59. What is the cube root of 6 to the nearest hundredth ?

60. What is the cube root of 450 to the nearest tenth? 61. What is the cube root of 0.07 to the nearest thousandth?

449. Miscellaneous Written Examples.

62. What is the 5th power of 15?

63. What is the 3d power of 5?

64. What is the square root of 85 to the nearest hundredth ? 65. What is the square root of 8 to the nearest thousandth? 66. What is the square root of

5

41

67. What is the square root of

to the nearest hundredth?

71

What is the value of each of the following expressions carried

MENSURATION.

ANGLES.

450. An Angle is the difference in direction of two lines.

If the lines meet, the point of meeting, B, is called the vertex; and the lines A B, BC, the sides of the angle.

451. If a straight line meets another so as to make the adjacent angles equal, each of these angles is a right angle; and the two lines are perpendicular to each other. A

Thus, ACD and DC B, being equal, are

C

B

right angles, and AB and D C are perpendicular to each other. A right angle contains 90°. (Art. 261.)

452. An Acute Angle is less than

E

a right angle; as ECB.

453. An Obtuse Angle is greater A

than a right angle; as ACE

Acute and obtuse angles are called oblique angles.

PLANE FIGURES.

454. A Plane Figure is a portion of a plane bounded by lines either straight or curved.

When the bounding lines are straight, the figure is a polygon, and the sum of the bounding lines is the perimeter.

455. Of Polygons, the simplest has three sides, and is called a triangle; one of four sides is called a quadrilateral ; one of five, a pentagon; one of six, a hexagon; one of eight, an octagon; one of ten, a decagon.

456. An Equilateral Polygon is one whose sides are equal each to each.

457. An Equiangular Polygon is one whose angles are equal each to each.

458. A Regular Polygon is one that is both equilateral and equiangular.

TRIANGLES.

459. A Triangle is a plane figure which is bounded by three lines; as ABC.

460. The base of a triangle (or of any other figure) is the side on which it is A supposed to stand; as A C

B

461. The altitude of a triangle is the perpendicular distance from the vertex of the angle opposite the base to the base, or to the base produced; as BD.

462. The area of a triangle is equal to half the product of its base and altitude.

NOTE. For demonstration of the principles of Mensuration, Geometry must be consulted.

1. The base of a triangle is 25 inches and the altitude 12 inches. What is the area? Ans. 150 sq. in.

2. The base is 18 ft. and the altitude 14 ft. What is the area?

3. The base is 10 yards and the altitude 15 yards. What is the area?

463. To find the area of a triangle when only the three sides are given,

From half the sum of the three sides subtract successively the three sides; find the square root of the product of these three remainders and the half sum.

4. Find the area of a triangle whose sides are respectively 5, 13, and 14 yards.

5+13 +14
2

= 16.

(16-5)x(16-13) × (16-14) × 16 =

V11 x 3 x 2 x 16 = √1056 = 32.5+ Ans. 32.5 sq. yd.

5. Find the area of a triangle whose sides are, respectively, 45, 55, and 60 feet.

6. What is the area of a triangle whose sides are, respectively, 117, 221, and 250 rods?

QUADRILATERALS.

464. Parallel Lines are such as have the Asame direction; as A B and C D.

465. A Trapezium is a quadrilateral which has no two of its sides parallel; as ABCD.

466. A Diagonal is a line joining the vertices of two angles of a polygon not adjacent; as DB.

D

B

D

B

467. A Trapezoid is a quadrilateral which has only two of its sides parallel; as EFGH.

468. The Altitude of a trapezoid is H the perpendicular distance between its parallel sides; as OP.

469. A Parallelogram is a quadrilateral whose opposite sides are parallel; as IKLM.

470. The Altitude of a parallelogram is the perpendicular distance from the opposite side to the base; as K N.

471. The area of a parallelogram is equal to the product of its base and altitude.

NOTE. For the Rectangle see Arts. 197-201.

7. What is the area of a parallelogram whose base is 25 yards and altitude 8 yards?

472. The area of a trapezoid is equal to half the product of its altitude and the sum of its parallel sides.

8. The parallel sides of a trapezoid are 13 and 19 feet, and its altitude is 9 feet; what is its area? Ans. 144 sq. ft.

9. How many square feet can be covered with a board whose length is 12 feet, the wider end being 2 feet and the narrower 20 inches in width?

473. The area of a trapezium can be found by finding the sum of the areas of the two triangles into which the trapezium is divided by a diagonal.