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Ex. 1. What is the area of a regular pentagon, of which the side is 250 feet, and the perpendicular from the centre to one side 172.05 feet?

Ans. 107531.25 sq. ft. 2. What is the area of a regular hexagon whose side is 356 yards, and whose perpendicular is 308.305 yards? Ans. 329269.74 yd.

3. The side of a regular octagonal enclosure is 60 yards; how many acres are included ? Ans. 3 acres 2 roods 14 rods 19 yards.

4. The side of a field, whose shape is that of a regular decagon, is 243 feet; what is its area?

Ans. 10 acres 1 rood 28 rods 24 yards 6.347 feet.

CIRCLES.

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627. A circle is a plane figure bounded by a line, every part of which is equally distant from a point within called the center; as A EFGBD.

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The circumference or periphery of a circle is the line that bounds it.

A radius is a line drawn from the center to the circumference; as CA, or C D.

A diameter is a line which passes through the center, and is terminated by the circumference; as A B.

An arc is any portion of the circumference; as AD, A E, or EGF.

The chord of an arc is the straight line joining its extremities; as EF, which is the chord of the arc EGF.

628. The segment of a circle is the portion included by an arc and its chord; as the space included by the arc EGF and the chord E F.

629. The sector of a circle is the portion included by two radii and the intercepted arc; as the space A CD A.

630. A zone is the space between two parallel chords of a circle ; as the space

AEFBA.

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632. A circular ring is the space included between the circumference of two concentric A cireles; as the space between the rings A B and CD.

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633. To find the circumference of a circle, the diameter being given.

Multiply the diameter by 3.141592.

Ex. 1. If the diameter of a circle is 144 feet, what is the circumference ?

Ans. 452.389248 feet. 2. If the diameter of the earth is 7964 miles, what is its circumference ?

Ans. 25019.6386887- miles. 3. Required the circumference of a circle, whose radius is 512 feet.

Ans. 4 furlongs 34 rods 5 yards 1 foot. 634. To find the diameter of a circle, the circumference being given.

Multiply the circumference by .318309.

Ex. 1. Required the diameter of a circle, whose circumference is 1043 feet.

Ans. 331.997+ feet. 2. If the circumference of a circle is 25000 miles, what is its diameter ?

Ans. 7957.74+ miles. 3. If the circumference of a round stick of timber is 50 inches, what is its diameter ?

Ans. 15.91519+ inches. 635. To find the area of a circle, the diameter, or the circum, ference, or both, being given.

Multiply the square of the diameter by .785398. Or,
Multiply the square of the circumference by .079577. Or,
Multiply half the diameter by half the circumference.
Ex. 1. If the diameter of a circle is 761 feet, what is the area ?

Ans. 454840.475158 feet. 2. There is a circular island, three miles in diameter; how many acres does it contain ?

Ans. 4523.89+ acres. 3. Required the area of a circle, of which the circumference is 1284 yards.

Ans. 27 acres 17 rods 0.8+ yards. 4. Required the area of a circle, of which the diameter is 169, and the circumference 532 inches. Ans. 17 yards 3 feet 13 inches.

636. To find the area of a sector of a circle. Multiply the length of the arc by half the radius of the circle. Or,

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

Ex. 1. Required the area of a sector, of which the arc is 79 and the radius of the circle 47 inches.

Ans. 1856.5 inches. 2. Required the area of a sector, of which the arc is 26°, and the radius of the circle 25 feet.

Ans. 141.8 square feet. 637. To find the area of the segment of a circle.

Find the area of the sector which has the same arc with the segment ; and also the area of the triangle formed by the chord and the radi drawn to its extremities. The difference of these areas, when the segment is less, and their sum, when the segment is greater, than the semicircle, will be the area of the segment. Or,

To two thirds of the product of the height of the segment by the chord add the cube of the height, divided by twice the chord.

B

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Ex. 1. Required the area of the segment ABC A, of which the arc ABC is 49.25°, the chord AC 10 feet, and the radii E A, E B, and EC, each 12 feet.

ft.

Ans. 7.35 sq.

E

2. Required the area of a segment whose height is 15 rods and whose chord is 24 rods. Ans. 1 acre 3 roods 30 rods 9.4 yards.

638. To find the area of a zone of a circle.

From the area of the whole circle subtract the areas of the segments on the sides of the zone.

Ex. 1. Required the area of a zone whose parallel sides are 23.25 and 20.8 feet, in a circle whose radius is 12 feet. Ans. 206+ sq. ft.

2. Required the area of a zone included between two chords of 16 feet each, the diameter of the circle being 20 feet. Ans. 224.7 sq. ft.

639. To find the area of a lune or crescent.

Find the difference of the areas of the two segments formed by the arcs of the lune and its chord.

Ex. 1. If the chord of two intersecting arcs is 72 feet, and the height of one of the segments is 30, and of the other 20 feet, what is the area of the crescent ?

Ans. 612 sq. ft. 640. To find the area of a circular ring. Multiply the sum of the diameters of the two circles by the difference of the diameters, and that product by .7854.

