Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 101 Vedanta Excel in Mathematics - Book 10 Mensuration (I): Pyramid Worked-out Examples Example 1: Based on the given right circular cone, answer the following question. (a) Write the formula to find the curved surface area of cone. (b) Find the measure of the radius. (c) Calculate its curved surface area, total surface area and volume. Solution: Here, height of the cone (h) = 4 cm and slant height (l) = 5 cm. (a) Curved surface area of cone (C.S.A.) = prl (b) Radius of the circular base (r) = l 2 – h2 = 52 – 42 = 25 – 16= 9 = 3 cm (c) Now, curved surface area of the cone = πrl = 22 7 × 3 cm × 5 cm = 47.14 cm2 Also, total surface area of the cone = πr (r + l) = 22 7 × 3 (3 + 5) = 75.43 cm2 And, volume of the cone = 1 3 πr2 h = 1 3 × 22 7 × 3 cm × 3 cm × 4 cm = 37.71 cm3 Example 2: If the total surface area of a cone is 704 cm2 and radius of its base is 7 cm, find the volume of the cone. Solution: Here, the radius of the base of the cone (r) = 7cm, The total surface area of the cone = 704 cm2 or, πr ( r + l ) = 704 cm2 or, 22 7 × 7 (7 + l) = 704 cm2 or, l = 25 cm Also, the vertical height of the cone (h) = l 2 – r2 = 252 – 72 = 24 cm Now, the volume of the cone = 1 3 πr2 h = 1 3 × 22 7 × 7 × 7 × 24cm3 = 1,232 cm3 So, the required volume of the cone is 1,232 cm3 . Example 3: If the volume of the given cone is 1848 cm3 , and the radius of its base is 14 cm, find its curved surface area. Solution: Here, the radius of the base of the cone ( r ) = 14 cm The volume of the cone = 1848 cm3 or, 1 3 πr2 h = 1848 cm3 5 cm4 cm 7cm 25cm h 14 cm Q O R P h
Vedanta Excel in Mathematics - Book 10 102 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Mensuration (I): Pyramid or, 1 3 × 22 7 × 14 × 14 × h = 1848 cm3 or, h = 9 cm Also, the slant height of the cone (l) = h2 + r2 = 92 + 142 = 16.64 cm Now, the curved surface area of the cone = πrl = 22 7 × 14 × 16.64 cm2 = 732.16 cm2 So, the required curved surface area of the cone is 732.16 cm2 . Example 4: Mrs. Karki designs a birthday hat with circumference of base 10π inch and vertical height of 12 inch. She attaches a piece of elastic along its diameter. (a) How long is the elastic? (b) How much paper does she need to make the hat? (c) How much cubic inch of the air does it hold? Solution: Here, circumference of the circular base = 10π inch or, 2π r = 10π ∴ = 5 inch (a) The length of elastic = diameter = 2r = 2 × 5 inch = 10 inch. (b) Vertical height of the hat (h) = 12 inch Now, the slant height of the cone (l) = h2 + r2 = 52 + 122 = 13 inch Also, C.S.A. of hat = πrl = 22 7 × 5 × 13 sq. inch = 204.29 sq. inch Hence, 204.29 sq. inch of paper is required to make the hat. (c) Volume = 1 3 πr2 h = 1 3 × 22 7 × 5 × 5 × 12 cu. inch = 314.29 cu. inch Hence, the cap holds 314.29 cu. inch of air. Example 6: Once a group of people went for a picnic at a hill side. Due to peak season, they did not get a proper accommodation there. The weather was fine so they decided to make a conical tent in a park. They went to a tent house near by the picnic sport and brought the canvas in rent that is enough to make the tent with base radius 11.2 m and height 6.6 m. (a) How many people were there in the group if each person required 8.96 square meter of space on the ground for the accommodation? (b) If they decided to share the rent of the canvas equally, what amount was paid by each person at Rs 15 per sq. m? Solution: Here, in the conical tent, radius of base (r) = 11.2 m and height (h) = 6.6 m (a) Now, area of circular base = πr2 = 22 7 × 11.2 × 11.2 m2 = 394.24 m2 Area of space on the ground required for each person = 8.96 m2 No. of people = Area of circular base Space required for each person = 394.24 m2 8.96 m2 = 44 Hence, there were 44 people.