AREAS Finding the area of a right triangle. 1. Find the area of a right triangle whose base is 4 yards and whose altitude is 3 yards. Altitude 4yd. Diagonal Base Observe: 1. That the diagonal divides the rectangle into two equal right triangles. 2. That the unit of measure is 1 sq. yd. 3. That the area of one of the right triangles is of the area of the rectangle; that is, of 4 × 3 × 1 sq. yd., or 6 sq. yd. The area of a right triangle is found by multiplying the unit of measure by half the product of the base and the altitude. Name the unit of measure, and find the area of each of the following right triangles : 1. Find the area of two right triangles, the base of one being 2 ft. and of the other 4 ft. and the alt. of each 3 ft. N 2ft. 3ft. M 4ft. Draw the triangles as shown in the figure. Observe: 1. That the unit of measure is 1 sq. ft. 2. That the area of the right triangle N is equal to of 2 × 3 × 1 sq. ft., or 3 sq. ft. 3. That the area of the right triangle M is equal to of 4 × 3 Therefore, the area of N plus the area of M is sq. ft., or 9 sq. ft. x 1 sq. ft., or 6 sq. ft. equal to 1 of 6 × 3 × 1 2. Find the area of a triangle whose base is 6 ft. and whose altitude is 3 ft. Observe that the area of the triangle in example 2 is equal to the area of the two right triangles in example 1, and is, therefore, equal to of 6 × 3 × 1 sq. ft., or 9 sq. ft. 3 ft. 6ft. Show by cutting and folding paper, as indicated in the following figures, that the area of each triangle is equal to one half the area of a rectangle, having the same base and altitude. The area of any triangle is found by multiplying the unit of measure by one half the product of the base and altitude. Find the area of the following triangles: 5. Base 10 ft., altitude 30 ft. 3. Base 20 ft., altitude 14 ft. 4. Altitude 8 ft., base 15 ft. 6. Altitude 50 ft., base 18 ft. Finding the area of a parallelogram. Find the area of a parallelogram whose base is 8 in. and altitude 3 in. Observe: 1. That the diagonal of the parallelogram divides it into two equal triangles. 2. That the area of each triangle is equal to of 8 × 3 × 1 sq. in., and the area of the parallelogram is equal to, or once the prod Diagonal 8 in. uct of the base and altitude; that is, 8 × 3 × 1 sq. in., or 24 sq. in. The area of a parallelogram is found by multiplying the unit of measure by the product of the base and altitude. Written Work Find the area in acres of: 1. A parallelogram whose base is 140 rd. and altitude 60 rd. 2. A rhomboid whose base is 90 rods and altitude 50 rods. 3. A rhombus whose base is 120 rods and altitude 100 rods. Find the altitude of : 4. A rhomboid whose area is 7.5 A., base 48 rd. 5. A rhomboid whose area is 6.125 A., base 140 rd. 6. Find the base of a parallelogram whose altitude is 601 rods and whose area is 30.25 acres. Finding the area of a trapezoid. Written Work 1. Find the area of a trapezoid whose parallel sides are 20 inches and 12 inches, and whose altitude is 8 inches. Draw the Examine the trapezoid ABCD. diagonal AC, dividing it into two triangles. Observe: 1. That the area of the trapezoid is equal to the area of its two triangles ABC and BACD. 2. That the area of triangle ABC equals of of 12 × 8 × 1 sq. in. 3. That the area of the triangle ACD equals 4. That the area of the trapezoid equals of (20+ 12) × 8 × 1 sq. in., or 128 square inches. The area of a trapezoid is found by multiplying the unit of measure by the product of the altitude and the sum of the parallel sides. 2. The parallel sides of a trapezoid are 38 inches and 62 inches respectively, and its altitude is 21 inches. Find its area. 3. The area of a trapezoid is 2.5 acres. The sum of its parallel sides is 80 rods. Find its altitude. 4. The area of a trapezoid is 41 A. If its altitude is 20 rd., and one of its parallel sides 38 rd., what is the other? Finding the area of a trapezium. Written Work 1. Find the area of a trapezium whose diagonal is 30 inches, and whose altitudes are 12 inches and 8 inches respectively. Observe: 1. That the area of the trapezium equals the area of its two triangles. 2. That the area of one triangle equals of 30 × 8 x 1 sq. in. 3. That the area of the other triangle equals of 30 x 12 x sq. in. 30 12in 4. That the area of the trapezium equals of 30 × 20 × 1 sq. in., or sq. in. 300 The area of a trapezium is found by dividing it into triangles and finding the sum of their areas. 2. The base line dividing a trapezium into two triangles is 40 ft. The altitude of one triangle is 10 ft., of the other is 12 ft. Find the area of the trapezium. 3. A trapezium is divided into two triangles by a line 28 ft. long. Find the area of the trapezium, if the altitude of one triangle is 8 ft. and of the other triangle 14 ft. THE CIRCLE Observe the figure. What is its shape? Observe that its boundary line changes its direction regularly circumference at every point. A circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within called the center. Diameter Radius The circumference of a circle is its bounding line. A diameter is a straight line passing through the center with both ends terminating in the circumference. A radius is a straight line extending from the center to the circumference. Measure carefully with a cord the distance around a circle 1 foot in diameter, and you will find it is about 3.1416 ft. in circumference. This relation of diameter to circumference is true of all circles. The circumference of a circle is found by multiplying the diameter by 3.1416. This ratio is represented by the symbol π (pi). The diameter of a circle is found by dividing the circumference by 3.1416. Observe 1. That the circle may be considered as made up of triangles whose bases form the circumference. 2. That the radius of the circle is equal to the altitudes of the triangles. 3. That the area of the circle is equal to the areas of all the triangles, or of the sum of their bases (circumference) by their altitude (radius). |