SOLIDS.

Transcription

1 SOLIDS Prisms Among the numerous objects we see around us, some have a regular shape while many others do not have a regular shape. Take, for example, a brick and a stone. A brick has a regular shape while a stone does not have a regular shape. A brick has an exact number of plane surfaces and an exact number of corners and edges; whereas a stone has an uneven surface with numerous corners and edges. Take another brick. It may be of different size, but has the same number of corners and edges. This is because of brick determines of definite geometrical shape. The shape determined by a brick is called a prism. The following are a few objects in the form of prisms. If we closely examine these objects, we find that a prism has a definite shape and has the same thickness from top to bottom. Let us examine the figure. It has in all six faces. The end faces (i.e., top and bottom faces) are square regions. The end faces are called bases. The lateral faces are rectangular in shape. Note that in a prism, the two end faces are congruent. A prism is named after the shape of its base. If the base of a prism is a square region, it is called a square prism. If the base of a prism is a triangular region, it is called a triangular prism, and so on. We have several kinds of prisms, namely, rectangular, pentagonal, hexagonal, trapezoidal prisms. Cross-Section Consider the prism in figure. It has the shape of a beam. Here, the top and the bottom faces of the beam are square regions. Suppose we cut the beam horizontally parallel to the base as shown in figure. The cut pieces are shown in figure (i) and (ii). We can easily see that the bottom face of piece (i) and top face of piece (ii) are the bottom and top respectively of the original beam. The other bases, the top face of (i) and bottom face of (ii) are formed by the cut we made. This base is known as the cross-section of the beam, of we can say the cross section of the square prism. (i) (ii)

2 A slice through a three-dimensional solid parallel to its base (or end) is called a cross-section. We can make any number of cross sections like this in a prism. They are lal of the same shape and size of the base of the prism. A prism is a solid shape whose cross-section parallel to an end-face are all identical. A Regular Prism A prism whose end faces are regular polygons is called a regular prism. A Right Prism A prism is said to be a right prism whose lateral faces are at right angles to its end faces. A cuboid and a cube are examples of right prisms. Note: In this chapter, we shall restrict our study to regular and right prisms only. Surface Area of a Right Prism (a) A Rectangular Prism Consider a rectangular prism with base a rectangle of dimensions l b and height h units. This prism has six faces-two end faces and four lateral faces. Area of one end face = l b sq units Area of both the end faces = lb sq units Now, Area of four lateral faces = lh + bh + lh + bh = lh + bh = (l + b)h = perimeter of the base height Adding (1) and (), we have Total surface area of the rectangular prism = [lb + (l + b)h] sq units. = (lb + bh + lh) sq units. (b) A Square Prism Consider a square prism whose base edges are a units each and height h units. Here, the end faces are square regions and the four lateral faces are rectangular regions.

3 Here, Area of both the end-faces = a sq units and Area of four lateral faces = 4 ah sq units = (4a)h sq units = perimeter of the base height Adding the two, we get Total surface area of the square prism = (a + 4ah) sq units. (c) A Triangular Prism Consider a triangular prism of height h units, whose base is a triangle with sides a, b and c units. This prism has three lateral faces which are rectangles of dimensions a h, b h and c h. Area of three lateral faces = (ah + bh + ch) sq units = (a + b + c)h sq units. = perimeter of the base height Area of both the end faces = area of triangle with sides a, b and c. So, Total surface area of the triangular prism = (perimeter of the base height) + (area of the base). Remark: The surface area of other prisms with different shapes of bases can easily be determined, using the following; Surface Area of a prism = perimeter of the base height of the prism + (area of the base) Illustration 1: The dimension of a rectangular beam are 0 cm 15 cm.5 m. Find the total surface area of the beam. Solution: Area of the base = (0 15) sq cm = 00 sq cm Area both the two end faces = 00 sq cm = 600 sq cm Illustration : perimeter of base Area of the four lateral faces = (0 + 15) cm = 70 cm. = perimeter of the base height = (70 50) sq cm [.5 m = 50 cm] = sq cm Adding (1) and (), we get Total surface area of the beam = ( ,500) sq cm = 5,100 sq cm. The base edges of a triangular prism are 10 cm, 1 cm and 18 cm. Its height is 80 cm. Calculate its total surface area. Solution: Area of the base = ss as bs c, where s = a + b + c = sq cm = sq cm = 40 sq cm Area of both the end faces = ( 40 ) sq cm = 80 sq cm (1) perimeter of the base = (a + b + c) cm = ( ) cm = 40 cm Area of the three lateral faces = perimeter of the base height = (40 80) sq cm () Adding (1) and (), we get Total surface area of the triangular prism = (80 +, 00) sq cm =, 1.1 sq cm.

