Math of Space: Surface Area and Volume
Paint the outside and you get surface area; fill the inside and you get volume — and a cone is just a pyramid that learned to be round.
Original notes — ready CBSE (NCERT, English Medium) • Class 9 • Mathematics • Book: Ganita Manjari — Part 2
Chapter 14: Math of Space: Surface Area and Volume — EG's own notes, in simple words
Book: Ganita Manjari — Part 2 (CBSE, Class 9) | Chapter 14 | Medium: English
Diagram (labels are printed on the image): A cuboid, a cylinder, a cone and a sphere side by side, each with its height and radius marked for surface area and volume.
Blocks, a ball and a cone — solid shapes whose surface and space we can measure.
What the chapter is about
Flat shapes have area; solid shapes have surface area and volume. Surface area is the total area of all the faces you could paint, and volume is the amount of space the solid fills — how much water it could hold. This chapter finds both for seven solids: cuboids, cylinders, cones, pyramids, spheres and hemispheres.
The formulas are not handed down as magic. A cylinder unrolls into a rectangle, so its curved surface is a rectangle's area. A cone's volume is exactly one-third of the cylinder that holds it. And Archimedes' beautiful result ties the sphere to its cylinder: a sphere fills exactly two-thirds of the cylinder that encloses it, in both surface area and volume.
The chapter's favourite problems are everyday ones: how many small balls fit in a box, how much water a tank holds, how much cloth covers a tent. Each one is just a careful choice of formula, followed by arithmetic you already know.
The chapter adds pyramidal shapes to the family. A pyramid has a polygon base and triangular faces meeting at one apex, and its volume is one-third of the base area times the height: V = (1/3) × base area × height. A square pyramid with a 6 cm by 6 cm base and 9 cm height therefore holds (1/3) × 36 × 9 = 108 cm³. The surface area is the base area plus the areas of all the triangular faces.
The cone fits the same pattern: think of it as a pyramid with a circular base, and its volume is the same one-third rule, (1/3)πr²h, with the circle's area πr² as the base. That is why a cone fills exactly one-third of the cylinder that holds it, and why the sphere's two-thirds rule for its enclosing cylinder feels like the same idea once more. Around us, tents, funnels, tanks and balls are all just these shapes, alone or joined — and a joined solid counts only its outside faces, never the faces glued together inside.
What you should learn
- Find surface area and volume of cuboids, cylinders, cones, spheres and hemispheres, and explain why the formulas work.
- Find the volume of a pyramid with V = (1/3) × base area × height.
- Treat a cone as a pyramid with a circular base.
- For joined solids, count only the outside faces.
Key Terms
- Surface area — The total area of all faces of a solid; for a closed solid it includes every face.
- Curved surface area — The area of the curved part of a solid only, excluding flat faces such as the base.
- Volume — The amount of space a solid occupies, measured in cubic units.
- Slant height — For a cone, the distance from the tip to a point on the base edge, written l, where l² = r² + h².
- Hemisphere — Half of a sphere, cut through its centre.
- Pyramid — A solid with a polygon base and triangular faces meeting at one apex; volume = (1/3) × base area × height.
- Cube — A cuboid with all edges equal: for edge a, surface area 6a² and volume a³.
Remember
- Cuboid: surface area = 2(lb + bh + hl); volume = lbh.
- Cylinder: curved area = 2πrh; volume = πr²h.
- Cone: curved area = πrl; volume = ⅓πr²h.
- Sphere: surface area = 4πr²; volume = ⁴⁄₃πr³. Hemisphere volume = ⅔πr³.
- One-third appears for pyramids and cones alike: both are one-third of the prism or cylinder that holds them.
- Total surface area counts flat faces too; curved surface area counts only the curved part.
Exam Tip / Common Mistake
Common mistake: using the curved surface area when the question asks for total surface area. A closed cylinder or hemisphere has flat faces too — add them.
