32. The PERIMETER of a figure is the sum of all its sides.

33. A figure all the sides of which are straight lines is called a rectilinear figure.

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34. A CIRCLE is a plane surface, enclosed by a curve line called the circumference, every part of which is equally distant from the centre.

35. The DIAMETER of a circle is a straight line passing through its centre and terminated by the circumference; as, C E.

36. The RADIUS of a circle is a straight line extending from the centre to the circumference of the circle; as, F C, or F E. The point F is the cen

tre.

37. A TANGENT is a straight line which touches the circumference only in one point, but which, when extended, does not cut it; as, GH.

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38. An ARC of a circle is any part of the circumference; as, A B or C E.

39. A CHORD is a straight line joining the extremities of an arc; as, A B.

40. A SEGMENT is any part of a circle bounded by an are and its chord; as the surface included between the chord A B and the arc A B.

41. A SECTOR is any part of a circle bounded by an arc and two radii drawn to its extremities.: as, the surface CDE.

42. A SEMICIRCUMFERENCE is one half the circumference; as, the line F G H.

43. A QUADRANT is one quarter of the circumference; as, the line F G or G H.

44. A SEMICIRCLE is a half of a circle bounded by a diameter and a semicircle; as, the surface FHK.

45. The CIRCUMFERENCE of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Hence, a semicircle contains 180 degrees, and a quadrant, 90 degrees.

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46. AN ELLIPSE is an oval figure, having two diameters or axes. The longer axis is called the transverse, and the shorter the conjugate axis, or diameter.

153. MENSURATION OF SURFACES. (Art. 56.)

The area of a SQUARE is equal to the square of one of its sides; consequently, the side of a square is equal to the square root of its area.

1. What is the area of a square field, each side of which is 40 rods? 15 rods? 20 rods? 18 rods 34 yards? 5 rd. 3 yd. 2 ft.?

2. What is the side of a square field whose area is 900 sq. rods? 15 acres? 25 acres?

3. The area of a circle is 361 sq. ft. How long is a square of equal area?

4. How many sq. ft. in the floor of a square room whose side is 15 ft. 8 in. ?

154. The area of a rectangle is found by multiplying its longer side by the shorter.

5. How many acres in a rectangular lot of land which is 20 rods long and 12 rods wide? What is one side of a square of equal area?

6. How many yards of plastering in the ceiling of a room which is 65 feet long and 38 feet wide?

7. How many feet of boards will it take to cover the four sides of a barn which is 57 feet long, 38 feet wide, and 18 feet high?

155. The area of any parallelogram is found by multiplying the base by the altitude.

156. The area of a trapezoid is found by multiplying the sum of its parallel sides by half the perpendicular distance between them.

8. How many square feet in a board that is 15 ft. long, one end of which is 15 and the other 10 inches wide? If one end is 5 and the other 15 inches wide? If one end is 25 and the other 23 inches wide?

157. A triangle is one half of a parallelogram of the same base and altitude. Therefore,

RULE 1. The area of a triangle is equal to the product of half the base multiplied by the altitude.

9. What is the area of a triangular field, one side of which is 15 rods, and the perpendicular to the corner opposite this side is 12 rods ?

NOTE. Any triangular field or other surface may be measured by this method. Or,

RULE 2. Measure the three sides of the field; add them together, and from half their sum subtract each side separately; then multiply the half sum and the three remainders together; the square root of this product will be the area.

10. The three sides of a triangular field measure 15, 20 and 25 rods respectively. What is the content of the field?

Solution. 15+20+25=30.

=10; 30-25=5.

30-15= 15; 30-20 30 X 15 × 10 × 5— the area. ×

158. QUADRILATERALS AND POLYGONS.

The area of any surface that is bounded by straight lines may be found by dividing it into triangles, and measuring each triangle separately.

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11. Let A B C D E represent a pasture. The side A B measures 20 rods, B C 30, C D 25, D E 15, and E A 25 rods; the diagonal A C 40, and A D 30 rods. What is the area of the field?

12. Required the area of the quadrilateral A B C D, in which the diagonal B D is 133, and the perpendiculars AF 37, and CE 44 yards.

13. If in the above quadrilateral the side A B measures 72, C B 46, C D 125, D A SO, and the diagonal B D 133 feet, what is the area? What would be the cost of such an area at 10 cents per square foot?

159. REGULAR POLYGONS.

Irregular polygons may be measured as in Art. 158, but to find the area of regular polygons,

Multiply half the perimeter by the perpendicular let fall from the centre upon one of its sides.

14. What is the area of a regular pentagon, each side of which is 250 feet, and the perpendicular from the centre 172.05 feet?

15. What is the area of a regular octagon, each side of which is 20 yards, and the perpendicular from the centre 24.14 yards?

160. THE CIRCLE.

The circumference of a circle is about 34, or, more nearly, 3.1416 times its diameter.

To find the area of a circle,

RULE 1. Multiply half the diameter by half the circumference. Or,

2. Multiply the square of the diameter by .7854. Or, 3. Multiply the square of the circumference by .0795775.

16. What is the circumference of a circle whose diameter is 15 inches? 25 in.? 23 ft.? 12 ft.?

17. What is the diameter of a circle whose circumference is 25 ft.? 37 ft.?

18. What is the circumference of a circle whose radius is 30 ft.?

19. What is the area of a circle the diameter of which is 15 inches? 25 in.? 23 ft.? 12 ft.?

20. Find the area of a circle whose circumference is 54 inches.

161. The area being given to find the diameter.

Since the area = (diameter) X.7854, the diameter == area.7854.

RULE. Divide the area by .7854, and take the square root of the quotient.

21. What is the diameter of a circle the area of which is 1 acre? of an acre? 4 acres?

162. To find the area of an ellipse,

Multiply the product of the two diameters by .7854.

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22. What is the area of an ellipse whose diameters are 20 and 24 feet?

23. What is the area of an ellipse the axes of which are 30 and 40 yards?

163. In every right-angled triangle, the square described on the hypothenuse is equal to the sum of the squares described upon the base and perpendicular. If E F G be a right-angled triangle, and right-angled at F, then will the square H, described on the hypothenuse EG, be equal to the sum of the squares I and K, described on the base F G and perpendicular E F. By counting the small squares, we find the number of small squares in the square H to be equal to the number of small squares in the squares I and K.

Hence, the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of both the other sides; and, therefore, the hypothenuse is equal to the square root of the sum of the squares of the other sides; and either side is equal to the square root of the difference between the square of the other side and the square of the hypothenuse.

24. The base of a right-angled triangle is 20 feet, and its height 30 feet. How long is the hypothenuse?

25. What is the distance between the opposite corners of a room 15 feet long and 12 feet wide?

26. The hypothenuse of a right-angled triangle being 75 ft., and the base 25 ft., what is its height?

27. How long is the diagonal of a square whose side is 8 feet?

28. If the diagonal of a square is 54 ft., what is one side of the square?