Chapter 8 Solids Pyramids This is a square pyramid. Draw this figure and write names of edges. Height and Slant Height
Right angles of Square Pyramid. Write 1 problem of page 193 Answer: Area of square pyramid = Area of Square + Area of 4 isosceles triangles.
Ie Area of Pyramid = 5x 5 + 4 x ½ (5 x 8) = 25 + 2 x 5 x 8 = 25 + 80 = 105 square cm. Write problem 5 of page 194 Answer Given that Lateral surface area is equal to base area. base 2 = 4 x½ x base x (Slant height) ie a 2 = 2 x a x l ie a = 2 l ie l =a / 2 now h = (a/2) 2 (a/2) 2 = 0, so we cant draw such a square pyramid. Write problem 1 of page 195 Height of Pyramid ( h) = (l) 2 (a/2) 2 = (18) 2 (12) 2 = (18+12)(18-12) = (30)(6) = 6 5 cm When lateral edge (e) = 30 cm and base (a) = 24 cm, then l = (e) 2 (a/2) 2
= (30) 2 (12) 2 = (30+12)(30-12) = 42 x 18 = 6 x 7 x 3 x 3 x 2 6 21 cm now h = (6 21) 2 12 2. = 6 17 cm Write problem 4 of page 196 Lateral height (l) = (25) 2 (30/2) 2. = (25) 2 (15) 2. = (25+15) (25-15) = 40 x 10 = 400 =20 cm. ====== When base edge is 40 cm. Lateral height (l) = (25) 2 (40/2) 2. = (25) 2 (20) 2. = (25+20) (25-20) = 45 x 5 = 225 =15 cm. ======
Volume of Pyramids Volume of a square pyramid = 1/3 a 2 h, a is the base edge, h is the height of the pyramid. Write problem 1 of page 197 Volume of a square pyramid = 1/3 a 2 h h = l 2 - (a/2) 2 = 15 2 - (10/2) 2 = (15+5)(15-5) = 20 x 10 = 10 2 cm Volume = 1/3 a 2 h = 1/3 10 2 x 10 2 = (1000 2) / 3 cm 3. Write problem 7 of page 198 Surface area of square pyramid = base area(a 2 ) + 4 Lateral area (2al) Base perimeter = 64 cm So one base length (a) = 64/4 = 16 cm. -------- Volume of square pyramid = 1/3 a 2 h 1/3 a 2 h = 1280 cu.cm 1/3 x 16 x 16 x h = 1280 h = (1280 x 3)/ (16 x 16) h = 15cm Lateral height (l) = (h) 2 + (a/2) 2. = (15) 2 + (8) 2. = 225 + 64 = 289 = 17 cm
------- Area of square pyramid = (a 2 ) +2al = 256 + 2 x 16 x 17 = 800 sq.cm Cone problem 1 This sector is rolled and made the cone. Given, Radius of the circle (R) = 10 cm. Ie Slant height of the Cone made from this sector is same as Radius of circle so Slant height = 10 cm. Now arc length is proportional to central angle. Total central angle = 360 0. So central angle 60 0 of this sector is 1/6 part of 360. With the same proportion the radius of cone is related to the radius of Sector radius.
So, Radius of cone = x 10 = cm =1.67cm ======= (3) What is the ratio of the base -radius and slant height of a cone made by rolling up a semicircle? Let radius of Semicircle = R Then Circumference =2 R/2 = R Let radius of the cone made by this Semicircle = r Then circumference of the base = 2 r then R = 2 r then R = 2r ie the relation of Semicircle radius and base radius of cone. Curved surface area Curved surface area of a cone is the area of the sector by which this cone is made. Area of sector (from 9 th std) = (X/360) x r 2, x is the central angle of Sector, r is the radius of sector. Arc length = 2 R x (X/360) 2 r = 2 R x (X/360) X = 360 x R/r = 360 x (l /r), l is the slant height So area of Sector becomes = (360 x (l/r) ) / 360 x r 2, = rl page 201 (1) What is the area of the curved surface of a cone of base radius 12 cm and slant height 25 cm?
Area of curved surface = rl = x 12 x 25 = 300 sq. Cm ========== (4) prove that for a cone made by rolling up a semicircle, the area of the curved surface is twice the base area. Area of curved surface = rl But R = 2r rl = rr = r x 2r Curved surface area = 2 r 2. = Twice of base area. Volume of Cone Volume of a cone with base radius r and height h = 1/3 ( r 2 h) Page 202 : (1) The base radius and height of a cylindrical block of wood are 15 cm and 40 cm. Find the volume of the largest cone made from this? Area of Cone = 1/3 ( r 2 h) = 1/3 x x 15 2 x 40 = 3000 Cubic Cm. (5) Two cones have the same volume and their base radii are in the ratio 4:5. What is the ratio of their heights? Since both cones have same volume 1/3 ( r 12 h 1 ) = 1/3 ( r 22 h 2 )---------(1) Ratio of radii is equal to 4 : 5 so r 1 = 4x and r 2 = 5x, so equation (1) becomes 1/3 ( (4x) 2 h 1 ) = 1/3 ( (5x) 2 h 2 ) 16h 1 = 25h 2
Sphere h 1 : h 2 = 25 : 16 ====== Surface area of sphere with radius r =4 r 2 Volume of Sphere with radius =(4/3) r 3. Surface area of Hemisphere with radius r =2 r 2 + r 2 2 = 3 r 2 Volume of Hemisphere = (2/3) r 3. (1) The surface area of solid sphere is 120 sq.cm. If it is cut into two halves, what would be the surface area of each hemisphere? Given that 4 r 2 = 120 r 2 = 30 Surface area of Sphere = 3 r 2 = 3 x 30 = 90 sq.cm ======= (5) How many litters of Petrol can hold in the petrol tank? Volume of petrol Tank = Volume of Spherical part + Volume of Cylindrical part Volume of Spherical part = (4/3) r 3 =(4/3) x 1 3 =(4/3) cu Meters Volume of Cylindrical part = r 2 h = x 1 2 x 6 =6 Cu Meters. Total volume = (4/3+6) =22/3 cu meteres =23.03 cu.m
=23.03 x 1000 liters =23030 liters. ===========