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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

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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

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2. That the perimeter of the solid forms

C

B

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.

[blocks in formation]

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 :

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.

=

=

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

4. Altitude of square pyramid=24 ft.; one side = 14 ft. 5. A church spire is in the form of a hexagonal pyramid, each side being 10 feet, and the slant height 65 feet. Find 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).

Comparative Surfaces of Cylinder and Sphere

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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?

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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.

The volume of a prism or of a cylinder is found by multiplying the unit of measure by the product of the numbers corresponding to the area of the base and the altitude.

Find the volume of :

1. A prism 4 inches square; altitude 8 inches.

2. A square prism, side 12 in.; altitude 24 in.

3. A hexagonal silo is 25 ft. high, 12 ft. on a side, and 10.3 ft. from the middle point of a side (measuring at the base) to the center of the base. Estimating 90 cu. ft. to a ton of ensilage, how many tons will the silo contain?

4. In the rotunda of a building there are 6 cylindrical marble columns, 18 in. in diameter and 18 ft. in height. Estimate the number of cubic feet in all.

PYRAMIDS AND CONES

1. Fill a hollow pyramid with sand. Empty it into a prism having the same base and altitude. How often must the pyramid be filled and emptied to fill the prism? The volume of a pyramid, then, is what part of the volume of the prism?

2. Measure in like manner with a cone the volume of a cylinder having the same dimensions. The volume of the cone is what part of the volume of the cylinder?

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