RULE.-1. Divide the area by .7854, and the square root of the quotient will be the diameter.

2. Divide the area by .07958, and the square root of the quotient will be the circumference.

To derive some of the rules in practical measurements, as has just been seen, involves a considerable knowledge of mathematics. Hence a number of practical rules will be given without any attempt to derive them.

EXAMPLES.

455. 1. The area of a circle is 350 sq. ft. What is its diameter? 2. Find the circumference of a circle whose area is 400 sq. rd. 3. What is the diameter of a circle which contains one acre?

456. To find the Side of a Square equal in area to a given Circle.

RULE.-Multiply the diameter of the circle by .8862, or the circumference by .2821.

The decimal .8862, or, approximately, the square root of .7854 is the side of a square equal in area to a circle whose diameter is 1; and .2821, the square root of .07958+, is the side of a square equal in area to a circle whose circumference is 1.

EXAMPLES.

1. The diameter of a circle is 20 ft. What is the side of a square of equal area?

2. A circular flower bed is 60 ft. in circumference. What must be the side of a square plot of the same area?

457. To find the Side of a Square inscribed in a given Circle.

458. A square is inscribed in a circle when the vertices of its angles are in the circumference.

It is seen that any inscribed square may be divided into two equal right triangles, having the diameter of the circle as a common hypotenuse.

D

A

B

, or AB

RULE.-Divide the square of the diameter of the circle by 2, and

extract the square root.

The side of a regular hexagon inscribed in a circle is equal to the radius of the circle.

EXAMPLES.

459. 1. Find the side of a square that can be cut from a circle 10 in. in diameter.

2. The diameter of a round piece of timber is 18 in. the side of a square piece that can be cut out of it?

THE ELLIPSE.

с

What is

460. An Ellipse is a plane figure bounded by a curved line, the sum of the distances from every point of which to two fixed points is equal to the line drawn through these points and termi- A nated by the curve. The two fixed points

are called the foci.

f

D

f

B

The transverse axis is the line passing through the foci and terminating in the curve; as AB. The conjugate axis is a line perpendicular to the transverse axis passing through the center and terminated by the curve.

461. To find the Area of an Ellipse.

RULE-Multiply half of the two axes together. and that product by 3.1416.

EXAMPLES.

1. Find the area of an ellipse whose axes are 12 ft. and 9 ft. 2. Find the area of an elliptical plot of ground 20 ft. long and 15 ft. wide.

462. To find the Cost of Painting, Plastering, and Kalsomining.

Painting, plastering and kalsomining are commonly estimated by the square yard.

Allowance is sometimes made for openings in the walls, as doors, windows, etc. Half the areas of the doors and windows is often deducted, but there is no established custom.

Laths are 1 in. wide and 4 ft. long. They are placed about in. apart for plastering, and 16 laths are reckoned to a sq. yd.

EXAMPLES.

1. What will it cost to plaster the walls and ceiling of a room 20 ft. long, 15 ft. wide, and 9 ft. high, at 25 a sq. yd., allowing

2. What will it cost to plaster the walls and ceiling of a room 18 ft. long, 14 ft. wide, and 9 ft. high, having 2 doors, 6 ft. by 3 ft., and 4 windows, 6 ft. by 3 ft., at 20o a sq. yd., deducting half the area of the doors and windows?

3. How much will it cost to kalsomine the walls and ceiling of a room 15 ft. long, 14 ft. wide, and 81 ft. high, with 2 coats, at 6o a sq. yd. each?

4. What will it cost to paint a barn 40 ft. long, 30 ft. wide, 18 ft. to the eaves, the gables being 10 ft. high, with 2 coats, at 79 a sq. yd. each?

463. To find the Cost of Roofing, Paving, etc.

Paving is usually computed by the square foot or square yard. Roofing is generally estimated by the square, which consists of 100 square feet.

Shingles are estimated by the thousand.

Shingles that average 4 in. in width and are laid 5 in. to the weather, 720 shingles to the square; 5 in. to the weather, 655; 6 in. to the weather, 600; 7 in. to the weather, 515; and 8 in. to the weather will require 450 shingles to the square. Making due allowance for waste and defects, it is customary, however, to count 1000 shingles to the square.

About 5 lbs. of nails are required for each 1000 of shingles.

EXAMPLES.

1. What will be the cost of slating a roof 36 ft. long, each side being 14 ft. from eaves to ridge, at $8.25 per square?

Solution: The area = 36 X 14 X 2 = 1008 sq. ft., or 10.08 squares. Cost= 10.08 $8.25 = $83.16.

