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Frustum: When a plane section is taken of a right prism parallel to its end (i.e., perpendicular to its axis), the section is known as a cross-section of the prism and the two positions of the prism are still prisms. If, however, the plane section taken is not parallel to the ends, the portion of the prism between the plane section and the base is called frustum. Volume of a frustum of a Prism: In figure ABCEFGHI represents a frustum of a prism whose cutting plane EFGH is inclined at an angle to the horizontal. In this case, the frustum can be taken as a prism with base ABEF and height BC. (i) Volume of the frustum ABEF is a trapezium whose area is Volume of frustum i.e. Lateral Surface Area of a Prism: If the cutting plane is inclined at an angle figure we have to the horizontal then from or or or Hence Total surface area = area of the base + area of the section + lateral surface area Note: Lateral surface area of the frustum is the combination of rectangle and trapeziums whose area can be calculated separately. Example: A hexagonal right prism, whose base is inscribed in a circle of radius 2m, is cut by a plane inclined at an angle of to the horizontal. Find the volume of the frustum and the area of the section when the heights of the frustum are 8m and 6m respectively. Solution: Area of cross-section Here Area of base Volume of frustum Area of the section Frustum of a Right Circular Cylinder If (i) is the radius and is the height of the frustum, then Volume of the frustum of the circular cylinder i.e. (ii) Curved surface area of the frustum (iii) Total surface area = curved surface area + area of the ends Example: A circular cylinder, having a radius 2.3m, is cut in the shape of a frustum with volume and the lateral surface area. . Find the Solution: (a) (b) Curved surface area Frustum of a Pyramid If a pyramid is cut through by a plane parallel to its base portion of the pyramid between that plane and the base is called frustum of the pyramid. Volume of Frustum of a Pyramid: A general formula for the volume of any pyramid can be derived in terms of the areas of the two bases and the height of the frustum. Consider any frustum of a pyramid altitude . Complete the pyramid in figure with the lower base of which the frustum Denote by , the volume of the small pyramid of is Let and , upper base and the is a part. , whose altitude is . Then the altitude . respectively, represents the volume of the frustum From the figure it is easily seen that we may write. and the pyramid . . Expressing this equality in terms of the dimensions, …...... (1) The pyramid may be considered as cut by the two parallel planes and . Hence & nbsp; (as if a pyramid is cut by two parallel planes, the areas of the sections are proportional to the squares of their distances from the vertex). Taking the square root of both sides, we have or Transposing to the L.H.S. of this equation and factorizing, Substituting the value of in (1), we have i.e. The volume of a frustum of a pyramid is equal to the one-third the product of the altitude and the sum of the upper base, the lower base and the square root of the product of two bases. Lateral Surface Area of Frustum of a Pyramid: Each of the faces such as CDML, of the frustum of a pyramid is a trapezium and the area of each trapezium will be half the sum of the parallel sides, CD and ML, multiplied by the slant distance between them. In the frustum of pyramid on a square base, let length of each side of the other end, denote the length of each side of the base, the the height of the frustum. Each face CDML is a trapezium, the lengths of the parallel sides and . Area i.e. Example: A frustum of a pyramid has rectangular ends, the sides of the base being 25dm and 36dm. If the area of the top face is 784sq.dm and the height of the frustum is 60dm, find its volume. Solution: Here Frustum of a Cone If a cone is cut by a plane parallel to its base, the portion of a solid between this plane and the base is known as frustum of a cone. The volume denoted by ABCD in figure is a frustum of the cone ABE. Volume of Frustum of a Cone: Since, we know that cone is a limit of a pyramid therefore; frustum of a cone will be the limit of frustum of a pyramid. But volume of a pyramid is Where Example: A cone 12cm high is cut 8cm from the vertex to form a frustum with a volume of 190cu.cm. Find the radius of the cone. Solution: Given that: Height of cone Height of frustum Volume of frustum Now volume of frustum cone or or Hence required radius of cone Curved Surface Area of a Frustum of a Cone: Since, a cone is the limiting case of a pyramid, therefore the lateral surface of frustum of a cone can be deduced from the slant surface of frustum of a pyramid, i.e., curved (lateral) surface of frustum of cone. , being the slant height of frustum, and being two radius of bases. Note: (1) Total surface area of frustum of a cone (2) To find the slant height of the cone, use Pythagorean theorem. Example: A material handling bucket is in the shape of the frustum of a right circular cone as shown in figure. Find the volume and the total surface area of the bucket. Solution: Slant height Lateral surface area Base areas Total surface area Volume Zone or Frustum of a Sphere The portion of a sphere intercepted between two parallel planes is called a zone (i.e. frustum). 1. The volume of the zone (or frustum) of a sphere may be found by taking the difference between segment EBD and the segment ABC (see figure) that is Where is the altitude, and are respectively the radii of the small circle (bases). (ii) The surface area of the zone or frustum is equal to the circumference of the great circle of the sphere times the altitude of the same. i.e. Where (iii) Total surface area of a zone Example: The sphere of radius 8cm is cut by two parallel planes, one passing 2cm from the center and other 6cm from the center. Find the area of the zone and the volume of the segment between two planes if both planes are on the same side of the center. Solution: Surface Area of the zone Now Example: A stone was rolled into a hemispherical basin 8cm diameter having depth of water in it, when the water immediately rose to the tip of the basin. What was the cubic content of the stone? Solution: Let the figure represents a vertical mid-section of the basin and stone. Let DE and ABindicate the level of the water before and after the rolling of the stone. Then If , then Now, the cubical content of the stone nearly. Segment of a Sphere: For a special segment of one base, the radius of the lower base is equal to zero. Therefore, In this case, the total surface area of the segment Example: Find the volume of a segment of a sphere whose height is is 8cm. Solution: Given that: Volume of the segment and the diameter of whose base