GCSE Angles in polygons Practice Questions

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

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

Substitute the number of sides into the rule.

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

State the sum of the interior angles.

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

Split the polygon into triangles from one vertex.

Drawing every diagonal from one vertex cuts an $n$-sided polygon into $n - 2$ triangles.

Multiply the number of triangles by the angle sum of a triangle.

Each triangle contributes $180^\circ$. Test it on a quadrilateral: $n = 4$ gives $2 \times 180^\circ = 360^\circ$, which is right, while $n \times 180^\circ$ would give $720^\circ$ and $(n - 2) \times 360^\circ$ would give $720^\circ$ as well. The last two options are one exterior and one interior angle of a REGULAR polygon, not a sum.

Select the correct option.

The interior angles of an $n$-sided polygon add up to $(n - 2) \times 180^\circ$.

Find the exterior angle.

The interior and exterior angles at a vertex add to $180^\circ$.

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

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

Divide 360 degrees by the exterior angle.

The polygon has $5$ sides.

Check the other options.

None of the other polygons has an interior angle of $108^\circ$.

Check the answer with the interior angle sum.

The interior angle sum divided by $5$ gives $108^\circ$, as required.

Select the correct polygon.

It is a regular pentagon.

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

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

Find the sum for polygon A.

Polygon $A$ splits into $7$ triangles.

Find the sum for polygon B.

Polygon $B$ splits into $10$ triangles.

Subtract to compare them.

Polygon $B$ has the larger sum, by $540^\circ$.

Explain the difference.

Each extra side adds one more triangle, so $3$ extra sides add $3$ lots of $180^\circ$.

Reject the equal option.

The sums are different, so they cannot be equal.

Reject the smaller option.

Polygon $B$ has more sides, so its interior angles must total MORE, not less.

Reject the other differences.

The gap is $540^\circ$, not $360^\circ$ and not $180^\circ$.

Check both sums are multiples of 180.

Both are whole numbers of triangles, as they must be.

Select the correct statement.

The sum for $B$ is $540^\circ$ greater than the sum for $A$.

Use the fact that an interior angle and its exterior angle lie on a straight line.

At every vertex the interior angle and the exterior angle make a straight line, so they add to $180^\circ$.

Write the interior angle in terms of the exterior angle.

The interior angle is $8$ times the exterior angle.

Substitute into the straight line fact.

This leaves an equation in $e$ alone.

Collect the terms.

$8e + e = 9e$.

Divide to find the exterior angle.

$180 \div 9 = 20$.

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

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

Divide 360 degrees by the exterior angle.

The polygon has $18$ sides.

Find the interior angle.

Each interior angle is $160^\circ$.

Check the multiple asked for.

The interior angle really is $8$ times the exterior angle.

Test the other options in turn.

For each of the other numbers of sides the interior angle is not $8$ times the exterior angle.

Check the exterior angles of the whole polygon add to 360 degrees.

The 18 equal exterior angles come to $360^\circ$, as they must.

Check the answer against the interior angle sum.

The 18 equal interior angles total $2880^\circ$, which is exactly $(18 - 2) \times 180^\circ$.

Check the answer is sensible.

Each interior angle of a convex polygon is less than $180^\circ$, and $160^\circ$ is.

Note the general result.

Whenever the interior angle is $8$ times the exterior angle, the polygon has $2 \times (8 + 1) = 18$ sides.

Select the correct statement.

The polygon has $18$ sides.