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Ex. 3. What is the length of an arc of 25°, in a circle whose radius is 44 rods ?

Ans., 19.198 rods. Ex. 4. What is the length of an arc of 11° 15', in a circle whose diameter is 1234 feet?

Ans., 121.147 feet. (97.) If the number of degrees in an arc is not given, it may be computed from the radius of the circle,

D and either the chord or height of the arc.

A

B Thus, let AB be the chord, and DE the height of the arc ADB, and C the center

C
of the circle. Then, in the right-angled tri-
angle ACE,

AE: sin. ACE,
AC : R::

CE : cos. ACE, either of which proportions will give the number of degrees in half the arc.

If only the chord and height of the arc are given, the diameter of the circle may be found. For, by Geometry, Prop. 22, Cor., B. IV.,

DE : AE :: AE : EF.

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F

Ex. 5. What is the length of an arc whose chord is 6 feet, in a circle whoso radius is 9 feet?

Ans., 6.117 feet.

PROBLEM IX. (98.) To find the area of a circle.

RULE I.

Multiply the circumference by half the radius.
For demonstration, see Geometry, Prop. 12, B. VI.

RULE II.

Multiply the square of the radius by 3.14159.
See Geometry, Prop 13, Cor. 3, B. VI.

Ex. 1. What is the area of a circle whose diameter is 18 feet?

Ans., 254.469 feet.

Ex. 2. What is the area of a circle whose circumference is 774 feet?

Ans., 435.766 feet. Ex. 3. What is the area of a circle whose radius is 125

yards?

Ans., 49087.38 yards.

PROBLEM X. (99.) To find the area of a sector of a circle.

RULE I.

Multiply the arc of the sector by half its radius.
See Geometry, Prop. 12, Cor., B. VI.

RULE II.

As 360 is to the number of degrees in the arc, so is the area of the circle to the area of the sector.

This follows from Geometry, Prop. 14, Cor. 2, B. III.

Ex. 1. What is the area of a sector whose arc is 22°, in a circle whose diameter is 125 feet?

The length of the arc is found to be 23.998.
Hence the area of the sector is 749.937.

Ex. 2. What is the area of a sector whose arc is 25°, in a circle whose radius is 44 rods?

Ans., 422.367 rods. Ex. 3. What is the area of a sector less than a semicircle. whose chord is 6 feet, in a circle whose radius is 9 feet?

Ans., 27.522 feet.

PROBLEM XI.

(100.) To find the area of a segment of a circle.

RULE.

Find the area of the sector which has the same arc, and also the area of the triangle formed by the chord of the segment and the radii of the sector.

Then take the sum of these areas if the segment is greater than a semicircle, but take their difference if it is less.

It is obvious that the segment AEB is

D equal to the sum of the sector ACBE and the triangle ACB, and that the segment A

B ADB is equal to the difference between the sector ACBD and the triangle ACB.

H Ex. 1. What is the area of a segment whose arc contains 280°, in a circle whose

E diameter is 50 ? The whole circle

1963.495 The sector

1527.163 The triangle

307.752 The segment

1834.915, Ans. Ex. 2. What is the area of a segment whose chord is 20 feet, and height 2 feet?

Ans., 26.8788 feet. Ex. 3. What is the area of a segment whose arc is 25°, a circle whose radius is 44 rods?

Ans. (101.) The area of the zone ABHG, included between two parallel chords, is equal to the difference between the segments GDII and ADB.

Ex. 4. What is the area of a zone, one side of which is 96, and the other side 60, and the distance between them 26 ?

Ans., 2136.7527. The radius of the circle in this example will be found to be 50.

in

PROBLEM XII.

(102.) To find the area of a ring included between the cir cumferences of two concentric circles.

RULE.

Take the difference between the areas of the two circles; or, Subtract the square of the less radius from the square of the greater, and multiply their difference by 3.14159. For, according to Geometry, Prop. 13, Cor. 3, B. VI.,

the area of the greater circle is equal to 7 R',

and the area of the smaller, Their difference, or the area of the ring, is a (R?—?).

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Ex. 1. The diameters of two concentric circles are 60 and 50. What is the area of the ring included between their circumferences ?

Ans., 863.938. Ex. 2. The diameters of two concentric circles are 320 and 280 What is the area of the ring included between their circumferences ?

Ans., 18849.55

PROBLEM XIII.

(103.) To find the area of an ellipse.

RULE.

Multiply the product of the semi-axes by 3.14159.
For demonstration, see Geometry, Ellipse, Prop. 21.

Ex. 1. What is the area of an ellipse whose major axis is 70 feet, and minor axis 60 feet?

Ans., 3298.67 feet. Ex. 2. What is the area of an ellipse whose axes are 34 and 310 ?

Ans., 82780.896

PROBLEM XIV.

(104.) To find the area of a parabola.

RULE.
Multiply the base by two thirds of the height.
For demonstration, see Geometry, Parabola, Prop. 12.

Ex. 1. What is the area of a parabola whose base is 18 feet, and height 5 feet?

Ans., 60 feet. Ex. 2. What is the area of a parabola whose base is 525 feet, and height 350 feet?

Ans., 122500 feet

MENSURATION OF SOLIDS.

(105.) The common measuring unit of solids is a cube, whose faces are squares of the same name; as, a cubic inch, a cubic foot, &c. This measuring unit is not, however, of

necessity a cube whose faces are squares of the same name. Thus a bushel may have the form of a cube, but its faces can only be expressed by means of some unit of a different denom. ination. The following is

The Table of Solid Measure.
1728 cubic inches = 1 cubic foot.

27 cubic feet = 1 cubic yard.
44921 cubic feet 1 cubic rod.
231 cubic inches 1 gallon (liquid measure).
268.8 cubic inches 1 gallon (dry measure).
2150.4 cubic inches = 1 bushel.

PROBLEM I.

(106.) To find the surface of a right prism.

RULE.

Multiply the perimeter of the base by the altitude for the convex surface. To this add the areas of the two ends when the entire surface is required.

See Geometry, Prop. 1, B. VIII.

Ex. 1. What is the entire surface of a parallelopiped whose altitude is 20 feet, breadth 4 feet, and depth 2 feet?

Ans., 256 square feet. Ex. 2. What is the entire surface of a pentagonal prism whose altitude is 25 feet 6 inches, and each side of its base 3 feet 9 inches ?

Ans., 526.513 square feet. Ex. 3. What is the entire surface of an octagonal prism whose altitude is 12 feet 9 inches, and each side of its base ? feet 5 inches ?

Ans., 302.898 square feet.

PROBLEM II.

(107.) To find the solidity of a prism.

RULE.

Multiply the area of the base by the altitude.
See Geometry, I'rop. 11, B. VIII.

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