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. OF THE CIRCLE. 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 A CBD 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 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 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. We first square the radius, and then multiply by 3.1416. OPERATION. 5x5=25 square feet. Ans. 78.5400 sq. 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. 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 multiplying the square of the diameter, which is 10, by 3.1416. Then multiplying this result by 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. OF THE CYLINDER. 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 OPERATION. 4× 3.1416=12,5664=area of base. 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. OF THE CONE. 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, 3.1416 x 10-31.416-circle of base. OPERATION. 31.416 x 25=785.4 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 cir cumference of its base. The altitude of a cone is the perpendicular dis tance from the vertex to its base. 2. Find the solidity of a cone, whose altitude is 24 feet, and the diameter of whose base is 12 feet... OPERATION. 6×6×3.1416=113.0976=area of base.` 113.0976 x 8 =904.7808 solidity. We first find the area of the base by multiplying the square of the radius by 3.1416 (Art. 243). Then multiply this area by 8, which is one-third of the altitude of the cone. The result is 904.7808 cubic feet. 3. Find the convex surface of a cone, whose slant height is 22 feet, and the radius of whose base is 3 feet. Ans. 4. Find the convex surface of a cone, whose slant height is 5 yards, and the circumference of whose base is 1100 feet. Ans. 8250 sq. ft. 5. Find the solidity of a cone, whose altitude is 20 yards, and the diameter of whose base is 15 feet. Ans. 6. Find the solid content of a cone, whose altitude is 32 feet, and the radius of whose base is 12 feet. Ans. 14476.4928 cu. ft. OF THE PYRAMID. 247. The convex surface of a pyramid is found by multiplying the perimeter of the base by half the slant height. The solidity of a pyramid is found by multiplying the area of the base by one-third the altitude. Q. How do you find the convex surface of a pyramid? The solidity of a pyramid ? EXAMPLES. 1. What is the convex surface of a pyramid, whose slant height is 10 feet, and perimeter of its base 100 feet? We multiply 100 feet by 5, which is half the slant height. OPERATION. 100 x 5 500 sq. feet. The perimeter of the base is the sum of the sides which form the base. |