Chapter 17. Mensuration of Pyramid
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1 44 Chapter 7 7. Pyramid: A pyramid is a solid whose base is a plane polygon and sides are triangles that meet in a common vertex. The triangular sides are called lateral faces. The common vertex is also called Apex. A pyramid is named according to the shape of its base. If the base is a triangle, square, hexagon etc. the pyramid is called as a triangular pyramid, a square pyramid, a hexagonal pyramid etc. respectively. Altitude (or height): The altitude of a pyramid is the perpendicular distance from the vertex to the base. Axis: The axis of a pyramid is the distance from the vertex to the centre of the base. 7. Right or Regular Pyramid: A pyramid whose base is a regular polygon and congruent isosceles triangles as lateral faces. In a regular pyramid the axis is perpendicular to the base. Thus in a regular pyramid th axis and the altitude are identical. Slant Height: The slant height of a regular pyramid is the length of the median through the apex of any lateral face. In the Fig 7. OG is the slant height. It is denoted by. Fig. 7. Fig. 7. Lateral edge: It is the common side where the two, faces meet. In the OA is the lateral edge. 7. The surface area and Volume of a Regular Pyramid: If a is the side of the polygon base, h is the height and is the slant height of a regular pyramid, then (i) Lateral surface area Sum of the triangular sides forming the pyramid, which are all equal in areas n (area of one triangle of base a and slant height )
2 45 n a (n a) x Perimeter of the base x slant height (ii) Total surface area Lateral surface area + area of the base (iii) Volume: The volume of a pyramid may be easily derived from the volume of a cube. By joining the centre O of the cube with all vertices, six equal pyramids are formed. The base of each pyramid is one of the faces of the cube. Hence the volume of each pyramid is one-sixth of the volume of the cube. The height of each pyramid is h a Volume of each pyramid the volume of the cube 6 a 6 a a. ah Volume of the pyramid area of the base x height Example : Find the volume, the lateral surface area and the total surface area of the square pyramid of perpendicular height 9.4cm and the length of the side of base.9cm. Solution: Here h 9.4cm, a.9cm
3 46 (i) Volume area of the base x height ah 8.56 x cu. cm (ii) For the lateral surface area we first calculate the slant height. In the right triangle OAB, OA a h 9.4 cm, AB.45 cm By Pythagorean theorem, h AB Now lateral surface area perimeter of the base x (iii) 4a x x.9 x sq. cm. Total surface area Lateral surface area + Area of the base (.9) sq. cm. Example : A right pyramid 0m high has a square base of which the diagonal is 0m. Find its slant surface. Solution: Here h 0m AB d 0m BC d 5m In the right triangle BCD,
4 47 CD BD So BC CD + BD 5 CD Or CD Side of the base 5 m a BE BD. 5 Now, in the right triangle OCD, OD OC + CD m The slant surface area perimeter of the base x 4a sq. m. Example : The base of a right pyramid is a regular hexagon of side 4m and its slant surfaces are inclined to the horizontal at an angle of 0 o. Find the volume. Solution: Here, a 4m θ B 0 o Area of the base n a 80 cot 4 n 6 x 4 cot 60 o sq. m In the right triangle ABC, BC a m Angle C 60 o
5 48 So, tan 60 o AB CB AB m AB Now, in the right triangle OAB tan 0 o h AB x h m OR AB AC BC Volume area of the base x height Example 4: x.86 x 9.5 cu. m The area of the base of a hexagonal pyramid is 54 sq. m. and the area of one of its face is 9 6 sq. m. Find the volume of the pyramid. Solution: Here, area of the base 54 sq. m Area of one side face 9 6 sq. m Volume area of the base x h (54 ) x h To find h, we have to find and AB. n a 80 Area of the hexagon cot 4 n 6 x a 54 cot x a 4 a 6 a 6m o
6 49 Area of one triangle, say, OCD a 9 6 x 6 x 6 m In the right triangle ABC, AC 6m, BC m So AB AC BC AB m Now, in the right triangle OAB OA OB AB h AB h Therefore, volume (54 ) x 6 cu. m. Exercise 7(A) Q.: Q.: Q.: Q.4; Q.5: Q.6: Q.7: Q.8: Q.9: Find the volume of a pyramid whose base is an equilateral triangle of side m and height 4m. Find the volume of a right pyramid whose base is a regular hexagon each side of which is 0m and height 50m. A regular hexagonal pyramid has the perimeter of its base cm and its altitude is 5cm. Find its volume. A pyramid with a base which is an equilateral triangle each side of which is m and has a volume of 0 cu. m find its height. A pyramid on a square base has every edge 00m long. Find the edge of a cube of equal volume. The faces of a pyramid on a square base are equilateral triangles. If each side of the base is 0m. Find the volume and the whole surface of the pyramid. Find the whole surface of a pyramid whose base is an equilateral triangle of side m and its slant height is 6m. The slant edge of a right regular hexagonal pyramid is 65 cm and the height is 56cm. Find the area of the base. Find the slant surface of a right pyramid whose height is 65m and whose base is a regular hexagon of side 48 m.
