Step-by-step solutions for Class 9 Mathematics Ganita Manjari Part 1I Chapter 14 Exercise Set 14.3 (Questions 1–9, Page 134). Covers curved surface area, total surface area, and volume of right circular cones, finding base radius and slant height, fabric requirement for conical tents with wastage, solid generated by rotating a right-angled triangle, and volume fraction of a conical cup filled to half depth.
For a right circular cone of base radius r and vertical height h, the slant height l represents the hypotenuse of the right-angled triangle formed by the axis and radius.
The curved (lateral) surface area of a right circular cone is obtained by unrolling its curved surface into a circular sector of radius l and arc length 2πr.
The total surface area (TSA) of a cone includes its curved surface area plus the area of its flat circular base.
The volume of a right circular cone is one-third the volume of a cylinder having the same base radius r and height h.
When a right triangle is rotated through 360° about one of its perpendicular legs of length h, the other leg forms the circular base radius r, and the hypotenuse forms the slant height l.
Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
r = 242 = 12 m
The total surface area of a cone is
TSA = πrl + πr² = πr(l + r)
TSA = 3.14 × 12 × (21 + 12)
Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.
The curved surface area of a cone is
The height of a cone is 16 cm, and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone.
First, find the slant height using the relation given in the chapter:
TSA = 3.14 × 12 × 20 + 3.14 × 12²
A cone has a height of 15 cm. If its volume is 1570 cm³, find the radius of the base.
The volume of a cone is
1570 = (13) × 3.14 × r² × 15
Since the radius is positive,
The curved surface area of a cone is 308 cm² and its slant height is 14 cm. Find (i) the radius of the base, (ii) the total surface area of the cone.
The curved surface area of a cone is
The total surface area of a cone is
A joker's cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
First, find the slant height using the relation
A joker's cap is open at the bottom, so only its curved surface needs to be covered.
The curved surface area of a cone is
CSA of one cap = (227) × 7 × 25
What length of tarpaulin 3 m wide is required to make a conical tent of height 8 m and base radius 6 m? Assume that the extra length of material required for stitching margins and wastage in cutting is 20 cm.
Extra length required = 20 cm = 0.2 m
First, find the slant height:
The tent has no floor, so only its curved surface needs to be covered.
Since the tarpaulin is 3 m wide,
A right triangle with sides 6 cm, 8 cm and 10 cm is rotated through 360° about the side of 8 cm. Find the volume and the curved surface area of the solid so formed.
Rotating the right triangle about the 8 cm leg generates a right circular cone with height h = 8 cm, base radius r = 6 cm, and slant height l = 10 cm.
When the triangle is rotated through 360° about the side of 8 cm, it forms a right circular cone.
The dimensions of the cone are:
The volume of a cone is
V = (13) × 3.14 × 6² × 8
The curved surface area of a cone is
Suppose you have a cup in the shape of a right circular cone. Fill it with water to half the depth of the cone. What fraction of the volume of the cup is occupied by the water?
When a conical cup is filled to half depth (h = H/2), the radius of the water surface is also halved (r = R/2), resulting in a water volume that is (1/2)³ = 1/8 of the cup's total volume.
Let the height and base radius of the cup be H and R, respectively.
The volume of the whole cone is
The water fills the cup to half its depth. Therefore, the height of the water-filled cone is
Since the water surface is parallel to the circular top of the cup, the water forms a smaller cone of the same shape. Its height is half the height of the whole cone, so its radius is also half the original radius:
The volume of the water is
Need to practice other sections in Chapter 14?
Return to Chapter 14 (Math of Space: Surface Area and Volume)→