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 103 Vedanta Excel in Mathematics - Book 10 Mensuration (I): Pyramid (b) Also, we know l 2 = r2 + h2 = (11.2 m)2 + (6.6 m)2 = 169 m2 = (13 m)2 ∴ l = 13 m C.S.A. = πrl = 22 7 × 11.2 × 13 m2 = 457.6 m2 Rent of canvas required to make tent = Area × Rate = 457.6 × Rs 15 = Rs 6,864 ∴ Share of each person for rent of the canvas = Rs 6864 44 = Rs 156 Hence, each person paid Rs 156 as the rent of the canvas for the tent. EXERCISE 5.2 General section 1. a) In the given right cone, the radius of circular base = r, vertical height = h and the slant height = l. Write the formulae to find: (i) circumference of base (ii) area of base (iii) curved surface area (iv) total surface area (v) volume (vi) relation among r, h and l. b) If a is the radius of the circular base and b is the slant height of a cone, write the formula to find its curved surface area. c) If x is the radius of the base and y is the slant height of a cone, write the formula to find its total surface area. d) The radius of the circular base of a cone of height y is x, write the formula to find its volume. 2. a) Calculate the curved surface area of the following right circular cones. (i) (ii) (iii) (iv) b) The slant height of a right circular cone with radius 7 cm is 15 cm. Find its curved surface area. c) The radius of the circular base of a cone is 14 cm and its slant height is 20cm, find the total surface area of the cone. d) The slant height of a circular cone with radius 21 cm is 25 cm, find the total surface area. 3. a) Find the volume of the following right cones. (i) (ii) (iii) (iv) h r l P O Q 7cm 25 cm 10cm 14 cm 28 cm 50 cm 30 cm 21 cm 6cm 7cm 7cm 24cm 35 cm 30 cm 42 cm 40 cm
Vedanta Excel in Mathematics - Book 10 104 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Mensuration (I): Pyramid b) The vertical height of a cone is 15 cm and the radius of its base is 7 cm, find the volume of the cone. c) If the vertical height of a cone with radius 21 cm is 12 cm, find the volume of the cone. d) A group of student is making a model of a mountain with clay in the shape of right circular cone. If the mountain is 3 feet tall and the radius of the base is 3.5 feet, what is the volume of clay needed to make the mountain? 4. a) The curved surface area of a cone is 220 cm2 . If the diameter of its circular base is 14 cm, find its slant height. b) If the total surface area of a cone is 594 cm2 and its slant height is 20 cm, find the radius of its circular base. c) The volume of a cone is 23.1 cm3 . If the radius of its base is 2.1 cm, find the vertical height of the cone. Creative section-A 5. a) The figure given alongside is a wooden cone. (i) Write the formula to find the curved surface area of a cone. (ii) Find the measure of its slant height. (iii) Compute the area of its curved surface. (iv) How much minimum cloth is required to cover its whole surfaces? b) Study the given cone and answer the following questions. (i) What is the formula to find the total surface area of a cone? (ii) What is the measure of its slant height? (iii) Find the area of its curved surface. (iv) Find the minimum paper required to cover its whole surface. c) A solid cone is shown aside. (i) Write the formula to find the area of its plane surface. (ii) Calculate the cost of painting the whole solid cone at the rate of 50 paisa per 4 sq. cm. d) The solid cone given alongside is made up of stone. (i) Write the relation among radius (r), height (h) and slant height (l) of a cone. (ii) Estimate the cost of colouring its surfaces at 25 paisa per 3 sq. cm. 6. a) The diameter of base of cone is 16 cm and its slant height is 17 cm. (i) Find the volume of the cone. (ii) Find the ratio between areas of its plane surface and curved surface. 24cm A O B 7cm 20cm P Q O 21cm M O N 28cm 35cm B O 65cm 56cm A P B O 16cm 17cm
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 105 Vedanta Excel in Mathematics - Book 10 Mensuration (I): Pyramid b) The slant height of a right circular cone is 35 cm and the circumference of its base 132 cm. (i) Find its volume. (ii) Find the ratio between the areas of its plane surface and curved surface. 