7 50 Q.0: The sides of the base of a square pyramid are each.5cm and height of the pyramid is 8.5cm. Find its volume and lateral surface. Q.: Find volume of a square pyramid whose every edge is 00cm long. Answers 7(A) Q. cu.m Q cu. m Q cu. m Q m Q m Q cu. m; 7. sq. m Q sq. m Q sq. cm Q sq. m Q cu. cm, 6.76 sq. cm Q cu.cm 7.4 Frustum of a Pyramid: When a pyramid is cut through by a plane parallel to its base, the portion of the pyramid between the cutting plane and the base is called a frustum of the pyramid. Each of the side face of the frustum of the pyramid is a trapezium. Slant height: The distance between the mid points of the sides of base and top. It is denoted by. 7.5 Volume and surface area of Frustum of a Regular Pyramid: Let, A by the area of the base, and A be the area of the top, a is the side of the base and b is the side of the top, is the slant height and h is the height of the frustum of a pyramid, then (i) Volume of the frustum of a pyramid A A A ) (ii) Lateral surface area Sum of the areas of all the trapezium faces, which are equals n (area of one trapezium, say, ABA B ) a + b n x (na + nb) x sum of the perimeters of the base and top x slant height (iii) Total surface area Lateral surface area + area of the base and the top Example 5 :
8 5 A frustum of a pyramid has rectangular ends, the sides of the base being 0m and m. If the area of the top face is 700 sq. m. and the height of the frustum is 50m; find its volume. Solution: Here A 0 x 640 sq. m, A 700 sq. m Volume h [A A A A ] 50 [ x 700] 50 [ ] 50 (009.) cu. m. Example 6 : A square pyramid m high is cut 8m from the vertex to form a frustum of a pyramid with a volume of 90 cu. m. Find the side of the base of the frustum of a pyramid. Solution: Here, volume of frustum of a pyramid 90 cu. m Height of pyramid h m Height of the frustum of a pyramid h 4m OC 8m If a and b are the sides of base and top, then since the right triangles OAB and OCD are similar. AB OA So CD OC a b 8 a OR b a OR b Now, Volume of the frustum of a pyramid h [A A A A ]
9 [a b ab] 4 4a a 90 [a ] 9 4 9a 90 [ ] 9 90 x 7 a a 8.m Exercise 7(B) Q. Find the volume and the total surface area of a frustum of a pyramid; the end being square of sides 8.6m and 4.8m respectively and the thickness of the frustum of a pyramid is 5m. Q. Find the lateral surface area and volume of frustum of a square pyramid. The sides of the base and top are 6m and 4m respectively and the slant height is 8m. Q. Find the net area of material required to make half dozen lamp shades each shaded as a hollow frustum of a square pyramid, having top and bottom sides of 0cm and 8cm respectively, and vertical height 6m. Q.4 The sides of the top and bottom ends of a frustum of a square pyramid are 6m and 5m respectively. Its height being 0m. it is capped at the top by a square pyramid m from the base to the apex. Find the number of cu. m in the frustum of a square pyramid and in the cape. Q.5 Find the cost of canvas, at the rate of Rs. 5 per square meter, required to make a tent in the form of a frustum of a square pyramid. The sides of the base and top are 6m and 4m respectively and the height is 8m, taking no account of waste. Q.6 A square pyramid 5cm height and side of the base cm is cut by a plane parallel to the base and 9cm from the base. Find the ratio of the values of the two parts thus formed. Q.7 What is the lateral area of a regular pyramid whose base is a square cm. on a side and whose slant height is 0cm? If a plane is passed parallel to the base and 4cm. from the vertex, what is the lateral area of the frustum? Answers 7(B)
