Area of regular polygons part three (area of a circle)
We have seen that as the number of sides of the regular polygon increase, the area of the polygon becomes closer and closer to the area of the circle that we have drawn it inside. When the polygon has a large number of sides, the perimeter of the polygon is approximately the same as the circumference of the circle. We know that the circumference of a circle is 2πr (where r is the radius of the circle). The length of the parallelogram is therefore approximately πr. As the number of sides of the parallelogram increase, the length of the height of the triangles also becomes closer to the radius of the circle. In other words, the height of the triangle becomes approximately equal to (≈) r. The area of the parallelogram calculated using these approximations is therefore πr * r = πr².
The area of a circle = πr²
Let’s look at our table again and then calculate the area of the circle with a radius of 10cm.
| Number of sides | Name | Number of triangles the polygon can be split into | Angle at the centre | Half of the angle at the centre (θ) | Height of right-angled triangle (10 * cos θ) | Base of right-angled triangle (10 * sin θ) | Area of right angled triangle (0.5 * base * height) | Area of polygon |
|
5 |
pentagon |
5 |
72.0 |
36.0 |
8.09 |
5.88 |
23.78 |
237.76 |
|
6 |
hexagon |
6 |
60.0 |
30.0 |
8.66 |
5.00 |
21.65 |
259.81 |
|
7 |
septagon |
7 |
51.4 |
25.7 |
9.01 |
4.34 |
19.55 |
273.64 |
|
8 |
octagon |
8 |
45.0 |
22.5 |
9.24 |
3.83 |
17.68 |
282.84 |
|
9 |
nonagon |
9 |
40.0 |
20.0 |
9.40 |
3.42 |
16.07 |
289.25 |
|
10 |
decagon |
10 |
36.0 |
18.0 |
9.51 |
3.09 |
14.69 |
293.89 |
|
20 |
20-gon |
20 |
18.0 |
9.0 |
9.88 |
1.56 |
7.73 |
309.02 |
|
30 |
30-gon |
30 |
12.0 |
6.0 |
9.95 |
1.05 |
5.20 |
311.87 |
|
40 |
40-gon |
40 |
9.0 |
4.5 |
9.97 |
0.78 |
3.91 |
312.87 |
|
100 |
100-gon |
100 |
3.6 |
1.8 |
10.00 |
0.31 |
1.57 |
313.95 |
Area of a circle with a radius of 10cm = πr² = π*10² = π*100 =314.16 cm²
We can look at the information in a graph. We can see that as the number of sides of the polygon increases, the area of the polygon (shown by the blue line) moves closer towards the area of the circle (the red line).
