6. The sum of the squares of the two given sides of a right-angled triangle is equal to the square of the hypothenuse.
7. The difference between the square of the hypothenuse, and given side of a right-angled triangle is equal to the square of the required side.
8. The area of a triangle equals half the product of the base multiplied by the perpendicular height;
9. Or, the area of a triangle equals half the product of the two sides, and the natural sine of the contained angle.
10. The side of any regular polygon multiplied by its apothem, or perpendicular, and by the number of its sides, half the product is the area.
| Table of the areas of regular polygons whose sides are unity. | |||||||
| Name of polygon. | No. of sides. | Apothem, or perpendicular. | Area, when side is one or unity. | Interior angle. | Central angle. | ||
| ° | ′ | ° | ′ | ||||
| Triangle | 3 | 0·2886751 | 0·4330127 | 60 | 0 | 120 | 0 |
| Square | 4 | 0·5 | 1· | 90 | 0 | 90 | 0 |
| Pentagon | 5 | 0·6881910 | 1·7204774 | 108 | 0 | 72 | 0 |
| Hexagon | 6 | 0·8660254 | 2·5980762 | 120 | 0 | 60 | 0 |
| Heptagon | 7 | 1·0382607 | 3·6339124 | 128 | 34 2 7 | 51 | 25 5 7 |
| Octagon | 8 | 1·2071068 | 4·8284271 | 135 | 0 | 45 | 0 |
| Nonagon | 9 | 1·3737387 | 6·1818242 | 140 | 0 | 40 | 0 |
| Decagon | 10 | 1·5388418 | 7·6942088 | 144 | 0 | 36 | 0 |
| Undecagon | 11 | 1·7028436 | 9·3656399 | 147 | 16 4 11 | 32 | 43 7 11 |
| Dodecagon | 12 | 1·8660254 | 11·1961524 | 150 | 0 | 30 | 0 |
The tabular area of the corresponding polygon multiplied by the square of the side of the given polygon, equals the area of the given polygon.
OF ELLIPSES, CONES, FRUSTRUMS, ETC.
1. The square root of half the sum of the squares of the two diameters of an ellipse multiplied by 3·1416 equals its circumference.