and found AC 36.21,

Aa 4.18

=

=

EC = 39.11, Bb 4, Dd 7.26, = required the area. 296.1292.

Ans.

To find the area of a regular polygon.

101. Let AB, denoted by s, represent one side of а regular polygon whose centre is C. Draw CA and CB, and from C draw CD perpendicular to AB. Then will CD be the apothem, and we shall have AD = BD.

Denote the number of sides of the polygon by n; then will the angle ACB, at the centre, be

(B. V., page 144, D. 2), and the angle ACD,

180°

which is half of ACB, will be equal to n

In the right-angled triangle ADC, we shall have, formula (3), Art. 37, Trig.,

But CAD, being the complement of ACD, we have

a formula by means of which the apothem may be computed.

But the area is equal to the perimeter multiplied by half the apothem (Book V., Prop. VIII.): hence the following

RULE.-Find the apothem, by the preceding formula; multiply the perimeter by half the apothem; the product will be the area required.

MENSURATION

The perimeter is equal to 120: hence, denoting the area

2. What is the area of an octagon, one of whose sides

is 20?

Ans. 1931.37.

The areas of some of the most important of the regular polygons have been computed by the preceding method, on the supposition that each side is equal to 1, and the results are given in the following

The areas of similar polygons are to each other as the squares of their homologous sides (Book IV., Prop. XXVII.).

Denoting the area of a regular polygon whose side is s by Q, and that of a similar polygon whose side is 1 by T, the tabular area, we have

Q: T ::

s2 : 12;

=

Ts2;

hence, the following

RULE.-Multiply the corresponding tabular area by the square of the given side; the product will be the area required.

Examples.

1. What is the area of a regular hexagon, each of whose sides is 20?

We have T = 2.5980762, and s2 400: hence,

2. Find the area of a pentagon, whose side is 25. Ans. 1075.298375.

3. Find the area of a decagon, whose side is 20.

Ans. 3077.68352.

To find the circumference of a circle, when the diameter is

given.

102. From the principle demonstrated in Book V., Prop. XVI., we may write the following

RULE. - Multiply the given diameter by 3.1416: the product will be the circumference required.

Examples.

1. What is the circumference of a circle, whose diameter is 25? Ans. 78.54.

2. If the diameter of the earth is 7921 miles, what is the circumference? 24884.6136.

Ans.

To find the diameter of a circle, when the circumference is

given.

103. From the preceding case, we may write the following

RULE.-Divide the given circumference by 3.1416; the quotient will be the diameter required.

Examples.

1.

What is the diameter of a circle, whose circumference is 11652.1944? Ans. 3709.

2. What is the diameter of a circle, whose circumference is 6850?

Ans. 2180.41.

To find the length of an are containing any number of

degrees.

104. The length of an arc of 1°, in a circle whose diameter is 1, is equal to the circumference, or 3.1416, divided by 360; that is, it is equal to 0.0087266: hence, the length of an arc of n degrees will be nx 0.0087266. To find the length of an arc containing n degrees, when the diameter is d, we employ the principle demonstrated in Book V., Prop. XIII., C. 2: hence, we may write the following

by

RULE-Multiply the number of degrees in the arc .0087266, and the product by the diameter of the circle; the result will be the length required.

Examples.

1. What is the length of an arc of 30 degrees, the diameter being 18 feet? Ans. 4.712364 ft.

2. What is the length of an arc of 12° 10', or 124°, the diameter being 20 feet? Ans. 2.123472 ft.

To find the area of a circle.

105. From the principle demonstrated in Book V., Prop. XV., we may write the following

RULE-Multiply the square of the radius by 3.1416; the product will be the area required;

Examples.

1. Find the area of a circle, whose diameter is 10 and circumference 31.416.

Ans. 78.54.

2. How many square yards in a circle whose diameter is 3 feet?

Ans. 1.069016.

3. What is the area of a circle whose circumference is 12 feet?

Ans. 11.4595.

To find the area of a circular sector.

106. From the principle demonstrated in Book V., Prop. XIV., C. 1 and 2, we may write the following

RULE.-I. Multiply half the length of the arc by the radius; or,