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8. A rectangular yard 40 ft. long and 30 ft. wide is surrounded by a walk 4 ft. wide; how many square feet in the walk? 9. If a rectangular field is 80 rd. long and 60 rd. wide, how much is the field worth, at $140 an acre?
10. A city lot is 5 rd. wide and 18 rd. long; what is it worth, at $1600 an acre?
11. There is a hollow square whose outer side is 120 rd. and the inner side 40 rd.; how many acres does it contain?
12. A rectangular field is 120 rd. long and 40 rd. wide; how many more acres will a square field of equal perimeter contain?
13. How many more feet in two sides of a square acre than in its diagonal?
14. There is a rectangular field whose sides are 125 rd. and 20 rd., respectively; what is the side of a square field of equal area?
245. The Parallelogram.
If the rectangle (Fig. 1) be divided by the line A E, and the triangle ADE be placed at the right of the figure, we shall have a parallelogram of the form ABDE (Fig. 2).
It is evident that the base, the altitude, and the area remain the same.
To find the area of a parallelogram, multiply the base by the altitude.
1. What is the area of a parallelogram whose base is 32 in. and altitude 25 in.?
2. How many acres in a parallelogram whose base is 120 rd. and altitude 15 rd.?
3. How many acres in a farm 320 rd. long and 45 rd. wide? 4. How many acres in a parallelogram 60 ch. long and 45 ch. wide?
246. The Triangle.
If the parallelogram (Fig. 2, ART. 245) be divided by a line drawn from B to E, it will form two equal triangles. It is evident that the width and altitude of each triangle are the same as the width and altitude of the parallelo
gram, but the area of each triangle is only one-half of the area of the parallelogram. Hence the
To find the area of a triangle, multiply the base by half the altitude.
1. Find the area of a triangle whose base is 120 ft. and altitude 60 ft.
2. How many acres in a triangular field whose base is 80 rd. and altitude 60 rd.?
3. How many acres in a triangular field whose base is 40 ch. and altitude 30 ch.?
4. What is the altitude of a triangle whose base is 360 ft. and area 1 acre?
NOTE.-In Geometry it is proved that if the three sides of a triangle are given and not the altitude, the area can be found as follows:
From half the sum of the three sides subtract each side sepa rately; multiply the half sum and the three remainders together, and extract the square root of the product; the result will be the area required.
5. What is the area of a triangle whose sides are, respectively, 6, 8, and 10 ft.?
6. What is the area of a triangle whose sides are each 90 ft.?
247. The Trapezoid.
If the triangle ADE be taken from the parallelogram (Fig. 2, ART. 245), the trapezoid ABCE will remain. The trapezoid ABCD consists of two triangles, ABD and BCD.
The area of ABD
DCX BC =
Adding, the area of the trapezoid = 1 (8+6) 4 = 28 sq. in. From this explanation we derive the
To find the area of a trapezoid, multiply the sum of the bases by half the altitude.
1. Find the area of a trapezoid whose bases are 44 and 36 in. respectively, and altitude 20 in.
2. Find the area in acres of a trapezoid whose bases are 80 and 50 rd. respectively, and altitude 40 rd.
3. The bases of a trapezoid are 32 and 24 ch. respectively, and the altitude is 40 ch.; how many acres in its area?
4. Find the side of a square equal in area to a trapezoid whose bases are 60 and 30 in. respectively, and altitude 20 in.
248. The Circle.
A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within called the centre.
The bounding line ADB E is called the circumference.
The radius is the distance from the centre to the circumference; as, CD, CF, etc.
The diameter is a straight line drawn through the centre and terminating in the circumference; as, A B, DE.
If we take a circle 6 inches in diameter and measure accurately the distance around it, we shall find the circumference to be about 18.8496 inches. Dividing the circumference, 18.8496 in., by the diameter, 6 in., we have a quotient of 3.1416.
I. To find the circumference of a circle, multiply the diameter by 3.1416.
The diameter =
II. To find the diameter of a circle, divide the circumference by 3.1416.
NOTE.-This quotient, 3.1416, is represented by the Greek letter π, called pi.
III. To find the diameter of a circle, multiply the circumference by .3183.
If the circumference be represented by C and the diameter by D, these rules may be briefly expressed, as follows:
С= 2лR; DCX.3183.
C = πD; D
Find the circumference when the
1. Diameter is 10 in. 2. Diameter is 40 rd.
Find the diameter when the
5. Circumference is 50 in. 6. Circumference is 100 ft.
3. Diameter is 305 ch.
7. Circumference is 314.16 rd. 8. Circumference is 6283.2 ch.
9. What is the circumference of the earth, its diameter being 7960 miles?
10. What is the circumference of the planet Jupiter, its diameter being 85000 miles?
11. How often does a car-wheel revolve in an hour, running at the rate of 30 miles an hour, if the wheel is 3 ft. in diameter?
12. What is the radius of a wheel which makes 17600 revolutions in going 40 miles?
13. How much farther would a man travel in going around a circular field 100 yards in diameter than in going around a square field whose area is 5625 square yards?
14. A cart whose wheels are 5 ft. in diameter and 5 ft. apart is turned around, making a complete circle: it is observed that the inner wheel has made one revolution; what is the circumference made by the outer wheel?
15. If in the last problem the inner wheel makes 4 revolutions, what is the circumference made by the outer wheel?