The altitude of a rhomboid or rhombus is the perpendi cular distance between the side taken as the base, and the side opposite. The altitude of a trapezoid is the perpendicular distance between its two parallel sides. The area of a square, rectangle, rhombus or rhomboid, is found by multiplying its base by its altitude. The area of a trapezoid is found by multiplying the sum of its parallel sides by half its altitude. Q. What is a quadrilateral? How many varieties? What is the square? Rectangle? Rhombus? Rhomboid? Trapezoid? What is the altitude of these figures? Draw each of them upon your slate. Point out the base of each. The altitude. How do you find the area of a square? Rectangle? Rhombus? Rhomboid? Trapezoid? EXAMPLES. 1. What is the area of a rectangle, the adjacent sides of which measure 20 feet and 15 feet? 2. What is the area of a trapezoid, whose parallel sides measure respectively 45 feet and 20 feet, and whose altitude is 6 feet? 4. What is the area of a rhombus, whose base is 75 feet, and altitude 17 feet? Ans. 1275 sq. ft. 5. What is the area of a trapezoid, whose parallel sides measure respectively 10 yards and 144 feet, and whose altitude is 70 feet? Ans. 6090 sq. ft. 243. A Circle is a plane figure, bounded by a curve-line, all of whose points are equally distant from a point within, called the centre. The curve ACBD is the circumference; O is the centre. The distance OD from the centre to the circumference is called the radius. A line passing through the centre, and terminated B on both sides in the circumference, is called the diameter. A B is the diameter. The diameter is twice the radius. The circumference of every circle is found by multiplying its diameter by 3.1416. The diameter of every circle is found by dividing its circumference by 3.1416. The area of every circle is found by multiplying the square of its radius by 3.1416. Q. What is a circle? Draw one upon your slate. Which is the circumference? Diameter? Radius? What is the radius equal to ? What is the circumference of a circle equal to? Its diameter? Its area? EXAMPLES. 1. Find the circumference of a circle whose radius is 10 feet. Multiply 3.1416 by twice the radius, since the diameter is twice the radius. OPERATION. 3.1416 20 Ans. 62.8320 feet. 2. Find the diameter of a circle, whose circumference is 1000 feet. 3. Find the area of a circle whose radius is 5 feet. 4. What is the area of a circle whose circumference is 1000 feet? Ans. 8824.754+ sq. yds. 5. Find the area of a circle whose diameter is 200 feet. Ans. 6. Find the area of a circle whose radius is 11 feet. Ans. 244. A Sphere is a solid round body, all the points of whose surface are equally distant from a point within, called the centre. The surface of a sphere is found by multiplying the square of its diameter by 3.1416. The solidity of a sphere is found by multiplying its surface by one-sixth of its diameter. Q. What is a sphere? How is its surface found? Its solidity? EXAMPLES. 1. What is the surface of the sphere whose diameter is 6 feet? 2. What is the solidity of a sphere whose radius is 5 feet? We first find the surface by multi plying the square of the diameter, which is 10, by 3.1416. Then multiplying this result by 10 or of the diameter, the solidity is 523.6 cubic feet. 3. What is the solidity of the sphere whose radius is 15 feet? Ans. 4. What is the surface of the sphere whose radius is 5 feet? Ans. 245. The convex surface of a cylinder is found by multiplying the circumference of its base by its altitude. The solidity of a cylinder is found by multiplying the area of its base by its altitude. Q. What is the surface of a cylinder equal to ? Its solid content? EXAMPLES. 1. Find the surface of a cylinder, the circumference of whose base is 10 feet, and altitude 5 feet. 2. Find the solid content of a cylinder, whose altitude is 15 feet, and the radius of whose base is 2 feet. We first find the area of the base by Art. 243, to be 12.5664 12.5664 × 15=188.496 solidity. square feet. Multiplying this by the altitude 15 feet, the result is 188.496 cubic feet. 3. Find the convex surface of a cylinder, whose altitude is 21 yards, and the radius of whose base is 3 feet. Ans. 131.9472 sq. yds. 4. Find the solidity of a cylinder, whose altitude is 32 feet, and the diameter of whose base is 2 yards. Ans. 33.5104 cub. yds. 5. Find the solidity of a cylinder, whose altitude is 10 yards, and the circumference of whose base is 1247 feet. Ans. 6. Find the convex surface of a cylinder, whose altitude is 12 feet, and the circumference of whose base is 14796 feet. Ans. 246. The convex surface of a cone is found by multiplying the circumference of its base by one-half its slant height.* The solidity of a cone is found by multiplying the area of its base by one-third of the altitude. Q. How is the convex surface of a cone found? The solidity? What is the slant height of a cone? Altitude? EXAMPLES. 1. Find the convex surface of a cone, whose slant height is 50 feet, and the diameter of whose base is 10 feet. We first find the circumfer ence of the base by Art. 243, OPERATION. 3.1416×10-31.416 circle of base. convex surface. and then multiply by 25, which is half the slant height. The convex surface is 785.4 square feet. *The slant height of a cone is the distance from its vertex to the circumference of its base. The altitude of a cone is the perpendicular distance from the vertex to its base. |