A water tank, a gas cylinder and an ice-cream cone are all solids. Exam questions ask how much each one holds and how much paint it needs.
This article puts every formula in one place. It adds shortcuts and five solved examples. Each example says why the method works.
- Volume is the space inside a solid. It is measured in cubic units.
- Surface area is the total outer covering. It is measured in square units.
- Learn three groups: box shapes, round shapes and recast problems.
- If the radius has a factor of 7, use π = 22/7 and cancel the 7 first.
- When a solid is melted and recast, its volume stays the same.
What do volume and surface area measure?
What is the difference between volume and surface area?
Volume is the space a solid fills. Think of the water inside a tank. It is measured in cubic units such as cm³ or m³.
Surface area is the area of the outside. Think of the paint on the tank. It is measured in square units such as cm² or m².
Sources: NCERT Class 8 Maths: Mensuration.
What are curved, lateral and total surface area?
Total surface area (TSA) covers every face, top and bottom too. Curved surface area (CSA) covers only the curved part of a cylinder, cone or sphere.
Lateral surface area means the four side walls of a box. It leaves out the top and bottom. For a cuboid it is 2h(l + b).
Sources: NCERT Class 8 Maths: Mensuration, cuboid and cylinder; Practice question: cuboid surface area.
How do you change cubic metres into litres?
Use 1 m³ = 1000 litres. Also 1 litre = 1000 cm³.
So a tank of 240 m³ holds 240,000 litres. Always make every length the same unit before you start.
Sources: NCERT Class 8 Maths: Mensuration, units of capacity.
How can you remember these formulas?
Group them. A cylinder is a stack of circles, so volume = base area × height. A cone holds one-third of the cylinder with the same base and height.
A hemisphere is half a sphere. Its volume is 2/3 π r³. Its curved part is 2π r². The flat lid adds π r², so the total is 3π r².
Sources: NCERT Class 10 Maths: Surface Areas and Volumes, cone and hemisphere; NCERT Class 8 Maths: Mensuration, cuboid and cylinder.
Which shortcuts save time in exams?
How do you find the longest pole in a room quickly?
The longest pole lies along the room’s space diagonal. Its length is √(l² + b² + h²).
Look for sets that make a perfect square. (1, 2, 2) gives 3. (2, 3, 6) gives 7. (3, 4, 12) gives 13. (4, 13, 16) gives 21.
Why does a cone need a slant height?
The cone’s side slopes, so its slant length l is longer than its height h. The radius, height and slant height form a right triangle. So l = √(r² + h²).
Learn these triples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17) and (20, 21, 29). Multiples also work. (21, 28, 35) is 7 × (3, 4, 5).
Sources: NCERT Class 10 Maths: Surface Areas and Volumes, cone and hemisphere; Practice question: cone curved surface area.
When should you take π as 22/7?
Use 22/7 when the radius or diameter has a factor of 7. The 7 cancels and the numbers stay small.
Example: for r = 21 cm, π r² = 22/7 × 21 × 21 = 22 × 3 × 21 = 1386 cm². Use 3.14 only when the question says so.
What happens when every length is multiplied by k?
Areas grow by k². Volumes grow by k³. This saves a full calculation.
- Double the edge of a cube: area × 4, volume × 8.
- Double the radius of a sphere: area × 4, volume × 8.
- Triple any length: area × 9, volume × 27.
How do you solve melt-and-recast problems?
Melting changes the shape, not the amount of metal. So volume before = volume after.
Write both volumes. Cancel what is common. Then solve for the unknown.
A cylinder recast into a cone of the same radius gives a cone three times as tall. A 4 cm cylinder becomes a 12 cm cone.
Sources: Practice question: cylinder melted into a cone; Practice question: what stays the same on melting.
Check 1
A cone with the same base and height as a cylinder holds one-third of its volume.
Check 2
Doubling a sphere’s radius doubles its surface area.
Check 3
The longest pole in a room 12 m long, 4 m wide and 3 m high is 13 m.
