Chapter 4 of 8

All 8 chapters
  1. 1Angles on a line and round a point, readunderstood
  2. 2Angles and parallel lines, readunderstood
  3. 3Triangles, readunderstood
  4. 4Angles in polygons, readunderstood
  5. 5Drawing with ruler and compasses, readunderstood
  6. 6Line symmetry and reflection, readunderstood
  7. 7Turning symmetry, readunderstood
  8. 8Solids, nets and views, readunderstood

M3.5.4 Chapter 4

Angles in polygons

Do the four angles of any four sided shape add up to the same number?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Show that the angles of a quadrilateral add up to 360°
  • Work out the angle sum of any polygon
  • Find each angle of a regular hexagon

A quadrilateral is a shape with four straight sides. A square, a rectangle and a kite are quadrilaterals. So are many shapes with no name.

A line that joins two corners that are not next to each other is a diagonal. A diagonal cuts a quadrilateral into two triangles.

Draw it first

Cut a quadrilateral into two triangles

  1. A four sided shape with corners that measure 80°, 100°, 70° and 110°. A dashed diagonal from the bottom left corner to the top right corner cuts it into two triangles, numbered 1 and 2. Under the picture it says 2 triangles × 180° = 360°.

    1 Draw a quadrilateral.

  2. A four sided shape with corners that measure 80°, 100°, 70° and 110°. A dashed diagonal from the bottom left corner to the top right corner cuts it into two triangles, numbered 1 and 2. Under the picture it says 2 triangles × 180° = 360°.12

    2 Draw one diagonal. It makes two triangles.

  3. A four sided shape with corners that measure 80°, 100°, 70° and 110°. A dashed diagonal from the bottom left corner to the top right corner cuts it into two triangles, numbered 1 and 2. Under the picture it says 2 triangles × 180° = 360°.80°100°70°110°2 triangles × 180° = 360°

    3 Each triangle has 180°, so the four angles add to 2 × 180° = 360°.

So the angles of every quadrilateral add to 360°. You can use this to find a missing angle.

Worked example

Three angles of a quadrilateral are 80°, 100° and 70°. What is the fourth angle, x?

  1. The four angles of a quadrilateral add to 360°.
  2. The three known angles add to 80° + 100° + 70° = 250°.
  3. x = 360° − 250° = 110°.

Answer: 110°

A four sided shape. Going round from the bottom left, the angles are 80°, 100°, 70° and x. The angle x is 110°.80°100°70°x
The four angles add to 80° + 100° + 70° + 110° = 360°.

A polygon is a flat shape with straight sides. Choose one corner. Draw diagonals from it to every corner it can reach. You get triangles.

A pentagon with five sides. Dashed diagonals from the top corner cut it into three triangles, numbered 1, 2 and 3.123
A pentagon has 5 sides and makes 3 triangles.
A hexagon with six sides. Dashed diagonals from one corner cut it into four triangles, numbered 1, 2, 3 and 4.1234
A hexagon has 6 sides and makes 4 triangles.

The number of triangles is always 2 less than the number of sides. So a polygon with n sides makes n − 2 triangles. Its angle sum is (n − 2) × 180°.

A table with three columns: sides, triangles and sum. It has rows for 3, 4, 5 and 6 sides. They make 1, 2, 3 and 4 triangles. The angle sums are 180°, 360°, 540° and 720°.sidestrianglessum31180°42360°53540°64720°
Every extra side adds one more triangle, and 180° more.

Worked example

What is the angle sum of a polygon with 8 sides?

  1. An 8 sided polygon makes 8 − 2 = 6 triangles.
  2. Each triangle has an angle sum of 180°.
  3. The angle sum is 6 × 180° = 1 080°.

Answer: 1 080°

An octagon with eight sides. Dashed diagonals from one corner cut it into six triangles, numbered 1 to 6.123456
An octagon makes 6 triangles. 6 × 180° = 1 080°.

A regular polygon has equal sides and equal angles. To find one angle, share the angle sum between all the corners.

