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1. What is the area of a regular pentagon, whose sides are each 35 feet, and the perpendicular 24.08 feet?
Ans. 2107 square feet. 2. What is the area of a regular hexagon, whose sides are each 20 feet, and the perpendicular 17.32 feet ?
339. A Circle is a plane figure bounded by a curved line, every part of which is equally distant from a point, called its center.
The Circumference or Periphery of a circle is the line which bounds it.
The Diameter of a circle is a line drawn through the center, and terminated by the circumference; as G H.
340. To find the CIRCUMFERENCE of a circle, the diameter being given.
RULE. -- Multiply the diameter by 3.141592, and the product is the circumference.
NOTE. — 3.141592 is the circumference of a circle whose diameter is 1. (Art. 291.)
1. What is the circumference of a circle, whose diameter is 50 feet ?
Ans. 157.0796+ feet. 2. A gentleman has a circular garden whose diameter is 100 rods; what is the length of the fence necessary to enclose it?
Ans. 314.15+ rods. 341. To find the DIAMETER of a circle, the circumference being given.
RULE. — Multiply the circumference by .318309, and the product will be the diameter.
NOTE.—.318309 is the diameter of a circle whose circumference is 1. (Art. 291.)
1. What is the diameter of a circle, whose circumference is 80 miles ?
Ans. 25.46+ miles. 2. If the circumference of a wheel is 62.84 feet, what is the diameter ?
Ans. 20+ feet.
337. What is the perimeter of a polygon ? — 338. The rule for finding the area of a regular polygon?
- 339. 'What is a circle? The circumference of a circle ? The diameter of a circle ? - 340. The rule for finding the circumference of a circle, the diameter being given ? - 341. The rule for finding the diameter of a circle, the circumference being given ?
342. To find the AREA of a circle, the diameter, the circumference, or both, being given.
RULE 1. Multiply half the diameter by half the circumference, and the product is the area.
RULE 2. Multiply the square of the diameter by .785398, and the product is the area.
Note. — .785398, or 4 of 3.141592, is the area of a circle whose diameter is 1. (Art. 291.) 1. If the diameter of a circle be 200 feet, what is the area?
Ans. 31415.92 square feet. 2. There is a certain farm, in the form of a circle, whose circumference is 400 rods ; how many acres does it contain ?
Ans. 79A. 2R. 12+p. 343. To find the side of a square equal in area to a given circle.
The square in the figure is supposed to have the same area as the circle.
RULE. Multiply the diameter by .886227, and the product is the side of an equal square.
Note. — .886227, or the square root of .785398, is the side of a square which is equivalent to a circle whose diameter is 1. (Art. 292.)
1. We have a round field 40 rods in diameter; what is the side of a square field that will contain the same quantity ?
Ans. 35.44+ rods. 2. I have a circular field 100 rods in circumference; what must be the side of a square field that shall contain the same
Ans. 28.2+ rods.
344. To find the SIDE of a square inscribed in a given circle.
A square is said to be inscribed in a circle when the vertices of its angles are in the circumference.
342. The rule for finding the area of a circle, when the diameter is given ? When the circumference is given ? When the diameter and circumference are both given ? - 343. The rule for finding the side of a square equal in area to a given circle ? — 344. When is a square said to be inscribed in a circle ?
RULE. — Multiply the diameter by .707106, and the product is the side ‘of the square inscribed.
NOTE. —.707106 is the side of the inscribed square, when the diameter of the circumscribed circle is l. (Art. 292.)
1. What is the thickness of a square stick of timber that may be hewn from a log 30 inches in diameter ?
Ans. 21.21+ inches. 2. How large a square field may be inscribed in a circle whose circumference is 100 rods?
Ans. 22.5+ rods square.
THE ELLIPSE. 345. An Ellipse is an oval figure having two diameters, or axes, the longer of which is called the transverse and the shorter the conjugate diameter.
346. To find the AREA of an ellipse.
RULE. - Multiply the two diameters together, and their product by .785398; the last product is the area.
