EPITOME OF MENSURATION.
OF THE CIRCLE, CYLINDER, SPHERE, ETC.
1. The circle contains a greater area than any other plane figure, bounded by an equal perimeter, or outline.
2. The areas of circles are to each other as the squares of their diameters; any circle twice the diameter of another contains four times the area of the other.
3. The diameter of a circle being 1, its circumference equals 3·1416.
4. The diameter of a circle is equal to ·31831 of its circumference.
5. The square of the diameter of a circle being 1, its area equals ·7854.
6. The square root of the area of a circle, multiplied by 1·12837, equals its diameter.
7. The diameter of a circle, multiplied by ·8862, or the circumference multiplied by ·2821, equals the side of a square of equal area.
8. The sum of the squares of half the chord, and versed sine, divided by the versed sine, the quotient equals the diameter of the corresponding circle.
9. The chord of the whole arc of a circle taken from eight times the chord of half the arc, one-third of the remainder equals the length of the arc.
10. Or, the number of degrees contained in the arc of a circle, multiplied by the diameter of the circle, and by ·008727, the product equals the length of the arc in equal terms of unity.
11. The length of the arc of the sector of a circle multiplied by its radius, half the product is the area.
12. The area of the segment of a circle equals the area of the sector, minus the area of a triangle whose vertex is the centre; and base equals the chord of the segment.
13. The sum of the diameters of two concentric circles multiplied by their difference, and by ·7854, equals the area of the ring, or space contained between them.
14. The sum of the thickness, and internal diameter of a cylindric ring multiplied by the square of its thickness, and by 2·4674, equals its solidity.
15. The circumference of a cylinder multiplied by its length, or height, equals its convex surface.
16. The area of the end of a cylinder multiplied by its length, equals its solid content.
17. The area of the internal diameter of a cylinder multiplied by its depth, equals its cubical capacity.
18. The square of the diameter of a cylinder multiplied by its length, and divided by any other required length, the square root of the quotient equals the diameter of the other cylinder of equal solidity, or capacity.
19. The square of the diameter of a sphere multiplied by 3·1416 equals its convex surface.
20. The cube of the diameter of a sphere multiplied by ·5236, equals its solid content.
21. The height of any spherical segment, or zone, multiplied by the diameter of the sphere, of which it is a part, and by 3·1416, equals the area, or convex surface of the segment;
22. Or, the height of the segment multiplied by the circumference of the sphere of which it is a part, equals the area.
23. The solidity of any spherical segment is equal to three times the square of the radius of its base, plus the square of its height, and multiplied by its height, and by ·5236.
24. The solidity of a spherical zone equals the sum of the squares of the radii of its two ends, and one-third the square of its height, multiplied by the height, and by 1·5708.
25. The solidity of the middle zone of a sphere equals the sum of the square of either end, and two-thirds the square of the height, multiplied by the height, and by ·7854.
26. The capacity of a cylinder 1 foot in diameter, and 1 foot in length, equals 4·895 imperial gallons.
27. The capacity of a cylinder 1 inch in diameter, and 1 foot in length, equals ·034 of an imperial gallon.
28. The capacity of a cylinder 1 inch in diameter, and 1 inch in length, equals ·002832 of an imperial gallon.
29. The capacity of a sphere 1 foot in diameter, equals 3·263 imperial gallons.
30. The capacity of a sphere 1 inch in diameter, equals ·001888 of an imperial gallon.
31. Hence the capacity of any other cylinder in imperial gallons is obtained by multiplying the square of its diameter by its length; or the capacity of any other sphere by the cube of its diameter, and by the number of imperial gallons contained as above in the unity of its measurement.
OF THE SQUARE, RECTANGLE, CUBE, ETC.
1. The side of a square equals the square root of its area.
2. The area of a square equals the square of one of its sides.
3. The diagonal of a square equals the square root of twice the square of its side.
4. The side of a square is equal to the square root of half the square of its diagonal.
5. The side of a square, equal to the diagonal of a given square, contains double the area of the given square.
6. The area of a rectangle equals its length multiplied by its breadth.
7. The length of a rectangle equals the area divided by the breadth; or the breadth equals the area divided by the length.
8. The side, or end of a rectangle, equals the square root of the sum of the diagonal, and opposite side to that required, multiplied by their difference.
9. The diagonal in a rectangle equals the square root of the sum of the squares of the base, and perpendicular.
10. The solidity of a cube equals the area of one of its sides multiplied by the length of one of its edges.
11. The edge of a cube equals the cube root of its solidity.
12. The capacity of a 12-inch cube equals 6·232 gallons.
| Surfaces, and solidities of the regular bodies, when the linear edge is 1. | |||
| No. of Sides. | Names. | Surfaces. | Solids. |
| 4 | Tetrahedron | 1·7320508 | 0·1178513 |
| 6 | Hexahedron | 6· | 1· |
| 8 | Octahedron | 3·4641016 | 0·4714045 |
| 12 | Dodecahedron | 20·6457788 | 7·6631189 |
| 20 | Icosahedron | 8·6602540 | 2·1816950 |
The tabular surface multiplied by the square of the linear edge, the product equals the surface required:
Or, the tabular solidity, multiplied by the cube of the linear edge, the product is the solidity required.
OF TRIANGLES, POLYGONS, ETC.
1. The complement of an angle is its defect from a right angle.
2. The supplement of an angle is its defect from two right angles.
3. The sine, tangent, and secant of an angle, are the cosine, cotangent and cosecant of the complement of that angle.
4. The hypothenuse of a right-angled triangle being made radii, its sides become the sines of the opposite angles, or the cosines of the adjacent angles.
5. The three angles of every triangle are equal to two right angles; hence the oblique angles of a right-angled triangle are each other’s complements.
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.
2. The product of the two axes of an ellipse multiplied by ·7854 equals its area.
3. The curve surface of a cone is equal to half the product of the circumference of its base multiplied by its slant side, to which, if the area of the base be added, the sum is the whole surface.
4. The solidity of a cone equals one-third of the product of its base multiplied by its altitude, or height.
5. The squares of the diameters of the two ends of the frustrum of a cone added to the product of the two diameters, and that sum multiplied by its height, and by ·2618, equals its solidity.
THE END.