Unit 13: Many Sides
Polygons and their angles.
Unit 13 of 28 in Problem-solving maths for kids. Its 7 lessons are Naming the Shapes, Walking Round the Outside, Regular and Irregular, Angle Sums With Letters, Symmetry, Building Polygons From Triangles and Diagonals and Tilings — below is everything each one explains, and a question or two from it to try.
The answers are marked in the app by value — 1/2, 0.5 and 2/4 all count — and every one is re-derived by sympy before it ships.
This unit opens with a fortnight’s trial of everything — no card needed — or with a family plan, bought in the iPhone app. The first two units of every track are free for ever. Try it in the app.
🔷 Naming the Shapes
Counting sides
A polygon is a closed shape made of straight sides. The names count them:
3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 10 decagon.
A polygon is regular when every side and every angle is the same. A stop sign is a regular octagon; a squashed one is still an octagon, just not regular.
Cut it into triangles
Pick one corner of a polygon and draw lines to every other corner.
A quadrilateral splits into 2 triangles. A pentagon into 3. A hexagon into 4. Always two fewer than the number of sides.
So the angles add to (n - 2) × 180°. That is not a formula to learn — it is a count of triangles.
Try it yourself
Which shape has eight sides?
- Octagon
- Hexagon
- Heptagon
- Decagon
What do the angles of a pentagon add up to, in degrees?
Answer them in the app
🚶 Walking Round the Outside
The exterior angles always make one turn
Imagine walking all the way round a polygon. At each corner you turn a little; by the time you get back you have turned all the way round exactly once.
So the exterior angles of any polygon add to 360° — no matter how many sides it has.
That is a much easier number to work with than the interior sum, and it is the shortcut for anything regular.
Try it yourself
What is each exterior angle of a regular hexagon, in degrees?
What is each interior angle of a regular hexagon, in degrees?
Answer them in the app
⬡ Regular and Irregular
Both conditions are needed
A regular polygon has all sides equal and all angles equal. One alone is not enough.
A rhombus has four equal sides and is not regular — its angles differ. A rectangle has four equal angles and is not regular either.
Only the square manages both, which is why the square is the regular quadrilateral.
Convex and concave
A polygon is convex if none of its corners point inwards — a rubber band stretched round it would touch every corner.
If one corner caves in, like an arrowhead, it is concave, and that dented corner is a reflex angle of more than 180°.
The angle-sum rule (n - 2) × 180 still works for both, which is a small surprise worth noticing.
Try it yourself
Which quadrilateral is regular?
- The square
- The rhombus
- The rectangle
- The kite
A regular octagon has perimeter 56. How long is each side?
Answer them in the app
🧮 Angle Sums With Letters
The sum is the equation
A pentagon has angles x, 2x, 2x, 3x and 2x. How big is x?
The five angles must add to (5 - 2) × 180 = 540. So 10x = 540 and x = 54.
The polygon supplies the total; unit 3 does the rest. Every question in this lesson is that same two-step.
Working backwards to the number of sides
Given the total, you can find how many sides there were.
If the angles add to 1800°, then (n - 2) × 180 = 1800, so n - 2 = 10 and n = 12.
Divide by 180 first, then add the 2 back. Forgetting that 2 is the classic slip.
Try it yourself
A pentagon has angles x, 2x, 2x, 3x and 2x. What is x, in degrees?
A quadrilateral has angles x, x + 10, x + 20 and x + 30. What is x, in degrees?
Answer them in the app
🦋 Symmetry
Folding and turning
A shape has line symmetry if you can fold it so the halves match exactly. Each such fold is a line of symmetry.
It has rotational symmetry if turning it lands it back on itself before a full turn. The number of times that happens is its order.
A regular polygon with n sides has n lines of symmetry and order n. Both numbers are the same, which is not a coincidence.
Irregular shapes have less
A rectangle that is not a square has two lines of symmetry — across and down, but not along the diagonals. Fold a rectangle corner to corner and the halves do not match.
A parallelogram has none at all, yet it still has rotational symmetry of order 2.
So the two kinds of symmetry really are independent. Having one does not give you the other.
Try it yourself
How many lines of symmetry does a regular hexagon have?
What is the order of rotational symmetry of a regular pentagon?
Answer them in the app
🔺 Building Polygons From Triangles
A regular polygon is a fan of triangles
Join every corner of a regular polygon to its centre. You get n identical triangles, meeting in a point.
So the angle at the centre of each is 360/n, and the triangles are isosceles — two sides are radii of the same circle.
That one picture gives you the interior angle, the area and the symmetry all at once.
The area follows too
A regular hexagon of side 6 is six equilateral triangles of side 6.
One of those has area 6 × 3 sqrt(3) / 2 = 9 sqrt(3).
Six of them is 54 sqrt(3). No special hexagon formula was needed — only the triangle you already knew.
Try it yourself
A regular hexagon is split into triangles from its centre. What is the angle at the centre of each, in degrees?
Those triangles are isosceles. What is each of the other two angles, in degrees?
Answer them in the app
🧩 Diagonals and Tilings
Counting diagonals
From each corner of a polygon you can draw a diagonal to every other corner except itself and its two neighbours — so n - 3 of them.
That gives n(n - 3) in total, but you have counted every diagonal twice, once from each end.
So the real count is n(n - 3)/2. A pentagon has 5, a hexagon has 9.
Which shapes tile a floor
Regular shapes fit round a point with no gaps only if their angle divides exactly into 360°.
Triangles: 60° goes into 360 six times. Squares: 90° goes four times. Hexagons: 120° goes three times.
Pentagons are 108°, and that does not divide 360. Which is why you have never seen a floor tiled with regular pentagons.
Try it yourself
How many diagonals does a hexagon have?
How many diagonals does a decagon (10 sides) have?
Answer them in the app