The polygon
Three sides or twenty-four — it is the same figure.
- Sides
- 6
- Side length
- 20,0 cm
- Circumference
- 120,0 cm
- Apothem
- 17,3 cm
- Area
- 1039,2 cm²
- Of the circle's area
- 82,7 %
Regnestykket
Perimeter × Apothem ÷ 2
120,0 cm × 17,3 cm ÷ 2 = 1039,2 cm²
Card: o · a ÷ 2
It is one figure, not fifteen
Triangle, square, pentagon, hexagon. They each have their own name, but they are made in exactly the same way: all sides the same length, all corners the same distance from the centre. That is why there is only one slider for all of them. Drag it, and you can see the names change without the figure doing anything new. It just gets one more corner. Everything you find out here therefore applies to all of them, and you only need to remember it once.
All the pieces are the same
Cut from the centre out to each corner, and you have as many triangles as the figure has sides. They are all the same. Each triangle has the figure's side as its base, and its height is the distance from the centre straight out to the side. That distance is called the apothem, and it is the copper-coloured line. Notice that it is shorter than the distance out to a corner. With a triangle it is much shorter. With twenty-four sides they are almost the same length.
Here it fits exactly
When the triangles were laid in a row, every other one was turned round. Then the straight edges line up two by two, and there are no gaps and nothing lying on top of anything else. The row is as tall as the apothem and as long as half the perimeter. Multiply the two numbers, and you have the area. It isn't roughly right: it matches exactly, with three sides and with twenty-four and with everything in between. That is why you get to work this one out by hand.
More sides, closer to a circle
Under the figure it shows how much of the circle around it the polygon fills. A triangle fills about forty-one per cent. A hexagon fills about eighty-three. At twenty-four sides the number is almost ninety-nine. The figure still has straight sides and sharp corners, but each side has become so short that the eye can no longer see them. It isn't a circle. It just looks like one.
The bridge to the circle
The circle in the other workshop is cut up in exactly the same way: pieces from the centre, every other one turned round, laid in a row. The only difference is that the circle's pieces have a round edge instead of a straight one, so there the row never becomes quite a rectangle. Here it becomes one straight away. You can see a circle as a polygon with infinitely many sides, where each side is infinitely short. Half the perimeter times the apothem then becomes half the circumference times the radius, and that is the same as pi times radius times radius.
It was the same proof twice
Notice what you didn't need to do. You didn't learn one formula for the triangle, another for the pentagon and a third for the circle. You cut a figure into pieces, rearranged the pieces into something you already knew, and read off the answer. That method holds far beyond the two figures here. When you later meet a shape you don't know, the first question is the same: can I cut it into pieces I can already measure?