1. What is the area of an ellipse whose transverse diameter is 14 inches, and its conjugate diameter 10 inches ?
Ans. 109.95+ square inches. 2. What is the area of an elliptical table, 8 feet long and 5 feet wide?
Ans. 31 square feet, 59+ square inches.
MENSURATION OF SOLIDS.
347. A Solid, or Body, is that which has length, breadth, and thickness.
Mensuration of solids includes two operations : first, to find their superficial contents, and second, their solidity or volume.
348. A Prism is a solid whose ends are any plane figures which are equal, similar, and parallel to each other, and whose sides are parallelograms.
344. The rule for finding the side of a square inscribed in a circle ? — 345. What is an ellipse? What is the longer diameter called ? The shorter ? 346. The rule for finding the area of an ellipse? - 347. What is a solid ? What two operations does mensuration of solids include ? 348. What is a prism?
It takes particular names, according to the figure of its base or ends, namely, triangular prism, square prism, pentagonal prism, &c.
The Base of a prism is either end; and of solids in general, the part upon which they are supposed to stand.
All prisms whose bases are parallelograms are comprehended under the general name Parallelopipedons or Parallelepipeds.
A Triangular Prism is one whose base is a triangle.
A Square Prism is one whose base is a square, and when all the sides are squares it is called a cube.
A Pentagonal Prism is one whose base is a pentagon.
349. To find the surface of a prism.
ROLE. -- Multiply the perimeter of its base by its hight, and to this product add the area of the two ends; the sum is the area of the prism.
1. What are the superficial contents of a triangular prism, the width of whose side is 3 feet, and its length 15 feet?
Ans. 142.79+ square feet. 2. What is the surface of a square prism, whose side is I feet wide, and its length 25 feet?
Ans. 1062 square feet. 350. To find the solidity of a prism.
RULE. Multiply the area of the base by the hight, and the product is the solidity.
348. What particular names does the prism take? What is the base of a prisin and of solids in general? What is a parallelopiped or parallelopipe. don? What is a triangular prism ? A square prism ? Á pentagonal prism ? - 349. The rule for finding the surface of a prism ? — 350. The rule for finding the solidity of a prism ?
1. What are the contents of a triangular prism, whose length is 20 feet, and the three sides of its triangular end or base 5, 4, and 3 feet?
Ans. 120 cubic feet. 2. How many cubic feet are there in a cube, whose sides are 8 feet?
Ans. 512 cubic feet. 3. What is the number of cubic feet in a room 30 feet long, 20 feet wide, and 10 feet high?
Ans. 6000 cubic feet,
THE CYLINDER. 351. A Cylinder is a round body, of uniform diameter, with circular ends or bases parallel to each other.
The Axis of a cylinder is a straight line drawn through it, from the center of one end to the center of the other.
352. To find the surface of a cylinder.
RULE. — Multiply the circumference of the base by the altitude, and to the product add the areas of the two ends; the sum will be the whole surface.
1. What is the surface of a cylinder, whose length is 4 feet, and the circumference 3 feet ?
Ans. 13.43+ square feet. 2. John Snow has a roller 12 feet long and 2 feet in diameter; what is its convex surface ?
Ans. 75.39+ square feet. 353. To find the solidity, or volume, of a cylinder.
RULE. - Multiply the area of the base by the altitude, and the product will be the solidity or volume.
1. What is the solidity of a cylinder 8 feet in length and 2 feet in diameter ?
Ans. 25.13+ cubic feet. 2. What is the solidity of a cylinder, whose diameter is 5 feet, and its altitude 20 feet?
Ans. 392.69+ cubic feet. THE PYRAMID AND CONE. 354. A Pyramid is a solid, standing on a triangular, square, or polygonal base, with its sides tapering uniformly to a point at the top, called the vertex.
The Slant Hight of a pyramid is a line drawn from the vertex to the middle of one of the sides of the base.
351. What is a cylinder? What is the axis of a cylinder ? - 352. The rule for finding the surface of a cylinder ? — 353. The rule for finding the solidity of a cylinder ? 354. What is a pyramid ? The slant hight of a pyramid ?