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Deriving the sum of interior angles in multiple ways

I can use reasoning to derive the sum of interior angles in multiple ways.

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Lesson 12 of 16

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Year 8

Deriving the sum of interior angles in multiple ways

I can use reasoning to derive the sum of interior angles in multiple ways.

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Lesson details

Key learning points

Not all vertices of component shapes contribute to the sum of the interior angles of a composite shape.
A numerical sequence can be used to derive the sum of interior angles.
A formula can be derived to find the sum of interior angles.

Keywords

Polygon - A polygon is a flat (2D), closed figure made up of straight line segments.
Interior angle - An interior angle is an angle formed inside a polygon by two of its edges.
Sum - The sum is the total when numbers are added together.

Common misconception

Interior angles always sum to 180 times the number of component triangles it is split into.

If line segments drawn make new vertices in the polygon, its angles won't contribute to the sum of interior angles in the initial polygon.

Following Task A, pupils could explore their own methods for splitting up a polygon into triangles and calculating the sum of interior angles.

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This content is © Oak National Academy Limited (2026), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
A is a flat (2D), closed figure made up of straight line segments.

Correct Answer: polygon

Q2.
What is the size of ∠BCD?

Correct answer: 73°

84°

95°

108°

Q3.
A hendecagon has 11 sides. The minimum number of triangles that it could be split into is .

Correct Answer: 9, nine

Q4.
A hexadecagon has 16 sides. Which calculation would find the sum of its interior angles?

Correct answer: $$14 \times 180$$

$$14 \times 360$$

$$16 \times 180$$

$$16 \times 360$$

Q5.
Angles around a point sum to °.

Correct Answer: 360, three hundred and sixty, 360°

Q6.
Evaluate the expression $$20(x − 2)$$ when $$x=5$$.

Correct Answer: 60, sixty

Assessment exit quiz

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6 Questions

Q1.
Each time the number of sides of a polygon increases by 1, the sum of the interior angles increases by °.

Correct Answer: 180, 180°, one hundred and eighty

Q2.
The interior angles in 19-sided polygon sum to 3060°. Therefore the interior angles in a 20-sided polygon sum to °.

Correct Answer: 3240, 3240°, 3 240, 3,240

Q3.
Which expression can be used to calculate the sum of interior angles in a polygon by substituting $$n$$ for the number of sides?

$$180n$$

$$180n - 2$$

Correct answer: $$180(n - 2)$$

$$180n + 2$$

$$180(n + 2)$$

Q4.
A triacontagon is a polygon with 30 sides. Which calculation finds the sum of its interior angles?

$$180 \times 30$$

Correct answer: $$180 \times (30 - 2)$$

$$180 \times (30 + 2)$$

Q5.
A tetracontagon is a polygon with 40 sides. Its interior angles sum to °.

Correct Answer: 6840, 6840°, 6,840, 6 840

Q6.
A polygon has interior angles which sum to 720°. The polygon has sides.

Correct Answer: 6, six

Lesson appears in

UnitMaths / Geometrical properties: polygons

Maths

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Deriving the sum of interior angles in multiple ways

Deriving the sum of interior angles in multiple ways

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.A is a flat (2D), closed figure made up of straight line segments.

Q2.What is the size of ∠BCD?

Q3.A hendecagon has 11 sides. The minimum number of triangles that it could be split into is .

Q4.A hexadecagon has 16 sides. Which calculation would find the sum of its interior angles?

Q5.Angles around a point sum to °.

Q6.Evaluate the expression $$20(x − 2)$$ when $$x=5$$.

Assessment exit quiz

6 Questions

Q1.Each time the number of sides of a polygon increases by 1, the sum of the interior angles increases by °.

Q2.The interior angles in 19-sided polygon sum to 3060°. Therefore the interior angles in a 20-sided polygon sum to °.

Q3.Which expression can be used to calculate the sum of interior angles in a polygon by substituting $$n$$ for the number of sides?

Q4.A triacontagon is a polygon with 30 sides. Which calculation finds the sum of its interior angles?

Q5.A tetracontagon is a polygon with 40 sides. Its interior angles sum to °.

Q6.A polygon has interior angles which sum to 720°. The polygon has sides.

Lesson appears in

UnitMaths / Geometrical properties: polygons

Maths

Q1.
A is a flat (2D), closed figure made up of straight line segments.

Q2.
What is the size of ∠BCD?

Q3.
A hendecagon has 11 sides. The minimum number of triangles that it could be split into is .

Q4.
A hexadecagon has 16 sides. Which calculation would find the sum of its interior angles?

Q5.
Angles around a point sum to °.

Q6.
Evaluate the expression $$20(x − 2)$$ when $$x=5$$.

Q1.
Each time the number of sides of a polygon increases by 1, the sum of the interior angles increases by °.

Q2.
The interior angles in 19-sided polygon sum to 3060°. Therefore the interior angles in a 20-sided polygon sum to °.

Q3.
Which expression can be used to calculate the sum of interior angles in a polygon by substituting $$n$$ for the number of sides?

Q4.
A triacontagon is a polygon with 30 sides. Which calculation finds the sum of its interior angles?

Q5.
A tetracontagon is a polygon with 40 sides. Its interior angles sum to °.

Q6.
A polygon has interior angles which sum to 720°. The polygon has sides.