Angles in polygons – Year 9 maths worksheet with answers to download as PDF

The sum of the interior angles of an \(n\)-sided polygon is \((n-2)\times180^\circ\). Use this rule to complete the following, showing each step.

1. Find the angle sum of a pentagon (\(n=5\)).
2. Find the angle sum of a heptagon (\(n=7\)).
3. Find the angle sum of a 12-sided polygon.
4. A polygon has an angle sum of \(1620^\circ\). How many sides does it have?

Work out the size of the angle marked \(x\) in each polygon. Write the angle sum first.

1. A quadrilateral has angles \(85^\circ\), \(102^\circ\), \(96^\circ\) and \(x\).
2. A pentagon has angles \(110^\circ\), \(95^\circ\), \(120^\circ\), \(105^\circ\) and \(x\).
3. A hexagon has angles \(130^\circ\), \(125^\circ\), \(118^\circ\), \(140^\circ\), \(112^\circ\) and \(x\).
4. A pentagon has angles \(90^\circ\), \(90^\circ\), \(90^\circ\), \(x\) and \(x\).

A regular polygon has equal sides and equal angles. Find the size of one interior angle of each regular polygon.

1. Regular hexagon
2. Regular octagon
3. Regular decagon (10 sides)
4. Regular 15-sided polygon, giving your answer as a whole number

The exterior angles of any polygon add up to \(360^\circ\), and at each vertex interior angle + exterior angle \(=180^\circ\). For a regular polygon each exterior angle is \(360\div n\).

1. Find one exterior angle of a regular pentagon.
2. Find one exterior angle of a regular 9-sided polygon.
3. A regular polygon has an exterior angle of \(30^\circ\). How many sides has it?
4. Find the interior angle of the polygon in part (c).

Find the number of sides of each polygon. Show how you decide.

1. A polygon has an angle sum of \(1440^\circ\).
2. A regular polygon has interior angles of \(140^\circ\).
3. A regular polygon has exterior angles of \(15^\circ\).
4. A polygon has an angle sum of \(2340^\circ\).

Form and solve an equation in each part.

1. The angles of a quadrilateral are \(x\), \(x+20\), \(2x\) and \(2x+40\) degrees. Find \(x\) and write down all four angles.
2. The five equal interior angles of a regular pentagon are each \((3x+6)^\circ\). Find \(x\).
3. Four angles of a hexagon are each \(100^\circ\). The other two are equal. Find them.

Maya is tiling a kitchen floor. The shop sells regular tiles in these shapes: equilateral triangle, square, regular pentagon, regular hexagon and regular octagon. For tiles to fit together with no gaps, the angles that meet at each point must add up to \(360^\circ\).

1. Find the interior angle of each of the five tile shapes.
2. Show that three hexagonal tiles meeting at a point leave no gap.
3. Show that two octagons and one square also fit together at a point.
4. Explain why regular pentagons alone will not tile the floor.
5. Maya covers a \(3.6\) m by \(2.4\) m floor with square tiles of side 30 cm. How many tiles does she need?

A stop sign is a regular octagon with each side 30 cm long. The council is ordering signs for a new housing estate.

1. Find the perimeter of one sign.
2. Find the size of each interior angle and each exterior angle.
3. A workman walks once around the edge of the sign, turning at each corner. Through what total angle does he turn?
4. Each sign costs £42.50 and the council orders 14. Find the total cost.

The cells of a honeycomb are regular hexagons. A beekeeper measures one cell and finds that each side is 6 mm long.

1. Find the perimeter of one cell.
2. Find the interior angle of a hexagon.
3. Explain why three cells meet at each corner without any gaps.
4. Each frame holds 3 200 cells. A hive has 9 frames. How many cells are in the hive?
5. A beekeeper says that the cells could not be regular pentagons. Is she right? Explain.

A classic football is covered with 12 black regular pentagons and 20 white regular hexagons. At each corner one pentagon meets two hexagons.

1. Write down the interior angle of a regular pentagon and of a regular hexagon.
2. Add the three angles that meet at a corner.
3. Why is the answer not \(360^\circ\)? What does this tell you about the surface?
4. Each pentagon has 5 sides and each hexagon has 6. Count the sides on all the panels together, then use the fact that each edge is shared by two panels to find the number of edges.

