in terms of n.
Use the difference
$Q$ has fewer sides, so its exterior angle is bigger: $\dfrac{360}{n} - \dfrac{180}{n} = 45$ , so $\dfrac{180}{n} = 45$ and $n = 4$ .
Think first. Which one is bigger? Write the equation.
Answer the question
$P$ has $2n = 8$ sides.
Think first. Is the question asking for P or Q?

Worked example · WAEC 2022

WAEC 2022 · Paper 2 · Q11 (a)

The exterior angles of a polygon are $42^\circ, 38^\circ, 57^\circ, x^\circ, (x + y)^\circ, (2x - 15)^\circ$ and $(3x - y)^\circ$ . If $x$ is $7^\circ$ less than $y$ , find the values of $x$ and $y$ .

Add the exterior angles
$42 + 38 + 57 + x + (x + y) + (2x - 15) + (3x - y) = 360$ . The $y$ terms cancel: $122 + 7x = 360$ .
Think first. What do the exterior angles of any polygon add up to?
Find x, then y
$7x = 238$ , so $x = 34$ . Then $y = x + 7 = 41$ .
Think first. Solve for x, then use 'x is 7° less than y'.
Check
The angles are $42^\circ, 38^\circ, 57^\circ, 34^\circ, 75^\circ, 53^\circ, 61^\circ$ , which add to $360^\circ$ ✓.
Think first. Do all seven angles add to 360°?

Special quadrilaterals

A quadrilateral’s angles add up to $360^\circ$ . The special ones have extra properties that questions use:

ParallelogramOpposite sides equal and parallel; opposite angles equal; diagonals bisect each other

RectangleA parallelogram with four right angles; diagonals equal

RhombusA parallelogram with four equal sides; diagonals cross at right angles and bisect its angles

TrapeziumOne pair of parallel sides; co-interior angles between them add to 180°

A square has all of these properties. In a parallelogram, next-door angles are co-interior, so they add up to $180^\circ$ .

Your turn

WAEC 2015 · Paper 2 · Q3 (a)

(a)
The ratio of the interior angle to the exterior angle of a regular polygon is $5 : 2$ . Find the number of sides of the polygon.

Worked solution (try it first)

(a)

At each corner, interior angle + exterior angle $= 180^\circ$ .
Sharing $180^\circ$ in the ratio $5 : 2$ (7 parts), the exterior angle is $\frac27 \times 180^\circ = \frac{360^\circ}{7}$ .
The number of sides is $360^\circ$ divided by the exterior angle: $360 \div \frac{360}{7} = 7$ sides.

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