Worked example

What is each angle of a regular hexagon?

  1. A hexagon has 6 sides.
  2. It makes 6 − 2 = 4 triangles.
  3. The angle sum is 4 × 180° = 720°.
  4. A regular hexagon has 6 equal angles.
  5. Each angle is 720° ÷ 6 = 120°.

Answer: 120°

A regular hexagon. All six sides are equal and each of the six angles is 120°.120°120°120°120°120°120°
Six angles of 120° make 720°.

See it three ways

With things

Draw a four sided shape on paper. Draw one diagonal. Cut along it. You have two triangles.

As a picture

A four sided shape cut by a dashed diagonal into two triangles, numbered 1 and 2.12

In symbols

2 triangles × 180° = 360°. So the four angles add to 360°.

Try it

Tear the corners off a quadrilateral

You need: a sheet of paper, a pencil.

  1. Draw a four sided shape on the paper. Make it any shape you like.
  2. Mark each corner with a small curved line.
  3. Tear off the four corners.
  4. Put the corners together with their points touching.

What you will see: The four corners fit round a point. That is a full turn, 360°. This works for every four sided shape.

If you cannot try it, this is what happens.
Four torn corners round a point. They are 80°, 100°, 70° and 110°. Together they fill a full turn.80°100°70°110°

A wrong idea many people have

The angles of a quadrilateral add to 180°, like a triangle.

Why it feels right: A triangle is the first shape we learn, and its angle sum is 180°.

The test that breaks it: Look at a rectangle. Its four angles are all 90°. Four angles of 90° add to 360°, not 180°.

What is true: A quadrilateral is two triangles joined along a diagonal. So its angle sum is 2 × 180° = 360°.

Key words

quadrilateral
a shape with four straight sides. Its angles add to 360°.
diagonal
a line that joins two corners that are not next to each other.
polygon
a flat shape with straight sides. With n sides, its angles add to (n − 2) × 180°.
regular polygon
a polygon with all sides equal and all angles equal.

Back to your guess

The answer is A: Yes. They always add up to 360°.

Draw a line from one corner to the opposite corner. It cuts the shape into two triangles. Each triangle has 180°, so the four angles add up to 2 × 180° = 360°.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Three angles of a quadrilateral are 100°, 90° and 60°. What is the fourth angle x?

    A four sided shape. Going round from the bottom left, the angles are 100°, 90°, 60° and x.100°90°60°x
    Tap your answer.

    1. Not this one. That is 100° + 90° + 60°, the three angles you know. Take it away from 360° to find x. Go to M3.5.4. Try another.

    2. Yes, that is right. 100° + 90° + 60° = 250°. Then 360° − 250° = 110°.

    3. Not this one. You may have used 180°, the angle sum of a triangle. The angles of a quadrilateral add to 360°. Go to M3.5.4. Try another.

    See the answer

    110°

    100° + 90° + 60° = 250°. Then 360° − 250° = 110°.

    • If you chose A: That is 100° + 90° + 60°, the three angles you know. Take it away from 360° to find x. Go to M3.5.4.
    • If you chose C: You may have used 180°, the angle sum of a triangle. The angles of a quadrilateral add to 360°. Go to M3.5.4.

  2. What is the angle sum of a pentagon?

    A pentagon with five sides. Dashed diagonals from one corner cut it into three triangles, numbered 1, 2 and 3.123
    Tap your answer.

    1. Not this one. You multiplied 5 sides by 180°. Take 2 away first. A pentagon makes 3 triangles, not 5. Go to M3.5.4. Try another.

    2. Not this one. That is the angle sum of a quadrilateral. A pentagon has one more side, so one more triangle. Go to M3.5.4. Try another.

    3. Yes, that is right. A pentagon makes 5 − 2 = 3 triangles. 3 × 180° = 540°.

    See the answer

    540°

    A pentagon makes 5 − 2 = 3 triangles. 3 × 180° = 540°.

