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. QUESTIONS.

427. What is a power of a number? 428. What is Involution? The first power? 429. What is an index? 430. Give the Rule for involving a number to any power. 432. What is a root? 433. What is Evolution? 434. How is a root indicated? 435. How does evolution compare with involution? 436. What is a perfect power? A perfect square? A perfect cube? 437. Of what number are the roots greater than the number itself? 438. What is meant by finding the square root of a number? 439. How many figures are there in the square of a number? 440. How then do we find the number of figures in the square root of a number? 441. Explain the operation of obtaining the square root of a number. 442. Give the Rule for finding the square root of a number. 443. Give the Rule for finding the square root of a fraction, or mixed number. 444. What is meant by finding the cube root of a number? 445. How many figures are there in the cube of a number? 446. How then can we find the number of figures in the cube root of a number? 447. Explain the operation of finding the cube root of a number. 448. Give the Rule for finding the cube root of a number.

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.

D

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. Thus, ACD and DC B, being equal, are right angles, and A B and D C are perpendicular to each other. A right angle contains 90°. (Art. 261.)

452. An Acute Angle is less than

a right angle; as EC B.

453. An Obtuse Angle is greater A

than a right angle; as ACE.

A

Acute and obtuse angles are called oblique angles.

PLANE FIGURES.

B

C

E

B

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

D

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 B D.

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 centimeters and the altitude 12 centimeters; what is the area? Ans. 150 sq. cm.

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

3. The base is 10 meters and the altitude 15 meters; 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 meters.

OPERATION.

5+13 +14
2

= 16.

(16-5)x(16-13) x (16-14) × 16=

11 x 3 x 2 X 16 = 1056 = 32.5+ Ans. 32.5 sq. m.

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.

B

с

D

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.

B

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

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 KN.

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 meters and altitude 8 meters?

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,