Angles in triangles and polygons
Polygon angle facts are not things to memorise separately: every one of them can be derived from the triangle, which is why deriving beats remembering.
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
- • Why a triangle totals 180°
- • Angle sum of any polygon
- • Regular polygons
- • Reasoning, not just answers
This lesson at a glance
- 30 minutes
- 21 parts to scroll through
- 4 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 21
Visual introductionPicture this
Angles in triangles and polygons
Polygon angle facts are not things to memorise separately: every one of them can be derived from the triangle, which is why deriving beats remembering.
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 21
Learning cycle 1 of 2
Part 1 · Why a triangle totals 180°
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 21
Learn
Why a triangle totals 180°
Draw a line through one vertex parallel to the opposite side. The two outer angles are alternate angles equal to the base angles, and together with the third angle they form a straight line of 180°. That single argument is the foundation for everything else in this lesson.
Reset break
Part 4 of 21
Pause
That's sitting 1 of 5 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 21
Learn
Angle sum of any polygon
A polygon with n sides can be split into n − 2 triangles from one vertex, so the interior angles total 180(n − 2)°. A hexagon gives 720° and an octagon 1,080°. The exterior angles of any convex polygon always total 360°, however many sides it has.
Quick check
Part 6 of 21
Quick check
Part 7 of 21
Reset break
Part 8 of 21
Pause
That's sitting 2 of 5 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 21
Learning cycle 2 of 2
Part 2 · Regular polygons
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 10 of 21
Learn
Regular polygons
In a regular polygon every exterior angle is 360 ÷ n, and each interior angle is 180 minus that. Working through the exterior angle is almost always quicker: if each interior angle is 156°, the exterior is 24° and n = 360 ÷ 24 = 15 sides.
Explore the idea
Part 11 of 21
Learn
Reasoning, not just answers
Full marks require the reason with the value: 'x = 68° because alternate angles between parallel lines are equal'. A chain of reasons should read like a short argument, each line justified by a named fact.
Reset break
Part 12 of 21
Pause
That's sitting 3 of 5 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.
Quick check
Part 13 of 21
Quick check
Part 14 of 21
Explore the idea
Part 15 of 21
Worked example
Worked answer: each interior angle of a regular polygon is 156°. How many sides? (3 marks)
Interior and exterior angles at a vertex lie on a straight line, so the exterior angle is 180 − 156 = 24° (1). The exterior angles of any polygon total 360°, so n = 360 ÷ 24 (1), giving 15 sides (1). Dividing 156 into 360 is a frequent error and gives a non-integer, which should itself signal the mistake.
Reset break
Part 16 of 21
Pause
That's sitting 4 of 5 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 17 of 21
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 18 of 21
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
The two outer ____ are alternate angles equal to the base angles, and together with the third angle they form a straight line of 180°.
A ____ with n sides can be split into n − 2 triangles from one vertex, so the interior angles total 180(n − 2)°.
Challenge round
Part 19 of 21
Game · Recall cards
What is the interior angle sum of a pentagon?
Card 1 of 4
Mastery quiz
Part 20 of 21
Marked quiz
End of lesson quiz: Angles 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 21 of 21
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