Chart / Table: Surface area and volume of solids
| Point | Detail |
|---|---|
| Cuboid (l, b, h) | Total surface area 2(lb + bh + hl); volume lbh |
| Cylinder (r, h) | Curved 2πrh; total 2πr(r + h); volume πr²h |
| Cone (r, h, l) | Curved πrl; total πr(r + l); volume ⅓πr²h |
| Sphere (r) | Surface area 4πr²; volume ⁴⁄₃πr³ |
| Hemisphere (r) | Curved 2πr²; total 3πr²; volume ⅔πr³ |
Worked method: volume of a square pyramid
- Base is 6 cm by 6 cm, so base area = 36 cm². Height = 9 cm.
- Volume = (1/3) × base area × height = (1/3) × 36 × 9.
- 36 × 9 = 324, and 324/3 = 108. Volume = 108 cm³.
Practice set — try it yourself
- A cuboid is 5 cm by 4 cm by 3 cm. Find its volume and total surface area.
Hint: Hint: Volume = lbh; surface area = 2(lb + bh + hl). - A cylinder has radius 7 cm and height 10 cm. Find its curved surface area (π = 22/7).
Hint: Hint: Curved area = 2πrh. - A cone has radius 3 cm and height 4 cm. Find its slant height and volume.
Hint: Hint: l² = r² + h²; volume = ⅓πr²h. - Find the volume of a sphere of radius 3 cm (π = 22/7, approximate).
Hint: Hint: Volume = ⁴⁄₃πr³. - A hemispherical bowl has radius 7 cm. How much water does it hold? (π = 22/7)
Hint: Hint: Volume of a hemisphere = ⅔πr³.
Questions with Answers
- What is the difference between surface area and volume?
Surface area is the total area of a solid's faces, measured in square units — what you would paint. Volume is the space the solid occupies, measured in cubic units — what it would hold. - Find the total surface area of a cylinder with r = 7 cm and h = 10 cm (π = 22/7).
Total = 2πr(r + h) = 2 × (22/7) × 7 × 17 = 748 cm². - Why is a cone's volume one-third of its cylinder's?
A cone with the same base and height as a cylinder fills exactly one-third of that cylinder — this can be shown by pouring water from the cone into the cylinder three times. - Find the volume of a sphere of radius 7 cm (π = 22/7).
Volume = ⁴⁄₃πr³ = (4/3) × (22/7) × 343 = 4312/3 ≈ 1437.3 cm³. - A solid is a cylinder with a hemisphere on top. What is its total surface area made of?
The cylinder's curved surface, the hemisphere's curved surface, and the cylinder's base — the joining circle is inside the solid and is not counted.
Extension questions
- A square pyramid has a 6 cm by 6 cm base and height 9 cm. Find its volume.
V = (1/3) × 36 × 9 = 108 cm³. - A cone has radius 3 cm and height 4 cm. Find its volume in terms of π.
V = (1/3)πr²h = (1/3) × π × 9 × 4 = 12π cm³. - A cube has edge 5 cm. Find its total surface area and volume.
Surface area = 6 × 25 = 150 cm². Volume = 125 cm³. - Find the total surface area of a hemisphere of radius 7 cm (π = 22/7).
Total = 3πr² = 3 × (22/7) × 49 = 462 cm². - Find the surface area of a sphere of radius 3 cm, in terms of π.
Surface area = 4πr² = 4 × π × 9 = 36π cm². - A solid is a cylinder with a cone on top, same base radius. Which faces count in its total surface area?
The cylinder's curved surface, the cone's curved surface, and the cylinder's base — the circle where cone meets cylinder is inside the solid and is not counted.
Paint the outside, fill the inside: cuboids lbh, cylinders πr²h, cones and pyramids one-third of that, spheres 4πr² and ⁴⁄₃πr³ — and joined solids count only what shows.
How to use this page: Read the summary, say the Key Terms in your own words, then try the practice set without looking. Use the Questions with Answers to check yourself afterwards. Ask your teacher or parent to check your answers.