    • If you chose A: You multiplied 5 sides by 180°. Take 2 away first. A pentagon makes 3 triangles, not 5. Go to M3.5.4.
    • If you chose B: That is the angle sum of a quadrilateral. A pentagon has one more side, so one more triangle. Go to M3.5.4.

  3. This polygon is a regular pentagon. What is the angle x?

    A regular pentagon with five equal sides. One of its angles is marked x.x
    Tap your answer.

    1. Yes, that is right. The angle sum is 540°. The 5 angles are equal, so each is 540° ÷ 5 = 108°.

    2. Not this one. That is the angle sum of all five angles. Share it between the 5 angles. Go to M3.5.4. Try another.

    3. Not this one. That is 360° ÷ 5. Use the angle sum of a pentagon, 540°, and share it between the 5 angles. Go to M3.5.4. Try another.

    See the answer

    108°

    The angle sum is 540°. The 5 angles are equal, so each is 540° ÷ 5 = 108°.

    • If you chose B: That is the angle sum of all five angles. Share it between the 5 angles. Go to M3.5.4.
    • If you chose C: That is 360° ÷ 5. Use the angle sum of a pentagon, 540°, and share it between the 5 angles. Go to M3.5.4.

  4. Draw all the diagonals from one corner of a polygon with 7 sides. How many triangles do you get?

    Tap your answer.

    1. Not this one. That is the number of sides. The corner cannot join itself or the 2 corners next to it. It draws 4 diagonals and makes 5 triangles. Go to M3.5.4. Try another.

    2. Yes, that is right. A polygon with n sides makes n − 2 triangles. 7 − 2 = 5.

    3. Not this one. You took away only 1. Take away 2: a polygon with n sides makes n − 2 triangles. Go to M3.5.4. Try another.

    See the answer

    5

    A polygon with n sides makes n − 2 triangles. 7 − 2 = 5.

    • If you chose A: That is the number of sides. The corner cannot join itself or the 2 corners next to it. It draws 4 diagonals and makes 5 triangles. Go to M3.5.4.
    • If you chose C: You took away only 1. Take away 2: a polygon with n sides makes n − 2 triangles. Go to M3.5.4.

  5. What is each angle of a regular hexagon?

    A regular hexagon with six equal sides, cut by dashed diagonals from one corner into four triangles, numbered 1 to 4.1234
    Tap your answer.

    1. Not this one. That is the sum of all six angles. Share it between the 6 corners. Go to M3.5.4. Try another.

    2. Not this one. 360° ÷ 6 is the turn at the centre, not the inside angle of the hexagon. Go to M3.5.4. Try another.

    3. Yes, that is right. A hexagon makes 4 triangles, so its angle sum is 4 × 180° = 720°. The 6 angles are equal, so each is 720° ÷ 6 = 120°.

    See the answer

    120°

    A hexagon makes 4 triangles, so its angle sum is 4 × 180° = 720°. The 6 angles are equal, so each is 720° ÷ 6 = 120°.

    • If you chose A: That is the sum of all six angles. Share it between the 6 corners. Go to M3.5.4.
    • If you chose B: 360° ÷ 6 is the turn at the centre, not the inside angle of the hexagon. Go to M3.5.4.

  6. What is the angle sum of a polygon with 9 sides?

    See the answer

    1 260°

    9 − 2 = 7 triangles. 7 × 180° = 1 260°.

  7. From an earlier chapter

    An isosceles triangle has an angle of 30° between its equal sides. What is each of the other two angles?

    See the answer

    75°

    The other two angles add to 180° − 30° = 150°. They are equal, so each is 150° ÷ 2 = 75°.

Keep it

Copy this into your exercise book.

A polygon with n sides makes n − 2 triangles. Its angle sum is (n − 2) × 180°. A quadrilateral adds to 360°. Each angle of a regular hexagon is 120°.

Draw: A hexagon cut into four triangles by diagonals from one corner. Write 4 × 180° = 720° beside it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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