Mensuration of Surfaces.

RULE.

1st. Square the diameter of each ring, and subtract the square of the less from that of the greater. 2nd. Multiply the difference of the squares by the decimal,7854, and the product will be the area.

2. What is the area of the ring when the diameters of the circles are 20 and 10?

Ans. 235,62.

3. If the diameters are 20 and 15, what will be the area included between the circumferences?

Ans. 137,445.

4. If the diameters are 16 and 10, what will be the

area included between the circumferences?

Ans. 122,5224.

Mensuration of Surfaces.

5. Two diameters are 21,75 and 9,5; required the

area of the circular ring.

Ans. 300,6609.

6. If the two diameters are 4 and 6, what is the area

of the ring.

Ans. 15,708.

PROBLEM XXII.

31. To find the area of an ellipse.

RULE.

Multiply the two axes together, and their product by the decimal 7854, and the result will be the required

2. Required the area of an ellipse whose axes are 24

and 18.

Ans. 339,2928.

QUEST.-31. How do you find the arc of an ellipse?

Mensuration of Solids.

3. What is the area of an ellipse whose axes are 35 and 25?

Ans. 687,225.

4. What is the area of an ellipse whose axes are 80 and 60?

Ans. 3769,92.

5. What is the area of an ellipse whose axes are 50 and 45?

Ans. 1767,15.

SECTION II.

OF THE MENSURATION OF SOLIDS.

1. The mensuration of solids is divided into two parts.

1st. The mensuration of the surfaces of solids and

2dly. The mensuration of their solidities.

We have already seen that the unit of measure for plane surfaces, is a square whose side is the unit of length. (See Part I. § V).

QUEST.-1. Into how many parts is the mensuration of solids divided? What is the unit of measure for plane surfaces?

Mensuration of Solids.

2. A curve line which is expressed by numbers is also referred to an unit of length, and its numerical value is the number of times which the line contains the unit.

If then, we suppose the linear unit to be reduced to a straight line, and a square constructed on this line, this square will be the unit of measure for curved surfaces.

3. The unit of solidity is a cube, whose edge is the unit in which the linear dimensions of the solid are expressed; and the face of this cube is the superficial unit in which the surface of the solid is estimated. Part I. X. Art. 20).

4. The following is a table of solid measure.

(See

QUEST.-2. How is a curve line expressed by numbers? If we suppose the linear unit to be reduced to a straight line, and a square constructed on this line, what is this square? 3. What is the unit of solidity? 4. Re peat the table of solid measure?

Mensuration of Solids.

OF POLYEDRONS, OR SOLIDS BOUNDED BY PLANES.

PROBLEM I.

5. To find the surface of a right prism.

RULE.

1. Multiply the perimeter of the base by the altitude and the product will be the convex surface: and to this add the area of the bases when the entire surface is required.

EXAMPLES.

1. Find the entire surface of the F regular prism whose base is the regular polygon ABCDE and al

titude AF-when each side of the base is 20 feet and the altitude AF, A 50 feet.

AB+BC+CD+DE+EA=100; and AF-50: then (AB+BC+CD+DE+EA) AF=convex surface

becomes

convex surface.

100 × 50=5000 square feet which is the

X

For the area of the end, we have

QUEST.-5. How do you find the surface of a right prism?