2. How much will it cost to pave a sidewalk 70 ft. long and 74 ft. wide, at $2.12 per sq. yd?

3. A roof is covered with shingles 4 in. wide, put 7 in. to the weather. What is the cost at $9 a thousand if the roof is 50 ft. long and each side is 19 ft. wide?

4. A barn is covered with shingles 4 in. wide, put 6 in. to the weather. What is the cost at $10 a thousand if the roof is 70 ft. long, each side being 30 ft., and the first course along the eaves being doubled?

464. To find the Cost of Carpeting.

In carpeting, the width of the carpet, the allowance for matching the figures, and whether the strips are to run lengthwise or crosswise, must all be considered.

1. Carpets are usually from yd. to 1 yd. in width; but matting, oilcloth, etc., are often much wider.

2. In matching figures it is often necessary to cut off or turn under one of the ends. To fit a room, if an exact number of strips is too wide, a part of one breadth is turned under.

RULE.-Multiply the number of yards in a strip, by the number of strips required, and this product by the price per yard.

If more convenient, the feet in length may first be multiplied by the number of strips and then reduced to lineal yards.

EXAMPLES.

1. What will it cost to carpet a room 20 ft. long and 17 ft. wide, with carpet 1 yd. wide, running lengthwise at 60% a yard?

Solution: Since the room is 17 ft. wide, it will require 6 strips. 20 ft. X 6 = 120 ft., or 40 yds. Cost 40 X $.60

=

=

$24.

2. What will it cost to carpet a room 18 ft. by 15 ft. with carpet yd. wide, at 759 per lineal yard?

3. Find the cost of a carpet 30 in. wide, at 85 per lineal yard, for a room 15 ft. long and 14 ft. wide, if the strips run lengthwise. Find the cost, if the strips run across the room.

4. How many yards of carpet, & yd. wide, will it take to carpet a room 22 ft., by 18 ft., the strips running lengthwise, allowing 6 in. waste in each strip for matching?

5. What will it cost to carpet a room 15 ft. 4 in. long by 13 ft. 9 in. wide, with carpet 30 in. wide, running lengthwise, at $1.25 per lineal yard, allowing 8 in. waste in each strip for matching?

465. To find the Cost of Papering.

Wall paper is sold by the roll, and in estimates a part of a roll is considered a whole roll.

A roll of paper is 8 yd. long and 18 in. wide.

1. Imported wall papers differ in width and the length of the roll.

2. Wall paper is often put up in double rolls, 16 yd. long. Double rolls are counted as 2 rolls each.

3. Borders and friezes are sold by the yard. They vary in width from 3 in. upward.

4. On account of waste, it is rarely possible to find the exact cost of papering a room.

RULE.-Measure the entire distance around the room in yards, and double the number of yards for the number of strips.

Then divide the number of strips required for the room by the number of strips that can be cut from each roll. The quotient will be the number of rolls.

EXAMPLES.

1. How many rolls of paper will cover the walls of a room 18 ft. by 15 ft., and 7 ft. 8 in. high?

=

2 (1815) 66 ft., or 22 yd. = 44 strips. 24 ÷ 73 3 strips.

44 ÷ 3

15 rolls.

EXPLANATION.

The distance around the room is 66 ft., or 22 yd. The number of strips equal 2 X 22 or 44. The number of full strips from a roll of 8 yd., or 24 ft. are 3. Hence, the number of whole rolls must be 44 ÷ 3 or 15 rolls.

2. What will the paper cost at 15o a roll, to paper the walls of a room 16 ft. by 14 ft.. and 8 ft. high, deducting 2 rolls for doors and windows?

3. How much will the paper cost for a room 21 ft. by 17 ft., 10 ft. high, at 18 a roll, making a reduction of 3 rolls for doors and windows?

4. What will the paper cost for a room 20 ft. long, 18 ft. wide, and 11 ft. high, at 20o a roll, deducting 4 rolls for doors and windows?

5. What will it cost to paper a room 25 ft. long, 20 ft. wide, and 11 ft. 6 in. high, with mop-board 12 in. wide, at 25¢ a roll, having also a border 18 in. wide, at 159 a yard, if the work can be done by 2 men in 1 day at $1.75 each per day?

6. How much will it cost to paper the walls and ceiling of a room 20 ft. long, 18 ft. wide, and 10 ft. 6 in. high, at 20o a roll, having also a border 15 in. wide, at 15 a yard, making a deduction of 4 rolls for doors, windows, etc., and paying $2.50 for the work?