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Introduction to Prisms A prism is a solid whose ends, or bases, are parallel congruent polygons, and whose sides or faces are parallelograms. A straight line moving parallel to itself, its extremities traveling round the outlines of plane figures, generates the prism. Prisms are named according to the shape of ends of bases. A prism with a square base is called a square prism and a prism with hexagonal base is called a hexagonal prism. Similarly, when the ends or bases of a prism are parallelogram, the prism is called a parallelepiped. As shown in the figure the sides or faces of the prism are , etc. These faces are parallelograms. If they were rectangles, angle would be and the prism would be aright prism. A prism that is not a right prism is called an oblique prism. A side of one of these parallelograms of these prisms is called a lateral edge. The altitude of a prism is the distance between the planes of the two bases (i.e. ). In a right prism, altitude is the same length as a lateral edge and this is not true for an oblique prism. Lateral Surface Area of a Prism The lateral surface area of a prism is the total of the area of the faces. Lateral Surface Area Perimeter of the base Height of the prism Perimeter of the base times the altitude Rule 1: The lateral area of the prism is equal to the perimeter of the base times the altitude. Rule 2: Total surface area of a prism is the sum of the lateral areas and the area of its base. Example: Find the area of the whole surface of a right triangular prism whose height is m and the sides of whose base are and m, respectively. Solution: In all there are five planes figures, i.e., two triangles and three rectangles. Since both the rectangles are of equal area. Area of both triangles Square meter Area of all three Rectangles Square meter Area of the whole surface Square meter Volume of a Prism Consider the right prism as shown in the figure. There are as many cubic units in each layer parallel to the base as there are square units in the area of the base. Also, there are many layers of cubic units in the prism as there are linear units in the altitudes . We can, therefore, find the total number of cubic units in the prism by multiplying the area of the base by the altitude. Volume of the Prism Area of the Base Height of the prism Rule: The volume of the prism equals its base times its altitude. Example: The base of a right prism is an equilateral triangle with a side of its volume. Solution: cm and its heights is cm, find Volume, Since, base is an equilateral triangle cm cm Cubic cm Example: The sides of a triangular prism are is cubic cm. What is its height? Solution: Now, cm, cm and cm respectively. The volume of the prism Square cm cm Types of a Prism Cube: Let The cube is a right prism with a square base and a height which is same as the side of the base. be the side of the cube, then 1. Volume of the cube area of base height i.e. 2. Total surface area of the cube area of six faces i.e. 3. The line joining the opposite corners of the cube is called the diagonal of the cube. The length of the diagonal of the cube Proof: In the given figure, the line is the diagonal of the cube. But Since and are the sides of the cube and each has length equal to , therefore Example: Three cubes of metal whose edges are in the ratio diagonal is Solution: are melted into a single cube whose cm. find the edges of three cubes. Let the edges of the cubes be Their volumes are: and and cu. cm and cu. cm cm Volume of the single cube Let be the edges of the cube then volume, edge cu. cm cm Now, Diagonal of the cube cm But, the diagonal of cube Hence the three edges of the cube are Rectangular Prism: and (1) Volume of the rectangular prism (2) Total surface area area of base height area of six faces (3) Length of the diagonal (as , ) Example: The length, width and thickness of a rectangular block are Find the volume, surface area and length of the diagonal of the block. Solution: Given that: cm, cm, and cm (1) Volume, cu. cm (2) Surface area, sq.cm (3) Length of diagonal cm cm respectively. Polygonal Prism: A prism with a polygon base is known as a polygonal prism. (a) Volume of the prism whose base is a rectangular polygon of height sides and height 1. When sides is given. 2. When radius of inscribed circles is given. 3. When radius (b) Lateral surface area 1. Perimeter of base When side of circumscribed circle is given. height is given. 2. When radius 3. When radius (c) Total surface area Lateral surface area of inscribed circles is given. of circumscribed circle is given. Area of base and top Example: A pentagonal prism which has its base circumscribed about a circle of radius height of dm is cast into a cube. Find the size of the cube. Solution: Here dm, dm, dm Since, volume of the material remains the same in both the cases Volume of the cube Volume of the pentagonal prism Now, volume of the pentagonal prism Now by the condition area of the base dm, and which has a or Taking both sides, we get Taking , we get dm.