4 Volume of a Right Prism We already know that every object occupies space and the measure of that space is called the volume of the object. Here in this section we shall determine the volume of a right prism. We have already learnt to determine the volume of a cuboid are l b h, its volume is given by lbh cubic units, i.e., Volume = lbh cu units = (lbh)h cu units = area of the base height We extend the formula (1) to the volume of a prism, whatsoever be the shape of its base. Thus Volume of a prism = area of the base height V = Ah Particular Cases: (a) If a rectangular prism has a base of dimension l b and height h, its volume is given by V = (l b)h cu units (b) If a square prism has a base of dimensions a a and height h, its volume is given by V = (a a)h cu units = a h cu units. (c) If a triangular prism has base area A and height h, its volume is given by V = Ah cu units. Illustration : One end of a prism is in the form of a trapezium. The parallel sides of the trapezium are 10 cm and 8 cm and the distance between them is 5 cm. The length of the prism is 6 cm. Find its volume. Solution: Area of trapezium = 1 (a + b)h sq units = 1 (10 + 8) 5 sq cm = 45 sq cm. Base area of the prism = 45 sq cm Length (or height) of the prism = 6 cm. Volume of the prism = base area of the prism length = (45 6) cu cm = 1, 60 cu cm. Pyramids You might have heard of great s of Egypt. These are considered to be one of the wonders of the world. The picture of one such is given on the right. Triangular (i) Pentagonal (ii) Hexagonal (iii) Rectangular (iv) Square (v)

5 A is a solid bounded by plane faces of which one called the base is a rectilinear figure and the remaining called the side faces are triangles, having a common vertex at some point outside the plane of the base. The simplest in which there is a triangular base and three triangular side (lateral) faces, is called a tetradedon. [see figure (ii)]. base base Height of the Pyramid The length of the perpendicular drawn from the vertex of a to its base is called the height of the. Axis of the Pyramid The straight line joining the vertex to the centre of the base is called the axis of the Pyramid. Regular Pyramid A is called a regular when its base is any regular polygon. Right Pyramid A is called a right when the foot of the perpendicular drawn from the vertex to the base is the centre of the slant height base, otherwise it is said to be oblique. height Slant Height of a Right and Regular Pyramid It is the straight line joining the vertex to the mid-point of one of the sides of the base. Kinds of Pyramids Pyramids are named after the shape of its base. A with base a square is called a square with base a rectangle is called a rectangular with base a triangle is called a triangular and so on. Regular Tetrahedron (or Congruent) A tetrahedron in which all the four faces are congruent and each is an equilateral triangle. The foot of the perpendicular from the vertex to the base is the centre of the base. Volume and Surface Area of a Right and Regular Pyramid 1 The volume (V) of a right and regular = area of the base height cu units. Surface area of the lateral faces of a right and regular 1 = perimeter of the base slant height sq units. Total surface area of a right and regular = surface area of the lateral faces + area of the base. For a regular tetrahedron, Area of the base = a sq units O Surface area of three lateral faces = a sq units. Total surface area of regular tetrahedron = 4 Height = a units a sq units. B D G C and Volume = a cu units, A

6 Where a is the length of each edge. Clearly, a regular tetrahedron has 4 congruent faces, each of which is an equilateral triangle. An octahedron is a solid figure contained by 8(esp. triangular) plane faces. Note: In this chapter, we shall restrict our study to right and regular s only. Illustration 4: Solution: Find the volume of a right and regular whose area of the base is 5 sq dm and height is 9 dm. We know that Volume of a = 1 (area of the base) height 1 Volume = 5 9 cu dm = 75 cu dm. Thus, the volume of the is 75 cu dm. Illustration 5: Find the area of the base of a right tetrahedron whose volume is 675 cu cm and height is 5 cm. Also, find the side of the base. Solution: We know that Volume of a tetrahedron = 1 (area of the base) height Area of the base = volume height = 675 sq cm = 81 sq cm 5 Let a be the side of the base. Since base of a tetrahedron is an equivalent triangle, we have Hence, Area of the base = a sq cm a = 81 a 9 a = 18 Hence, the side of the base is 18 cm long.