Check 4
A solid keeps the same surface area after it is melted and recast.
How do you solve five typical problems step by step?
Example 1: A room is 12 m long, 4 m wide and 3 m high. Find its volume, wall area and longest pole.
Step 1. Volume = l × b × h = 12 × 4 × 3 = 144 m³.
Step 2. Wall area = 2h(l + b) = 2 × 3 × (12 + 4) = 96 m².
Step 3. Longest pole = √(144 + 16 + 9) = √169 = 13 m.
Why this works. The pole runs corner to corner. It first crosses the floor diagonal √160. Then it rises 3 m, which gives √(160 + 9) = 13.
Example 2: A tank has radius 7 m and height 10 m. Find its capacity and the sheet for its side and ends.
Step 1. Base area = π r² = 22/7 × 49 = 154 m². Volume = 154 × 10 = 1540 m³, which is 1,540,000 litres.
Step 2. Curved side = 2π r h = 2 × 22/7 × 7 × 10 = 440 m².
Step 3. Total = 2π r (r + h) = 44 × 17 = 748 m². Check: 440 + 2 × 154 = 748.
Why this works. Volume stacks the base circle 10 m high. The curved side opens out into a flat rectangle 2π r wide and h tall.
Example 3: A cone has radius 21 cm and height 28 cm. Find its slant height, CSA, TSA and volume.
Step 1. l = √(21² + 28²) = √(441 + 784) = √1225 = 35 cm.
Step 2. CSA = π r l = 22/7 × 21 × 35 = 2310 cm².
Step 3. Base = π r² = 1386 cm². TSA = 2310 + 1386 = 3696 cm².
Step 4. Volume = ⅓ π r² h = ⅓ × 1386 × 28 = 12,936 cm³.
Why this works. The sides 21 and 28 are 7 × (3, 4) and the slant is 7 × 5. The flat base adds one circle to the CSA.
Example 4: A hemisphere-shaped bowl has radius 21 cm. Find its volume and total surface area.
Step 1. Sphere volume = 4/3 × 22/7 × 21³ = 38,808 cm³. The hemisphere is half of that.
Step 2. Volume = 19,404 cm³, which is about 19.4 litres.
Step 3. TSA = 3π r² = 3 × 1386 = 4158 cm². The curved part alone is 2 × 1386 = 2772 cm².
Why this works. The curved part is half of 4π r². The lid is a circle of area π r². So 2π r² + π r² = 3π r².
Example 5: A sphere of radius 9 cm is melted into small spheres of radius 3 cm. How many are made?
Step 1. Volume is proportional to r³. Big sphere : small sphere = 9³ : 3³ = 729 : 27.
Step 2. Number made = 729 ÷ 27 = 27 small spheres.
Why this works. All the metal is reused, so the volumes match. The 4/3 π part is the same on both sides and cancels.
Try this. Cubes of edge 3, 4 and 5 cm melt into one cube. Total volume = 27 + 64 + 125 = 216 cm³. The new edge is 6 cm. Its TSA is 6 × 36 = 216 cm².
How do you check your answer?
What quick check catches big mistakes?
Fit the solid inside a box. A sphere of radius 21 cm fits in a 42 cm cube. That cube holds 42³ = 74,088 cm³.
The sphere’s volume of 38,808 cm³ is about half of that. This is right, because a sphere fills about 52 percent of its cube.
Also check units. Area has two lengths multiplied. Volume has three.
Check 5
The total surface area of a hemisphere is 3π r².
Check 6
The CSA of a cone is π r h.
Exam traps
Slant trap. The CSA of a cone uses l, not h. Find l first.
Lid trap. The TSA of a hemisphere is 3π r², not 2π r². Do not forget the lid.
Unit trap. Put every length in one unit first. Remember 1 m³ = 1000 litres.
Scale trap. Double the radius and the area becomes four times as large, but the volume becomes eight times.
Pole trap. The longest pole is the space diagonal, not the floor diagonal.
MCQ on Volume and Surface Area
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