1. Find the surface of the bottom of a hexagonal silo that is 12 feet on a side, the distance from the middle point of the side to the center of the bottom being 10.3 ft.

2. How far from the corner is the center of a square field that is 40 rods on a side? (Draw the figure.)

3. Find the area of an

equilateral triangular design that is 15 inches on a side. (Divide into two right triangles.)

SOLIDS

A solid is anything that has length, breadth, and thickness. The faces of a solid are the surfaces that bound it.

The lateral or convex surface of a solid is the area of its sides, or faces.

The volume of a solid is the number of cubic units it contains.

A prism is a solid whose ends are equal and parallel polygons, and whose sides are parallelograms. Prisms are named from their bases, as triangular, square, rectangular, pentagonal, hexagonal, etc.

TRIANGULAR
PRISM

SQUARE

PRISM

PENTANGULAR PRISM RECTANGULAR PRISM

A cylinder is a solid with circular ends and uniform diameter. The ends are the bases, and the curved surface is the convex surface.

The altitude of a prism or of a cylinder is the perpendicu lar distance between the bases.

The slant height of a pyramid is the altitude of the triangles that bound it.

A cone is a solid whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex.

The altitude of a pyramid, or of a cone, is the perpendicular distance from the vertex to the

base.

CYLINDER

CONE

The slant height of a cone is the distance between the vertex and any point in the circumference of

the base.

A globe or sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center.

SPHERE

SURFACES OF SOLIDS

Surfaces of Prisms and Cylinders

Observe: 1. That if a piece of paper is fitted to cover the convex surface of a prism or a cylinder, and then unrolled, its form will be that of a rectangle, as ABCD.

2. That the perimeter of the solid forms

one side of the rectangle, and the altitude of the solid the other

D

A

C

The convex surface of a prism or of a cylinder is found by multiplying the unit of measure by the product of the perimeter and the B altitude.

Find the convex surface of a regular prism of:

1. 5 sides; 1 side 10 ft.; height 5 ft.

2. 3 sides; 1 side 20 in.; height 42 in.

3. A steam boiler, diameter 3 ft.; length 10 ft. Entire surface

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?

4. A water pail, diameter 11 in.; height 15 in. Entire surface?

Surfaces of Pyramids and Cones

Observe: 1. That the convex surface of a pyramid is composed of triangles.

2. That the convex surface of a cone may also be considered as made up of small triangles.

3. That the bases of the triangles in both pyramid and cone form the perimeter of the base of the figure, and the altitude of the triangles the slant height. Hence,

The convex surface of a pyramid or of a cone is found by multiplying the unit of measure by one half the product of the perimeter and the slant height.

Find the convex surface of a pyramid or a cone if :

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1. Diameter of base of cone 9 ft.; slant height = 12 ft. 2. One side of a square pyramid = 16 ft.; slant height 24 ft.

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3. One side of a square pyramid 5 ft.; altitude = 16 ft.

4. Altitude of square pyramid:

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24 ft.; one side 14 ft.

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Find

5. A church spire is in the form of a hexagonal pyramid, each side being 10 feet, and the slant height 65 feet. the cost of painting it at 25¢ per square yard.

cone.

6. A spire on the corner of a church is in the form of a Its base is 12 feet in diameter and its slant height 24 feet. Find the cost of covering it with tin at $13 per square (100 sq. ft. 1 square).

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Examine the solids. What is the height of the cylinder? What is the diameter of the cylinder? What is the diameter of the sphere? How does the diameter of each compare with the height of the cylinder? Observe that the dimensions are equal.

Geometry shows that the surface of a sphere is equal to the convex surface of a cylinder whose height and diameter are each equal to the diameter of the sphere.

To show this, wind a hard wax cord around a cylinder 1 in. in height and 1 in. in diameter until its convex surface is covered. Unwind the cord from the cylinder on to a sphere 1 in. in diameter as shown in the illustration. When one half the surface of the sphere is covered with the cord, one half of the convex surface of the cylinder is uncovered. Hence,

The surface of any sphere equals the convex surface of a cylinder of equal dimensions.

It may also be shown by geometry that

The surface of a sphere equals the square of the diameter multiplied by 3.1416, or d2 (representing the diameter by d and 3.1416 by π).

Find the surface of:

1. A globe, D. 12 in.

2. A ball, R. 1 in.

3. A sphere, D. 13 in.

4. A ball, D. 4 in.

5. How much will it cost to paint a dome in the form of a hemisphere, 20 ft. in diameter, at 25 cents per square yard?

Observe: 1. That the solids are all 4 in. high.

2. That the first row in the rectangular prism contains 4 cu. in.

3. That if the first row in each solid contains 4 cu. in., the volume of each solid is 4 times 4 cu. in., or 16 cu. in.