MEASURES OF SOLIDS

466. A Solid is anything that has length, breadth, and thickness or height. A solid is also called a Body.

The planes which bound a solid are called its faces, and their intersections, its edges.

467. A Prism is a solid whose two ends are equal polygons and

whose sides are parallelograms.

Prisms, from their bases, are named triangular, quadrangular, pentagonal, hexagonal, etc.

468. A Rectangular Solid, Parallelopiped, or Square Prism, is a prism whose six faces are all rectangles.

469. A Cube is a parallelopiped whose six sides are squares.

470. A Cylinder is a regular solid bounded by a uniformly curved surface with equal and parallel circles for its bases.

471. A Pyramid is a solid whose base is a polygon and whose faces are triangles meeting in a common point called the vertex.

472. A Cone is a solid whose base is a circle and whose curved surface tapers uniformly to a point called the vertex.

473. A Frustum of a pyramid or cone is the part of a pyramid or cone that remains after cutting off the top by a plane parallel to the base.

474. A Sphere is a solid bounded by a curved surface every point of which is equally distant from a point within called the center.

The Diameter of a sphere is a line passing through its center and terminating in the surface.

The Radius is half the diameter, or the distance from the centre to the surface of a sphere.

A

SPHERE.

The Circumference of a sphere is the greatest distance around it. The Altitude of a solid is the perpendicular distance from its highest point to the plane of the base; as, AC (in cone or pyramid).

The Slant height of a pyramid is the altitude of any lateral face. The Slant height of a cone is the distance from the vertex to the circumference of the base.

SURFACE OF SOLIDS.

475. The Lateral Surface of a solid is all of the surface except that of its base or bases.

The lateral surface of a cylinder and a cone may properly be called the convex surface.

476. The Entire Surface of a solid includes the lateral surface and also the surface of the base or bases.

477. To find the Lateral Surface of a Prism or Cylinder.

If the lateral surface of a prism or cylinder were unfolded, it would form a rectangle, as ABCD, whose length is the perimeter of the base, and whose altitude is the height of the prism or cylinder. Hence,

C

RULE.-Multiply the perimeter of the A

base by the altitude.

EXAMPLES.

B

D

1. What is the lateral surface of a triangular prism whose sides

are each 3 ft., and whose height is 5 ft.?

2. What is the convex surface of a cylinder whose diameter is 4 ft., and whose length is 7 ft. ?

3. What is the entire surface of a square prism whose altitude 15 ft. and the side of the base 5 ft. ?

4. What is the entire surface of a cylinder whose length is 12 ft., and the radius 2 ft.?

478. To find the Lateral Surface of a Pyramid or Cone.

It is seen that the lateral surface of a pyramid is composed of triangles whose bases form the perimeter, and whose height is the slant height of the pyramid. This is also true when the pyramid has an infinite number of sides and coincides with a cone. Hence,

RULE -Multiply the perimeter of the base by half the slant height.

EXAMPLES.

1. What is the lateral surface of a triangular pyramid, the sides of whose base are each 6 ft., and whose slant height is 20 ft.?

2. What will be the cost of painting a hexagonal church steeple at 25 per square yard, the sides of whose base are each 5 ft, and whose slant height is 40 ft.?

3. What is the lateral surface of a cone whose base is 7 ft. in diameter, and slant height 56 ft.?

4. What is the convex surface of a cone whose slant height is 48 ft., and whose base has a radius of 3 ft.?

5. Find the entire surface of a square pyramid the side of whose base is 8 ft., and slant height 25 ft.

479. To find the Lateral Surface of a Frustum of a Pyramid or Cone.

It is seen that the lateral surface of a frustum of a pyramid is composed of trapezoids whose parallel sides form the perimeter of the bases, and whose altitude is the slant height of the frustum. This is also true when the frustum has an infinite number of lateral faces, or trapezoids, and coincides with the frustum of a cone. Hence,

RULE.-Multiply the sum of the perimeters of the two bases by half the slant height.

EXAMPLES.

1. Find the lateral surface of the frustum of a square pyramid whose slant height is 20 ft., the ide of the lower base 10 ft. and of the upper base 6 ft.

2. What is the convex surface of the frustum of a cone whose slant height is 14 ft., the diameter of the lower base 8 ft. and of the upper base 5 ft.?

3. How many square feet in the lateral surface of the frustum of a pentagonal pyramid whose slant height is 10 ft., and each side of whose lower base is 6 ft. and of the upper base 4 ft.?

4. What is the entire surface of the frustum of a cone whose slant height is 8 ft., the diameter of the lower base being 5 ft. and of the upper base 4 ft.?

480. To find the Convex Surface of a Sphere.

The following rules are derived from principles in geometry: RULE.-Multiply the circumference by the diameter; or, multiply the square of the radius by 4 times 3.1416.

The convex surface of a sphere may also be found by multiplying the square of the diameter by 3.1416.

EXAMPLES.

1. What is the convex surface of a sphere whose diameter is 12 inches?

2. What is the convex surface of a sphere whose radius is 5 inches?

3. What is the convex surface of a cannon-ball whose circumference is 25 inches ?

4. What would it cost to plate with silver a sphere 15 in. in diameter, at $2.25 a square foot ?

VOLUME OF SOLIDS.

481. The Volume of any body is the number of cubic units it contains.

The volume of a body is also called the solid contents, contents, or capacity. 482. To find the Volume of a Rectangular Solid.

It is seen that the volume of a rectangular solid is the number of small cubes it contains. Each layer contains the number in each row multiplied by the number of rows, and this multiplied by the number of layers, will give the number; but this equals the product of the length, breadth, and height, or thickness. Hence,

5

The length, breadth, and thickness must be expressed in units of the same denomination.

EXAMPLES,

1. How many cubic feet in a block 6 ft. long, 5 ft. wide, and 3 ft. thick?

2. How many cubic feet of water in a rectangular reservoir 16 ft. long, 14 ft. wide, 7 ft. deep.

3. What will it cost to dig a cellar 38 ft. long, 30 ft. wide, and 5 ft. 6 in. deep, at $1.25 a cubic yard?

4. How many cubic feet of air in a room 16 ft. long, 14 ft. 9 in. wide, and 8 ft. 6 in. high?

5. A rectangular block of stone contains 182 cubic feet. It is 4 ft. wide and 31 ft. thick. How long is the block?

483. To find the Volume of a Prism or Cylinder.

It is seen that if a prism or cylinder were 1 inch high it would contain as many cubic inches as there are square A inches in the area of the base; and if 2 in. high, the volume would be 2 times as many; if 3 in. high, 3 times as many, etc. Hence,

RULE.-Multiply the area of the base by the alti

tude.

For area of the base see Art. 452.

E

B

EXAMPLES.

1. Find the solid contents of a prism whose base is 8 in. square and whose height is 15 inches ?

2. Find the volume of a cylinder whose diameter is 10 in. and whose height is 2 ft.

3. How many cubic feet of earth must be removed to dig a cistern 8 ft. in diameter and 13 ft. deep?

484. To find the Volume of a Pyramid or Cone.

If we take two vessels, the one a prism and the other a pyramid, having equal bases and altitudes, the pyramid will hold just one-third as much as the prism. This is also true when the prism and pyramid have an infinite number of sides and coincide with the cylinder and cone. Hence,

RULE.-Multiply the area of the base by one third of the alti

tude.