Class 9 Mathematics

Chapter 14 Solutions — Math of Space: Surface Area and Volume

Open Solutions PDF →Also seeTextbook page→

Open Solutions PDFReads in your browser→Also seeTextbook page→

Get the full experience in the app

Free practice, offline PDFs & solutions.

Already installed? Open in the app

Solutions

Overview

Step-by-step NCERT solutions for Math of Space: Surface Area and Volume (Chapter 14, NCERT Class 9 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Math of Space: Surface Area and Volume textbook chapter.

Solved

What these solutions cover

All 52 questions in Math of Space: Surface Area and Volume are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise Set 14.1

The volume of a cube is 64 cm³. What is its total surface area?
How many small cubes with side 20 cm can be packed tight in a cubical box with side 2 m?
The dimensions of a godown are 40 m × 25 m × 10 m. If it is filled with cuboidal boxes, each of dimensions 2 m × 1.25 m × 1 m, then find the number of boxes.
Two cubes each of volume 125 cm³ are joined end to end. Find the surface area of the resulting cuboid.
A cube of side 4 cm is cut into cubes of side 1 cm. What is the ratio of the surface areas of the original cube and all the cut-out cubes?
The surface areas of the three faces of a cuboid that meet at one of the corners of the cuboid are 6 cm², 15 cm², and 10 cm² respectively. What is the volume of the cuboid?
7.
A cube of side 5 cm is painted on all its faces. If it is sliced into 1 cm³ cubes, how many of these 1 cm³ cubes have
- (i) exactly three faces painted?
- (ii) exactly two faces painted?
- (iii) exactly one face painted?
- (iv) no face painted?
8.
Find a cuboid with edges whose lengths are integers (in cm), given that it has a total surface area of exactly 100 cm².
- (i) Is there more than one such cuboid?
- (ii) Can you find them all?
- (iii) Show that you have found them all.

Exercise Set 14.2

Two cylinders, A and B, are given. The radius of cylinder B is twice that of cylinder A, and the height of cylinder B is half that of cylinder A. Find the ratio of the curved surface area of A to the curved surface area of B. Also find the ratio of the volume of A to the volume of B.
2.
The radii of two cylinders are in the ratio 2:3, and their heights are in the ratio 3:2. Find
- (a) the ratio of their volumes, and
- (b) the ratio of their curved surface areas.
The edge of a cube measures r cm. The largest possible right circular cylinder is cut out of the cube. What do you think is the volume of the cylinder (in cm³)?
The radius of the base of a cylinder is increased by 10%. At the same time, the height of the cylinder is decreased by x%. Given that the volume of the cylinder remains unchanged, find the value of x.
5.
A solid metallic cube of side 12 cm is melted and recast into solid cylindrical rods, each having radius 2 cm and height 12 cm. Find:
- (i) the volume of the cube,
- (ii) the volume of one cylindrical rod,
- (iii) the approximate number of complete cylindrical rods that can be formed. (π ≈ 22/7)

Exercise Set 14.3

Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.
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.
A cone has a height of 15 cm. If its volume is 1570 cm³, find the radius of the base.
5.
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.
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.
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 that is required for stitching margins and wastage in cutting is 20 cm.
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.
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?

Exercise Set 14.4

A ball bearing has a radius of 0.7 cm. Find its surface area.
Two solid spheres made of the same metal have weights 5920 g and 740 g. Determine the radius of the larger sphere, if the diameter of the smaller one is 5 cm.
The diameter of the moon is approximately one fourth the diameter of the earth. Given that the moon and earth are both roughly spherical, find the ratio of their surface areas.
4.
Find
- (i) the curved surface area and
- (ii) the total surface area of a hemisphere of radius 21 cm.
Metal spheres, each of radius 2 cm, are packed into a rectangular box of internal dimensions 16 cm × 8 cm × 8 cm. When 16 spheres are packed, the box is filled with preservative liquid. Find the volume of this liquid. Round your answer to the nearest integer.
The radius of a sphere is increased by 10%. Show that the volume increases by approximately 33.1%.
The radius of a sphere is increased by x%. The volume of the sphere increases by 72.8%. Find the value of x.
The hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 35.2 m, find the cost of painting it, given the cost of painting is ₹10 per 100 cm².

