Geometrical Representation of Square and Cube Roots. 553. We will illustrate square root by giving a Geometrical representation of the square root of 1225. The square root of 1225 is 35. The square of (30 + 5) = 302 + 2 (30 × 5) + 52. (§ 537) (§ 536) The 302 may be represented by a square (Fig. 1) 30 in. on a side. The 2 (30 × 5) may be represented by two strips 30 in. long and 5 in. wide of Fig. 2, which are added to two adjacent sides of Fig. 1. FIG. 1. FIG. 2. FIG. 3. The 52 may be represented by the small square of Fig. 3 required to make Fig. 2 a complete square. In extracting the square root of 1225, the large square, which is 30 in. on a side, is first removed, and a surface of 325 sq. in. remains. This surface consists of two equal rectangles, each 30 in. long, and a small square whose side is equal to the width of the rectangles. The width of the rectangles is found by dividing the 325 sq. in. by the sum of their lengths; that is, by 60 in., which gives 5 in. Hence, the entire length of the surfaces added is 30 in. + 30 in. + 5 in. = 65 in., and the width is 5 in. Therefore, the total area is (65 × 5) sq. in. = 325 sq. in. 554. We will illustrate cube root by giving a Geometrical representation of the cube root of 42,875. The cube root of 42,875 is 35. (§ 547) The cube of (30+ 5) = 303 + 3 (302 × 5) + 3 (30 × 52) + 53. (§ 546) The 303 may be represented by a cube whose edge is 30 in. (Fig. 1). The 3 (302 × 5) may be represented by three rectangular solids, each 30 in. long, 30 in. wide, and 5 in. thick, to be added to three adjacent faces of Fig. 1. The 3 (3052) may be represented by three equal rectangular solids, 30 in. long, 5 in. wide, and 5 in. thick, to be added to Fig. 2. The 53 may be represented by the small cube required to complete the cube of Fig. 3. FIG. 1. FIG. 2. FIG. 3. FIG. 4. In extracting the cube root of 42,875, the large cube (Fig. 1), whose edge is 30 in., is first removed. = 15,875 cu. in. There remain (42,875 -27,000) cu. in. The greatest part of this is contained in the three rectangular solids which are added to Fig. 1, and are each 30 in. long and 30 in. wide. The thickness of these solids is found by dividing the 15,875 cu. in. by the sum of the three faces, each of which is 30 in. square; that is, by 2700 sq. in. The result is 5 in. There are also the three rectangular solids which are added to Fig. 2, and which are 30 in. long and 5 in. wide; and a cube which is added to Fig. 3, and which is 5 in. long and 5 in. wide. Hence, the sum of the products of two dimensions of all these solids is For the larger rectangular solids, 3 (30 × 30) sq. in. = For the small cube, 2700 sq. in. 450 sq. in. (5 × 5) sq. in. = 25 sq. in. 3175 sq. in. This number multiplied by the third dimension gives (5 × 3175) CHAPTER XVI. MENSURATION. 555. We have already considered Areas of Rectangles and Circles; and Volumes and Surfaces of Rectangular Solids, Spheres, and Right Cylinders. 556. A polygon is a plane figure bounded by straight lines. A polygon of three sides is a triangle; of four sides, a quadrilateral; of five sides, a pentagon; of six sides, a hexagon; of eight sides, an octagon; of ten sides, a decagon; of twelve sides, a dodecagon; and so on. 557. The area of any polygon may be found by dividing it into triangles and finding the sum of their areas. 558. A vertex of a polygon is the point of intersection of two adjacent sides. 559. A diagonal of a polygon is a straight line joining any two vertices not adjacent. 560. A regular polygon is a polygon with all its sides equal and all its angles equal. The centre of a regular polygon is a point equidistant from the vertices and also equidistant from the sides. The radius of a regular polygon is the distance from the centre to any vertex. The radii of a regular polygon divide the polygon into equal isosceles triangles; that is, into triangles having two sides equal. The apothem of a regular polygon is the distance from the centre to any side. 561. The area of a regular polygon apothem). = (perimeter X 562. The apothem of a regular polygon bears a constant ratio to one side. The following table shows the ratio of the apothem to one side in the most common regular polygons: 563. A trapezium is a quadrilateral with no two of its sides parallel. NOTE. Two lines are parallel if all points of one are equally distant from the other. 564. A trapezoid is a quadrilateral with two of its sides parallel, but the other two sides not parallel. 565. A parallelogram is a quadrilateral with its opposite sides parallel. 566. A rhomboid is a parallelogram with its angles not right angles. 567. A rhombus is a parallelogram with its angles not right angles, but with all its sides equal. Rhomboid. Parallelograms. Rhombus. Square. Rectangle. 568. The altitude of a parallelogram or of a trapezoid is the shortest distance between its parallel sides regarded as bases. 569. The area of any parallelogram = base X altitude. 570. The area of a rhombus also = half the product of its diagonals. 571. The area of a trapezoid tude). Triangles. = (sum of bases X alti 4AA Right. Isosceles. Equilateral. Scalene. 572. A right triangle is a triangle one of whose angles is a right angle. The hypotenuse is the side opposite the right angle, and the other two sides, called legs, are the base and the perpendicular. 573. Other kinds of triangles are, isosceles, with two sides equal; equilateral, with three sides equal; scalene, with no two sides equal. The altitude of a triangle is the shortest distance from the vertex to the base or the base produced. 574. When the base and altitude are given, The area of the triangle = (base × altitude). |