Ex. 1. What is the area of the ring formed by two circles whose diameters are 10 and 20 yards ?

Ans. 235.62 sq. yd. 2. In the centre of a circular pond there is an island 128 yards in diameter; what is the area of the pond, provided the exact distance from any part of the outer side of the pond to the center of the island is 784 yards ? Ans. 1 acre 1 rood 14 rods 17 yards 7.4 feet.

641. To find the side of a square that shall equal the area of a circle of a given diameter or circumference.

Multiply the diameter of the circle by .886227. Or,
Multiply the circumference of the circle by .282094.

Ex. 1. I have a round field, 50 rods in diameter; what is the side of a square field that shall contain the same area?

Ans. 44.31135+ rods. 2. I have a circular field 360 rods in circumference; what must be the side of a square field that shall contain the same area ?

Ans. 101.55+ rods. 3. John Smith had a farm which was 10,000 rods in circumference, which he sold at $ 71.75 per acre. He purchased another farm con

taining the same quantity of land in the form of a square; required the length of one of its sides.

Ans. 2820.94+ rods. 642. To find the diameter of a circle that shall contain the area of a given square.

Muliiply the side of the given square by 1.12838.

Ex. 1. The side of a square is 44.31135 roos; required the diameter of a circular field containing the same area.

613. To find the side of the largest equilateral triangle that can be inscribed in a circle of a given diameter or circumference.

Multiply the given diameter by .866025. Or,
Multiply the given circumference by .275664.

Ex. 1. There is a certain piece of round timber 30 inches in diameter; required the side of an equilateral triangular beam that may be hewn from it.

Ans. 25.98+ inches. 2. How large an equilateral triangle may be inscribed in a circle whose circumference is 5000 feet ?

Ans. 1378.320 feet. 3. Required the side of an equilateral triangular beam, that may be hewn from a round piece of timber 80 inches in circumference.

Ans. 22.05+ inches. 614. To find the side of the largest square that can be inscribed in a circle of a given diameter or circumference.

Multiply the given diameter by .707106. Or,
Multiply the given circumference by .225079.

NOTE. — To find the circumference of a circle required to exactly admit a square of a given side, divide the given side by .225079.

Ex. 1. I have a piece of timber 30 inches in diameter; how large a square stick can be hewn from it ?

Ans. 21.21+ in. square. 2. Required the side of a square that may be inscribed in a circle 80 feet in diameter.

Ans. 56.56848+ feet. 3. I have a circular field whose circumference is 5000 rods; what is the side of the largest square field that can be made in it ?

Ans. 1125.395+ rods. 4. How large a square stick may be hewn from a piece of round timber 100 inches in circumference ? Ans. 22.5+ inches square.

5. What must be the circumference of a tree that, when hewn, shall be 18 inches square ?

Ans. 79.97+ inches. 6. I have a garden which is 20 rods square; required, in feet, the circumference of a circle that will enclose this garden.

Ans. 1466.15+ feet. 645. To find the diameter of the three largest equal circles that can be inscribed in a circle of a given diameter.

Divide the given diameter by 2.155.

Ex. 1. Required the diameter of each of the largest three circles that can be inscribed in a circle 86.2 inches in diameter.

Ans. 40 inches.

E C

G

H

B

FD

ELLIPSE. 646. An ellipse is a plane figure bounded by a curve, from any point of which the sum of the distances to two fixed points is equal to a given distance. The two fixed points are called the foci, as G,H in the ellipse A ECBDFA. The major or transverse axis of an ellipse is its longest diameter, as A B. The minor or conjugate axis of an ellipse is its shortest diameter, as C D. The segment of an ellipse is a portion cut off from the ellipse, as F A EF.

647. To find the area of an ellipse, the two diameters being given.

Multiply the two diameters together, and that product by .785398.

Ex. 1. What is the area of an ellipse, whose two diameters are 24 and 18 inches ?

Ans. 339.2919 inches. 2. What is the area of an elliptical pond, whose longest diameter is 33 feet 5 inches, and whose shortest diameter 20 feet 3 inches?

Ans. 59 sq. yd. 67 sq. in.

MENSURATION OF SOLIDS.

PRISMS AND CYLINDERS.

A

B

648. A prism is a figure whose ends or bases are any plane figures which are equal and similar, and parallel to each other, and whose sides are parallelograns.

A triangular prism is one whose base is a triangle; as the figure A B.

A square prism is one whose base is a square ; a pentagonal prism, one whose base is a pentagon ; and so on, according to the figure of the ends or bases.

A parallelopiped is a prism whose ends or bases, as well as its sides, are parallelograms.

649. A cylinder is a round body of uniform diameter, and which has circular bases parallel to each other; as the figure C D.

The perimeter of a prism or cylinder is the line that bounds its end or base ; and the altitude or height is the distance between the ends or bases.

The convex surface of a prism or cylinder is the entire surface, exclusive of the two ends or bases.

650. To find the surface of a prism, or of a cylinder.

Multiply the perimeter of the given prism or cylinuler by the height, and to the product add the area of the tuo ends. Ex. 1. Required the surface of a triangular prism, of which the dis

D

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