End-of-Chapter Exercises

Estimate how many scoops of ice cream can be obtained from a cuboidal container of dimensions 10 cm × 15 cm × 20 cm.
A cube of integer side length a is made of unit cubes. Write an expression giving the number of unit cubes to be added to make a cube of side length a + 1.
3.
Solve the following:
- (i) Could a person drink enough water in a lifetime to fill an entire room the size of your classroom?
- (ii) Estimate the number of bricks used to build the walls of your classroom.
You are given two options: (A) one chocolate cube of side 50 mm, (B) hundred chocolate cubes of side 10 mm. Which option gives you more chocolate?
If all the people in the world are crowded together in one location, how much area would it cover?
A school provides milk to its students in cylindrical glasses, each with a diameter of 7 cm. If each glass is filled with milk to a height of 12 cm, how many litres of milk are needed to serve 1600 students? Use 1000 cm³ = 1 litre.
The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height and the volume of the cone.
8.
Take Earth to be a perfect sphere with radius 6370 km. Take Jupiter to be a perfect sphere with radius 69,900 km. Take the Sun to be a perfect sphere with radius 6,95,700 km. Compute approximately:
- (i) the ratio of the volume of Jupiter to the volume of the Earth;
- (ii) the ratio of the volume of the Sun to the volume of the Earth. (Calculator can be used.)
Show that the volume of a sphere is equal to 2/3 of the volume of the smallest cylinder which encloses it.
10.
Suppose the Earth is perfectly spherical. A string is wrapped tightly around the equator of the Earth. Another string, 1 metre longer, is placed around the Earth so that it forms a larger circle, staying the same distance above the ground everywhere. How high above the ground is the second string? Repeat this for:
- (i) the Moon
- (ii) Jupiter
- (iii) a volleyball. Are you surprised by the three…
We have a cylinder with a base radius of r cm and height h cm. A square pyramid is fitted inside it. The square base of the pyramid lies on the base of the cylinder, its corners on the boundary of the cylinder. The apex of the pyramid lies on the top of the cylinder. Find the ratio of the volume of this pyramid to the volume of the cylinder.
12.
What is the change in volume when:
- (i) the length of a cuboid with dimensions l, w, h is increased by 1 unit?
- (ii) the radius of a cylinder with dimensions r, h is increased by 1 unit? *(iii) the radius of a sphere is decreased by 1 unit?
Given a cube with volume V, express its total surface area S in terms of V.
Given a cube with total surface area S, express its volume V in terms of S.
A cylindrical glass of height 25 cm and radius 4 cm has water up to a height of 16 cm. A crow wants to drink water from this glass. The water must be at a height of 20 cm for the crow to reach it. There are some marbles lying around. How many marbles of radius 1 cm should the crow drop into the glass to make the water reach the required height?
Find the volume of ink in a new ball point pen. Take the necessary measurements and make approximations, as needed.
Solve: (i) A ball of chapati dough of radius 6 cm is prepared. Estimate how many chapatis can be made from it? *(ii) Cut a coconut/muskmelon in half and find out the approximate volume of edible coconut flesh/fruit by taking the necessary measurements.
18.
Looking at the Ganita Manjari, Grade 9, Part 2 textbook, Sheela wonders:
- (i) If all the pages of this textbook were laid out side-by-side on the floor would they cover the entire classroom floor?
- (ii) What is the maximum number of textbooks that can fit in an empty storeroom of dimensions 15 ft × 20 ft × 30 ft?
If the entire human population decided to climb into one giant cube, how long would the side have to be?
20.
The Earth's surface has an estimated volume of 1.38 billion km³ of water. Suppose the Earth is a perfect sphere and all this water forms a uniform layer completely covering the Earth's surface, like a thin water bubble. Estimate the thickness of this water layer. The Earth's radius is ~6371 km.
- (i) Write an expression that gives the thickness of this water layer.
- (ii) Simplify the expression in…
Give the dimension of a cuboid whose volume is halved when its surface area is doubled.
Project: Find the volume of your house making necessary approximations. Present how you solved it.

Keep solving