Cut shapes into triangles, gather their corners and find the missing angle.
Start here. A polygon is a flat shape with straight sides, like a triangle, a square or a hexagon. The angles inside the corners are called the inside angles. This game asks one question: what do they add up to?
Triangles first. The three angles of any triangle add up to 180°. You can check this by tearing off the three corners and placing them side by side. They fit together on a straight line, and a straight line is 180°. Press Gather the corners to see it.
Quadrilaterals. Cut a four-sided shape from one corner to the opposite corner and you get 2 triangles. That is 2 × 180° = 360°. The four corners fit round a point, and a full turn is 360°.
Any polygon. Pick one corner and draw lines from it to every corner that is not next to it. A shape with n sides always splits into n − 2 triangles. So the angles add up to (n − 2) × 180°. A pentagon is 3 × 180° = 540°. A hexagon is 4 × 180° = 720°.
Regular polygons. If all the sides and all the angles are equal, share the total equally. A regular pentagon has 540 ÷ 5 = 108° at each corner. A regular hexagon has 720 ÷ 6 = 120°.
Outside angles. Extend one side past a corner. The turn you make to follow the next side is the outside angle. The inside angle and the outside angle make a straight line, so they add up to 180°. If you walk all the way round a polygon, you turn one full turn, so the outside angles of any polygon add up to 360°. A regular polygon with 9 sides has outside angles of 360 ÷ 9 = 40°.
Try this above. In Play, press + until you have 6 sides and count the triangles. Then press Equal corners to see 120° at every corner. At Challenge me, walk round the outside and see 360 ÷ n.
When you draw lines from one corner, you cannot draw to the corner itself or to the two corners next to it, because those lines are already sides. That leaves n − 3 lines, which cut the shape into n − 2 triangles.
Yes, for any polygon whose sides do not cross each other. This game uses shapes with no dents, called convex polygons, so you can always cut from any corner.
If you walk along the sides and turn at each corner, you end up facing the way you started after going all the way round. That is one full turn, which is 360°.
Pick the level, not the age. "Just starting" uses triangles and quadrilaterals. "I know the basics" uses the (n − 2) × 180° rule. "Challenge me" uses regular polygons and outside angles.
No. It is free and needs no account. Your stars are saved in your own browser.