Angle sums in triangles and polygons
Every polygon, no matter how many sides, can be sliced into triangles — and that single fact unlocks its whole angle sum.
Part of your national curriculum
- Geometry and measures: Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon
Lesson overview
What you'll learn in this lesson
Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon
Key learning points
- • Triangles first
- • Worked example: polygon sum
- • Worked example: missing angle
This lesson at a glance
- 25 minutes
- 17 parts to scroll through
- 3 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 17
Visual introductionPicture this
Angle sums in triangles and polygons
Every polygon, no matter how many sides, can be sliced into triangles — and that single fact unlocks its whole angle sum.
In a nutshell
Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon
Learning cycle
Part 2 of 17
Learning cycle 1 of 2
Part 1 · Triangles first
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 17
Learn
Triangles first
The angles in any triangle always sum to 180°. This can be shown by tearing the three corners off a paper triangle and placing them together to form a straight line.
Reset break
Part 4 of 17
Pause
That's sitting 1 of 4 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Explore the idea
Part 5 of 17
Learn
Worked example: polygon sum
A polygon can be split into (n-2) triangles, where n is the number of sides. For a hexagon, n = 6, so it splits into 4 triangles: angle sum = 4 × 180° = 720°.
Quick check
Part 6 of 17
Quick check
Part 7 of 17
Reset break
Part 8 of 17
Pause
That's sitting 2 of 4 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 9 of 17
Learning cycle 2 of 2
Part 2 · Worked example: missing angle
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 10 of 17
Learn
Worked example: missing angle
A quadrilateral has three angles of 90°, 85° and 100°. Since the sum is 360°, the fourth angle = 360 - (90+85+100) = 360 - 275 = 85°.
Quick check
Part 11 of 17
Reset break
Part 12 of 17
Pause
That's sitting 3 of 4 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Challenge round
Part 13 of 17
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 14 of 17
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
The ____ in any triangle always sum to 180°.
This can be shown by tearing the three corners off a paper ____ and placing them together to form a straight line.
A ____ can be split into (n-2) triangles, where n is the number of sides.
Challenge round
Part 15 of 17
Game · Recall cards
What is the interior angle sum of a pentagon?
Card 1 of 4
Mastery quiz
Part 16 of 17
Marked quiz
End of lesson quiz: Angle sums in triangles and polygons
4 questions, marked with the reasoning shown. No timer.
1. What is the interior angle sum of a pentagon?
2. A triangle has angles 48° and 67°. Find the third.
3. Each interior angle of a regular hexagon is:
4. What do the angles in a triangle always sum to?
Lesson round-up
Part 17 of 17
Lesson round-up
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Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon