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From half the sum of the three sides subtract each side separately; 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. ?

6

4

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= 1 ABXBC= X 4.
The area of BCD IDC XBC= X 4.
Adding, the area of the trapezoid = } (8+6) 4=28 sq. in.
From this explanation we derive the

8

RULE. To find the area of a trapezoid, multiply the sum of the bases by half the altitude.

WRITTEN EXERCISES. 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.

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A

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 ADBE 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, D E.

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.

Hence the

E

RULE.

I. To find the circumference of a circle, multiply the diameter by 3.1416.

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 a, called pi. circumference

1 The diameter =

= circumference X 3.1416

3.1416 1 But

.3183. Hence 3.1416

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:

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WRITTEN EXERCISES. Find the circumference when the1. Diameter is 10 in.

3. Diameter is 305 ch. 2. Diameter is 40 rd.

4. Diameter is 456 ft.

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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 ?

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219. A circle may be regarded as made up of a large number of triangles, the sum of whose bases forms the circumference of the circle, and whose altitude is the radius of the circle. Hence the

RULE. I. To find the area of a circle, multiply the circumference by half the radius.

Area = Circumference X Radius. Since Circumference = 2 X Radius X N, Area = 2 X Radius X 7 X Radius, or,

Area = r X Radius', or a RP. Hence,

II. To find the area of a circle, multiply the square of the radius by 3.1416.

WRITTEN EXERCISES. Find the area of a circle whose1. Radius is 10 ft.

3. Circumference is 18 ch. 2. Diameter is 40 rd.

4. Radius is 3 ft. 6 in.

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5. A horse is tied to a stake by a rope 18 ft. long; over how many square yards can he graze?

6. A has a garden that is 40 rods square, and B has a circular garden 40 rods in diameter; what is the difference in area ?

7. F has a circular fish-pond 30 yards in diameter: he makes a circular gravel walk around it 6 ft. wide; what is the area of the walk ?

8. There is a circular farm whose diameter is 30 chains; what is it worth, at $55 an acre ?

D

250. A square is inscribed in a circle when each of its angles is in the circumference.

B

WRITTEN EXERCISES. 1. Find the side of the largest square that can be cut out of a circular board 16 in. in diameter. Since A B C is a right triangle,

A B? + BC = A C°.
But A B = BC.
Hence 2 A B2 AC?, or 256.

A B2 or,

AB = V128 11.361+ in., Ans. Hence the

128,

RULE. To find the side of the largest square that can be cut from a circle, divide the square of the diameter by 2 and extract the

square root.

2. Find the side of a square that can be cut from a circle 80 inches in diameter.

3. Find the side of a square field that can be made from a circular field 40 rd. in radius.

4. How large a square can be cut out of a circular yard 100 ft. in circumference ?

5. Find the difference between the area of a circular field 60 rd. in circumference and that of the largest square field that can be cut from it.

6. The radius of a circle is 1.5 inches; what is the side of the inscribed square?

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