GCSE · Maths · AQA · Spec 8300 · Foundation

Angle sum in a triangle and polygons

Draw any triangle, tall, skinny, lopsided or almost flat, and add its three angles. You get 180° every time. You can't break it. Why not?

Angles in a triangle

Stretch it. Squash it. Add it up.

angle A 61°. angle B 36°. angle C 83°. Classification: every triangle. Relationship: angle A + angle B + angle C = 180°

angle A61°angle B36°angle C83°drag A or C

every triangle

angle A + angle B + angle C = 180°

Drag A along the base and C up or down its track. Make the triangle tall and thin, right-angled, obtuse or almost flat, and add the three readings each time.

Watch out: Each reading is rounded to the nearest degree, so now and then the three add to 179° or 181°. Measuring can only ever say “about 180°”. To know the total is exactly 180° for every triangle, you need a reason.

Why it is always 180°

Problem

Take any triangle ABC, with angle a at A, angle b at B and angle c at C. Prove that a + b + c = 180°.

Using the 180° fact

Supply the missing steps

(a) Two angles of a triangle are 47° and 68°. Find the third angle and give a reason. (b) The angles of another triangle are x, 2x and 3x + 30°. Find x and the size of each angle.

(a) Add the two known angles: 47° + 68° = 115°
missing step

Which line is step 2?

Third angle = 360° − 115° = 245°Third angle = 180° − 115° = 65°Third angle = 180° − 68° = 112°

From triangles to any polygon

2 × 180° = 360°

Pick one corner and draw the only diagonal from it. It cuts the four-sided shape into 2 triangles. The triangles' corners are the shape's corners, so their angles together make up exactly the quadrilateral's angles: 2 × 180° = 360°.

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Polygon angle sums

How big is an octagon's total?

An octagon has 8 sides and 8 interior angles.

Which is closest to what you think about the sum of its interior angles?

The angles of any polygon add up to 360°, so an octagon's do too.It has 8 sides, so it splits into 8 triangles from one corner: 8 × 180° = 1440°.Join the centre to every corner to make 8 triangles, so the total is 8 × 180° = 1440°.From one corner it splits into 6 triangles, so the total is 6 × 180° = 1080°.

How sure are you?

I'm sureI think soI'm guessing

The angles of any polygon add up to 360°, so an octagon's do too.

This is a common misconception

360° belongs to the quadrilateral, which splits into just 2 triangles. An octagon splits into more triangles than that, and every extra triangle brings another 180°, so its eight angles share far more than 360°.

It has 8 sides, so it splits into 8 triangles from one corner: 8 × 180° = 1440°.

This is a common misconception

Count the triangles you actually get. From one corner, diagonals go only to the 5 corners that are not next door, and they make 6 triangles, not 8. The count is always two fewer than the number of sides.

Join the centre to every corner to make 8 triangles, so the total is 8 × 180° = 1440°.

This is a common misconception

Those 8 triangles are real, but their angles include the ones meeting at the centre. Those make a full turn, 360°, and they are not angles of the octagon. Take them away: 1440° − 360° = 1080°.

From one corner it splits into 6 triangles, so the total is 6 × 180° = 1080°.

This matches the evidence

Yes. Eight sides give 8 − 2 = 6 triangles from one corner, and 6 × 180° = 1080°.

An octagon's angles total 1080°. Count triangles from one corner, two fewer than the sides, and give each one 180°.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One fact about triangles — and every polygon is built from it.

What you need to know

The interior angles of every triangle add up to 180°, whatever its shape.
Proof: draw a line through one corner parallel to the opposite side; two pairs of alternate angles copy the other two angles onto that line, and angles on a straight line add up to 180°.
To find a missing angle in a triangle, subtract the known angles from 180° and give the reason: angles in a triangle add up to 180°.
From one corner, an n-sided polygon splits into n − 2 triangles, so its interior angles add up to (n − 2) × 180°.
Each interior angle of a regular polygon is the angle sum divided by the number of angles, n.

The big picture

The three interior angles of any triangle add up to 180°. This lesson lets you test that by reshaping a triangle, proves it with one parallel line, uses it to find missing angles, and then builds every polygon out of triangles: an n-sided polygon splits into n − 2 triangles from one corner, so its angles total (n − 2) × 180°, and a regular polygon shares that total equally between its n angles.

Key points

1Triangle: 180°, for every shape of triangle.

2The proof uses alternate angles and angles on a straight line.

3Missing angle = 180° − the known angles, with the reason written in words.

4n sides → n − 2 triangles → (n − 2) × 180°.

5Regular polygon: share the angle sum equally between the n angles.

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Worked example

Problem

A heptagon (7 sides) has five interior angles of 130° each. Its other two angles are equal. Find the size of each of those two angles.

⚠ Watch out

Treating every polygon as if its angles add up to 360°, or counting one triangle per side. From one corner a 12-sided polygon makes 10 triangles, not 12, so its angles total 10 × 180° = 1800°, not 360° and not 12 × 180° = 2160°.

🧠

Memory hook

One triangle, one 180°. Every polygon is triangles in disguise: take the number of sides, subtract 2, and multiply by 180°.

✓

Check yourself

Without looking back: a triangle has angles of 38° and 94°. What is the third angle, and what reason do you give? Then work out each interior angle of a regular polygon with 9 sides.

Flashcards

(8)

What do the interior angles of any triangle add up to?

180°, whatever the shape of the triangle.

In the proof of the triangle angle sum, what line do you draw first?

A line through one corner, parallel to the opposite side.

Which angle facts complete the proof?

Two pairs of alternate angles are equal, and angles on a straight line add up to 180°.

You have found a missing angle in a triangle. What reason do you write?

Angles in a triangle add up to 180°.

How many triangles does an n-sided polygon split into from one corner, and why?

n − 2: the diagonals from a corner can't go to that corner or its two neighbours.

How do you turn a polygon's triangle count into its angle sum?

Multiply by 180°, one lot for each triangle: (n − 2) × 180°.

How do you find each interior angle of a regular polygon?

Find the angle sum, then divide it by the number of angles, n, because they are all equal.

Why don't all polygons have angles adding up to 360°?

360° is the quadrilateral's total (2 triangles). Each extra side adds another triangle, so another 180°.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Supply the missing steps

How big is an octagon's total?

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