10 5 Q. 9.6 cu. m ; 9.88 sq. m Q. 0.5 cu. m ; sq. m Q sq. cm Q4. Volume of frustum V 06 cu. m. Volume of cape V 44 cu. m Number of cubic m V + V 50 cu. m. Q5. Rs Q6. : 4.6 Q7. 40 sq. cm ; 80 sq. cm. Summary. Lateral surface area of regular pyramid (perimeter of the base) x slant height. Total surface area of regular pyramid Lateral surface area + area of the base. Volume of pyramid (area of the base) x height 4. Volume of the frustum of a pyramid A A A ] 5. Lateral surface area of frustum of a pyramid (sum of the perimeters of base and top) x slant height i.e. (P + P ) x 6. Total surface area of frustum of a pyramid lateral surface + area of the base and top
11 54 Short Questions Write the short answers of the following. Q.: Q.: Q.: Q.4: Q.5: Q.6: Q.7: Q.8: Define pyramids. Find the volume of a pyramid whose base is an equilateral triangle of side m and whose height is 4 m. Find the whole surface of a pyramid whose base is an equilateral triangle of side m and its slant height is 6m. Find the volume of a pyramid with a square base of side 0 cm. and height 5 cm. Find the volume of a pentagonal based pyramid whose area of base is 5 sq.cm and height is 5 cm. A square pyramid has a volume of 60 cu.cm and the side of the base is 6 cm. Find height of the pyramid. Find the volume of a square pyramid if the side of the base is cm. and perpendicular height is 0 cm. The height of pyramid with square base is cm. and its volume is 00 cu.cm. Find length of side of square base Answers Q cu.cm Q sq.m Q cu.cm. Q5. 75 cu.cm Q6. 5 cm. Q7. 0 cu.cm. Q8. 5cm
12 55 Objective Type Questions Q. Each questions has four possible answers. Choose the correct answer and encircle it.. A solid figure whose base is a plane polygon and sides an triangles that meet in a common vertex is known as (a) pyramid (b) cube (c) frustum of a pyramid (d) None of these. If the base of pyramid is hexagon, the pyramid is called (a) triangular pyramid (b) square pyramid (c) hexagonal pyramid (d) pentagonal pyramid. If the base of pyramid is square, the pyramid is called (a) square pyramid (b) hexagonal pyramid (c) triangular pyramid (d) Rectangular pyramid 4. If area of base of pyramid is A and height h then volume of pyramid (a) Ah (b) Ah (c) Ah 6 (d) Ah 5. Volume of a pyramid whose area of base 6a and height h is (a) a h (b) a h (c) a h (d) a h 6. Lateral surface area of regular pyramid if perimeter of base is P and slant height is (a) P (b) P (c) P (d) P 6 7. The length of median through vertex (apex) of any lateral surface of a regular pyramid is (a) length of diagonal (b) slant height (c) height (d) axis 8. Volume of frustum of pyramid is (a) h [A A A A ] (b) A A A ] (c) h [A A A A ] (d) h[a A AA ] 9. Each of the side face of frustum of the pyramid is a (a) triangle (b) rectangle (c) trapezium (d) square 0. If P and P are perimeters of base and top of frustum of pyramid respectively then lateral surface area is
13 56 (a) PP (c) (P +P )h (b) (P +P ) (d) (P +P ) Answers Q.. a. c. a 4. a 5. b 6. c 7. b 8. a 9. c 0. b
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