7 ASSIGNMENTS SUBJECTIVE LEVEL I 1. A prism whose base is an equilateral triangle has a height 1 cm. The area of the base is 64 sq cm. Find its volume and the area of lateral faces.. The base edges of a triangular prism are 5 cm, 1 cm and 1 cm. If the height of the prism is 0 cm, find its volume of the prism.. The base of a right is an equilateral triangle with side 0 cm. If the slant height of the is 0 cm, find its total surface area. 4. Find the surface area of the lateral faces of a regular tetrahedron whose length of each edge is 10 cm long. 5. Find the cost of a solid iron whose base is a square of side cm and height 1 cm at Rs 0 per cu cm. 6. A prism whose base is an right angled isosceles triangle has a height 4 cm and the area of the base is sq.cm. Find its volume and the area of lateral faces. 7. Find the total surface area of regular tetrahedron of height = units 8. Find the total surface area of a square prism of height 4 cm, where the side of square is cm. 9. The base of a prism is a trapezium of sides cm and 11 cm, and the distance between the parallel sides is 1 cm. Find the volume and total surface area of the prism if the height of the prism is 17 cm. 10. Find the slant height of the right regular of height 5 cm, if the volume and perimeter of the are 0 cubic cm and 9 cm respectively and the total surface area is 1 sq. cm. / m. LEVEL II 1. The base of a right 15 m high is an equilateral triangle. If each side of the base is 5m, find its volume.. Show that the surface area and volume of a regular tetrahedron of height h cm are h sq cm and 8 h cu cm respectively.. A triangular prism whose base is an equilateral triangle, is 70 cm high. The perimeter of one of its lateral face is 180 cm. Find the volume and total surface area of the prism. 4. A glass paper weight is a prism has 5 cm length. The perimeter of one of its end faces is 60 cm. Find its volume and total surface area. 5. A regular hexagonal prism has 5 cm length. The perimeter of one of its end faces is 60 cm. Find its volume and total surface area.

8 OBJECTIVE LEVEL I 1. If one edge of the base of a square prism is 7 cm and its height is 1.5 m, its total surface area is (A) sq. m (B) sq. m (C) sq. m (D) sq. m. The total surface area of a trapezoidal prism whose each end face is a trapezium, where the height (or length) of the prism is 10 cm is (A) sq. m (B) sq. m (C) sq. m (D) sq. m. The base area of a prism is 6 sq cm and its height is 50 cm. The volume of this prism is (A) Cubic m (B) Cubic m (C) Cubic m (D) Cubic m 4. The area of one end face of a triangular prism is 64 sq. cm and its length is 60 cm. Its volume is (A) Cubic m (B) Cubic m (C) Cubic m (D) Cubic m 5. The height of a right and regular whose volume is 1.cu m and its base area is 180 sq cm is (A) sq. m (B) sq. m (C) sq. m (D) sq. m 6. The base of a right 10 m high is a square. If each side of the base is 5 m, its volume is (A) Cubic m (B) Cubic m (C) Cubic m (D) Cubic m 7. The base of a right 10 m high is an equilateral triangle. If each side of the base is 6 m, Its volume is (A) Cubic m (B) Cubic m (C) Cubic m (D) Cubic m 8. The base of a right is an equilateral triangle with side 10 cm. If its slant height is 14 cm, its total surface area is (A) sq. m (B) sq. m (C) sq. m (D) sq. m ANSWERS SUBJECTIVE LEVEL I 1. Volume = 768 ci cm; area of the lateral faces = 576 sq cm. 600 cu cm. ( ) sq cm sq cm 5. Rs 0. LEVEL II cu m. Volume = 7, 000 cu cm; total surface area = (4, ) sq cm cu cm 5. Volume =, 750 cu cm; total surface area = (1, ) sq cm.