A school garden has a flowerbed in the shape of a regular pentagon \(ABCDE\) with sides of 2 m. A fence will go round the edge and costs £12 per metre.

1. Find the cost of the fence.
2. Write down the interior angle at \(A\).
3. The gardener lays a path along the diagonal \(AC\). Triangle \(ABC\) is isosceles with \(AB=BC\). Find angle \(BAC\).
4. The interior angle \(BAE\) is \(108^\circ\). Find angle \(CAE\), between the path \(AC\) and the side \(AE\).

A robot is programmed to move forward 2 m, then turn left through \(40^\circ\), and to repeat this command until it returns to its start.

1. How many times does the robot repeat the command?
2. What regular polygon does the path make?
3. Find the total distance the robot travels.
4. The program is changed so that the robot turns through \(72^\circ\) instead. What shape is made and what is its interior angle?

A farmer's field has five corners. A surveyor measures four of the interior angles as \(98^\circ\), \(121^\circ\), \(104^\circ\) and \(117^\circ\). The five sides measure 40 m, 55 m, 62 m, 48 m and 51 m.

1. Work out the fifth interior angle.
2. Find the perimeter of the field.
3. Fencing costs £7.50 per metre. Find the cost of fencing the whole field.
4. The farmer leaves a 4 m gap for a gate. Find the new cost.

A church has a window in the shape of a regular 12-sided polygon. Twelve identical triangular panes meet at the centre.

1. Find each exterior angle of the window.
2. Find each interior angle of the window.
3. Each pane is an isosceles triangle with the apex at the centre. Find the apex angle.
4. Find the two equal base angles of each pane and check that they add up to the interior angle of the window.

A teacher gives the class a fact file about regular polygons. Some entries are missing.

1. Complete the table.
2. Which regular polygon has an interior angle exactly twice its exterior angle? Show your working.
3. Which regular polygon has an interior angle that is 5 times its exterior angle?
4. Explain why no regular polygon has an interior angle of \(170^\circ\) and an exterior angle of \(20^\circ\) at the same time as having fewer than 15 sides.

\(ABCDE\) is a regular pentagon. The diagonals \(AC\) and \(AD\) are drawn from \(A\).

1. Write down angle \(ABC\).
2. Triangle \(ABC\) is isosceles. Find angle \(BAC\) and angle \(BCA\).
3. Find angle \(CAD\) using the fact that angle \(BAE=108^\circ\) and that triangle \(ADE\) is congruent to triangle \(ABC\).
4. Show that the three angles at \(A\) in triangles \(ABC\), \(ACD\) and \(ADE\) add up to \(108^\circ\).
5. Triangle \(ACD\) is isosceles with \(AC=AD\). Find angle \(ACD\).

Four pupils make claims about polygons. Decide which are true. If a claim is false, explain why and give the correct fact.

1. Which pupils are correct?
2. Correct the wrong claims.
3. What is the largest whole-number exterior angle that a regular polygon can have, and which polygon is it?

A carpenter is building a garden gazebo with a regular octagonal frame. The rails meet at each corner with a mitre joint: each rail is cut at half the interior angle.

1. Find the interior angle of a regular octagon.
2. What angle does the carpenter cut on each rail?
3. Each side is 1.5 m. Find the perimeter of the frame.
4. Posts are placed every 0.5 m around the perimeter. How many posts are needed?
5. Each post costs £3.50. Find the total cost of the posts.

A designer is sketching a seven-sided coin. She starts with a regular heptagon with sides of 2.4 cm. Give angles to 1 decimal place where necessary.

1. Find the sum of the interior angles of a heptagon.
2. Find each interior angle and each exterior angle.
3. Find the perimeter of the sketch.
4. Seven of these coins are placed edge to edge around a central point so that one corner of each touches the centre. Explain why this cannot fit with no gaps.
5. How big is the gap?

A polygon has \(n\) sides. All its interior angles are \(150^\circ\) except one, which is \(90^\circ\).

1. Write down an expression for the sum of the interior angles in terms of \(n\).
2. Write down a second expression for the sum using the angle sizes.
3. Form and solve an equation to find \(n\).
4. Check your answer by finding the sum both ways.