This is Lesson 4, Florence — the last full teaching lesson of the week. The triangle gave you 180°. Now we go one shape further: to quadrilaterals (four sides) and then to polygons with any number of sides at all.
Here is the good news. Almost nothing new has to be memorised today. Every rule in this lesson is the triangle's 180° rule, used again. Three ideas — four-sided shapes, a pattern that fits any polygon, and one last rule that works for every polygon there is.
Small steps, answers tapped or typed, a quiet nudge if one isn't right, and a part to drag. Tap Continue when you're ready.
A quadrilateral is any shape with four straight sides — a square, a kite, or something with no special name at all.
Draw one diagonal across it, corner to corner, and you split it into two triangles. Each triangle's angles add to 180°, and together the two triangles fill exactly the four corners of the quadrilateral. Two lots of 180° is 360° — so the four angles of any quadrilateral add up to 360°.
Try it. Drag any corner of the shape below — pull it into a kite, a long dart, anything you like — and the four angles still add to 360° every single time.
The four angles of any quadrilateral add up to 360°.
85° + 100° + 75° + x = 360°
Add the three you know, then take the total away from 360°.
85° + 100° + 75° = 260°, and 360° − 260° = 100°
The four angles of a quadrilateral add up to 360°.
Add the three you know: 90° + 90° + 100° = 280°.
Then 360° − 280° = 80°.
The four angles add up to 360°.
70° + 110° + 65° = 245°.
Then 360° − 245° = 115°.
The four angles add up to 360°.
120° + 95° + 88° = 303°.
Then 360° − 303° = 57°.
A polygon is any closed shape made of straight sides. The triangle and the quadrilateral are simply the first two.
The diagonal trick keeps working. From one corner of any polygon, you can fan out diagonals that cut it into triangles — and you always get two fewer triangles than the shape has sides. A pentagon (5 sides) makes 3 triangles, a hexagon (6 sides) makes 4, and so on.
Each triangle is 180°. So the angles of a polygon with n sides add up to (n − 2) × 180°. That one formula covers every polygon there is.
First, the total. A pentagon has 5 sides, so n = 5, and it splits into (n − 2) = 3 triangles.
(5 − 2) × 180° = 3 × 180° = 540°
All five angles add to 540°. Add the four you are given.
100° + 115° + 120° + 108° = 443°
Take that away from 540°.
x = 540° − 443° = 97°
Use (n − 2) × 180° with n = 6.
(6 − 2) × 180° = 4 × 180° = 720°.
Use (n − 2) × 180° with n = 7.
(7 − 2) × 180° = 5 × 180° = 900°.
A pentagon's angles add up to 540°.
90° + 110° + 130° + 100° = 430°.
Then 540° − 430° = 110°.
At each corner of a polygon, carry one side on a little way past the corner. The angle between that extension and the next side is the exterior angle at that corner.
Here is the surprising part. Imagine walking right around the outside of the polygon. At every corner you turn by the exterior angle — and by the time you are back where you started, facing your first direction again, you have turned through one full circle. So the exterior angles of any polygon add up to 360° — three sides or thirty, it makes no difference.
If the polygon is regular — all its sides and angles equal — then those 360° are shared out evenly, so each exterior angle is simply 360° ÷ n.
The exterior angles of any polygon add up to 360°. An octagon has eight of them.
all 8 exterior angles → 360°
The octagon is regular, so all eight exterior angles are equal. Share the 360° equally between them.
x = 360° ÷ 8 = 45°
The exterior angles add up to 360°, shared equally.
So each is 360° ÷ 9 = 40°.
The exterior angles add up to 360°, shared equally between 6 corners.
So each is 360° ÷ 6 = 60°.
This last part has two steps. It brings back the straight-line rule from Lesson 1 — so it uses the start and the end of the week at once. Here is the first step.
The exterior angles add up to 360°, shared equally between 5 corners.
So each is 360° ÷ 5 = 72°.
You found each exterior angle is 72°. At any corner, the exterior angle and the interior angle sit together on a straight line.
The exterior angle and the interior angle lie together on a straight line.
Angles on a straight line add up to 180°, so the interior angle is 180° − 72° = 108°.
That's Lesson 4 done, Florence — and the four teaching lessons of the week with it.
You can now find missing angles in quadrilaterals and in polygons of any size at all. And the whole of it rests on one idea you have had since Lesson 3: a triangle is 180°. Quadrilaterals are two triangles, pentagons are three, and the exterior-angle rule is just one full turn, shared out. Almost nothing today was new — it was the same idea, stretched.
One lesson left this week, and it is a gentler one: no new rules, just a chance to use everything together. Whenever you're ready for it.
You can close this page now — or step back through any part you'd like to see again.