7. a) The diameter of circular base of a right cone is 10 cm and its total surface area is 90π cm2 . Find its (i) slant height (ii) vertical height (iii) volume b) The diameter of circular base of a right cone is 60 cm and its volume is 12,000π cm3 . Find its (i) height (ii) slant height (iii) total surface area c) The slant height of a right circular cone is 50 cm and its total surface area is 2,816 cm2 . Find its (i) radius of base (ii) vertical height (iii) volume 8. a) A solid cone is made up of iron. Its vertical height is 24 cm and slant height is 25 cm. (i) Write the formula to find the volume of a cone. (ii) Find the radius of the cone. (iii) Find the cost of iron required to make the cone at the rate of Rs 80 per kg. Given that 1 cm3 of iron = 7.874 g. b) The solid cone is made up of silver. It's radius of the circular base is 21 cm and the curved surface area is 2,310 cm2 . (i) Write the formula to find the curved surface area of a cone. (ii) Find the slant height of the cone. (iii) Find the cost of silver required to make the cone at the rate of Rs 20,000 per kg; given that, 1 cm3 of silver = 10.5 g. Creative section-B 9. a) On the occasion of daughter’s birthday, Rahul designs a birthday hat with circumference of base 44 cm and vertical height of 24 cm. He attaches a piece of elastic along its diameter. (i) How long is the elastic? (ii) How much paper does he require to make the hat? (iii) How much cubic centimetres of the air does it hold? b) A Vietnamese leaf hat is in the shape of a right circular cone. The circumference of its circular base is 66 inch and a slant height of 13.7 inch. (i) How long is its radius? (ii) How many sq. inch of material is required to make the hat? (iii) How much cu.inch of the air does it hold? 10. a) Once Sumesh and his friends went for a picnic at a hill side. Due to peak season, they did not get a proper accommodation there. The weather was fine so they decided to make a conical tent in a park. They went to a tent house near by the picnic sport and brought the canvas enough to make the tent with base radius 10.5 m and height 10 m for rent. 132cm 35cm
Vedanta Excel in Mathematics - Book 10 106 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Mensuration (I): Pyramid (i) How many people were there in the group if each person required 7.7 square meter of space on the ground for the accommodation? (ii) If they paid the rent of the canvas equally, how much amount was added to each person’s bill at Rs 30 per sq. m? b) Due to heavy floods in a province of Nepal, thousands of people were victimized last year. Immediately an organization of 1,200 schools collectively offered to the province government to provide the canvas for 2,500 conical tents each of having base radius 5.6 m and height 4.2 m. They decided to share the whole expenditure equally. (i) How much canvas was provided for the tents? (ii) If the canvas used to make the tents cost Rs 450 per square meter, what was the amount shared by each school to set-up the tents? Project Work and Activity Section 11. a) Make the groups of your at least 5 friends. Draw two circles of same radii as large as possible on the chart paper and cut them out. Cut three sectors of different sizes and roll to form the cones. Measure the radii of base and slant heights. Then, find the curved surface area of each cone. b) Make a hollow right circular cone and a hollow right-circular cylinder of the same height and base radius such that it is open at the top and closed at the bottom. Fill the cone up to the brim with sand and pour it into the cylinder. Repeat the same process until the cylinder gets completely filled up with the sand. Write a short note on the relationship between the volumes of cone and cylinder. Multiple Choice Section 12. Tick (√) the correct alternative. (a) In a square based-pyramid, each side of base is a and slant height is l, then the area of its triangular surfaces is (A) al (B) 2al (C) a2 + 2al (D) a2 – 2l (b) The relationship among the base side a, height h and slant height l in a cone is (A) a2 + h2 = l 2 (B) a2 + h2 = l 2 (C) a2 + h2 = l 2 (D) a2 – h2 = l 2 (c) How much canvas is required to make a square based pyramid shaped tent having base length 18 feet and slant height 15 feet? (A) 270 sq. ft. (B) 540 sq. ft. (C) 864 sq. ft. (D) 1,296 sq. ft. (d) The formula to calculate the total surface area of right cone is (A) πrl (B) 2πrl (C) πr (r + l) (D) 2πr (r + l) (e) A cone with base radius is 12 cm has 21 cm height. Its volume is (A) 3,168 cm3 (B) 5,544 cm3 (C) 12 cm3 (D) 12 cm3