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REET L1+L2 · Maths

Surface Area and Volume

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REET Maths questions on Surface Area and Volume are directly formula-based, but they often combine two solid shapes (cutting a cylinder out of a cube, melting spheres together, and so on). Here are all the formulas for a cube, cuboid, cylinder, cone and sphere, along with REET's recurring question patterns.

  • 4 questions in REET 2024
  • Class 8-10 level
  • 10 min read
Verified 17 September 2026

How REET asks this topic

None of REET 2024's four questions were solved by plugging a number into a single formula — each combined two ideas: finding a cube's side from its surface area and then its diagonal, finding the ratio of radii for cylinders of equal volume, cutting the largest possible cylinder out of a cube, and melting three cubes into a single new cube (volume conservation). So along with memorising the formulas, you need to know which formula and which approach applies to which situation.

Cube and Cuboid formulas

QuantityCube (side a)Cuboid (length l, breadth b, height h)
Total surface area (TSA)6a²2(lb + bh + hl)
Lateral surface area (LSA)4a²2h(l + b)
Volumea³l × b × h
Diagonala√3√(l² + b² + h²)

Cylinder, cone, sphere and hemisphere formulas

SolidCurved surface (CSA)Total surface (TSA)Volume
Cylinder (radius r, height h)2πrh2πr(r + h)πr²h
Cone (radius r, height h, slant height l = √(r² + h²))πrlπr(r + l)(1/3)πr²h
Sphere (radius r)4πr²4πr² (no flat base)(4/3)πr³
Hemisphere (radius r)2πr²3πr²(2/3)πr³

Volume conservation questions

When a solid is melted or recast into a different shape (like a cube being melted into a sphere, or several small cubes melted into one large cube), the total volume doesn't change — only the shape does. Questions built on this principle are solved by setting the volumes of both solids equal and forming an equation.

Example: if a cylinder of radius 3 cm and height 4 cm is melted to form a sphere, find the sphere's radius. Volume of the cylinder = πr²h = π × 3² × 4 = 36π. Volume of the sphere = (4/3)πR³, so (4/3)πR³ = 36π ⇒ R³ = 27 ⇒ R = 3 cm.

The largest solid cut from a cube, and the equal-volume rule

Largest cylinder cut from a cube of side a
Radius = a/2, height = a
Largest sphere that fits inside a cube of side a
Radius = a/2
Two cylinders of equal volume (radii r₁, r₂ and heights h₁, h₂)
r₁²h₁ = r₂²h₂
Total surface of a composite shape made of a cone and hemisphere (like an ice-cream cone)
CSA(cone) + CSA(hemisphere) = πrl + 2πr²

Being careful with unit conversion

In surface area and volume questions, the length is often given in metres while breadth-height are given in centimetres, or the other way around. Before substituting values into a formula, converting every measurement to the same unit is essential, otherwise the answer will be completely wrong.

Another important conversion worth remembering in practical questions: 1 litre = 1,000 cubic centimetres (cm³). In questions asking for the capacity of a water tank or a container (in litres), the volume is first found in cubic centimetres and then divided by 1,000.

Questions on combinations of solids

REET and other exams sometimes ask for the surface area or volume of a shape formed by joining two solids together (like an ice-cream shape made by placing a hemisphere on a cone, or a capsule/tablet shape made by joining a hemisphere to each end of a cylinder). In such questions, only the visible (outer) surface of each solid is added — the surface that gets hidden where the two solids join is never counted.

Example: a cylinder of radius 3.5 cm has a hemisphere of the same radius, 3.5 cm, joined to each of its two ends, and the cylinder's height is 6 cm. The total surface area of this 'capsule' shape = the cylinder's curved surface + the curved surfaces of both hemispheres (the flat ends of the cylinder aren't included, because they're covered by the hemispheres). For volume, though, the volumes of all three parts (the cylinder plus both hemispheres) are simply added together, because this is solid content, not surface.

How to teach children

Classroom activity

Ask children to bring a cuboid (a matchbox), a cylinder (a tin can) and a sphere (a ball) from home, have them measure and work out the surface area and volume, and then build a paper net (unfolded shape) and count the surfaces. Groups can be given different solids and results compared in class.

Teaching aids

Paper models and nets of solid shapes, and an experiment pouring water from one container into another so children can see that volume (quantity) doesn't change when the shape changes. Sand or lentils work just as well for this demonstration.

Common misconception

Students often swap the surface area and volume formulas around (like writing a cylinder's volume as 2πrh), and confuse 'curved surface area' with 'total surface area'. Some children plug different units (metres, centimetres) straight into a formula without converting them.

Assessment idea

Have students measure the diameter and height of a real water bottle, work out its volume, and compare it to the actual capacity (in millilitres) printed on the bottle — this helps them see the gap between calculation and reality.

Memory trick / exam tip

A sphere is the only solid whose curved surface and total surface are the same (4πr²), because it has no flat base — whereas for a cylinder, cone or hemisphere, the base area has to be added separately to get the TSA. For composite solids, also remember: the joining surface doesn't get counted in surface area, but every part's full volume is counted in volume — this distinction is exactly where exams often create confusion.

Questions asked in REET 2024 (with solutions)

Question 111

If the surface area of a cube is 600 cm², then the length of its diagonal will be :

  1. A3√10 सेमी
  2. B10√3 सेमी
  3. C10√2 सेमी
  4. D10 सेमी

Explanation

घन का कुल पृष्ठीय क्षेत्रफल = 6a²। 6a² = 600 से a² = 100, अतः a = 10 सेमी। घन का विकर्ण = a√3 = 10√3 सेमी। 10√2 सेमी तो एक फलक (face) का विकर्ण होता है।

Question 116

If the ratio of heights of two cylinders of equal volume is 1 : 3 then the ratio of their radii will be :

  1. A1 : 3
  2. B√3 : 1
  3. C1 : √2
  4. D√5 : 2

Explanation

बेलन का आयतन = πr²h। आयतन बराबर होने पर πr₁²h₁ = πr₂²h₂, अतः r₁²/r₂² = h₂/h₁ = 3/1। इसलिए r₁ : r₂ = √3 : 1। ऊँचाई कम होने पर समान आयतन के लिए त्रिज्या अधिक होनी चाहिए।

Question 102

The curved surface area of the largest cylinder that can be made by cutting from a cube of side 8 cm will be

  1. A32π सेमी²
  2. B64π सेमी²
  3. C128π सेमी²
  4. D16π सेमी²

Explanation

8 सेमी भुजा वाले घन से काटे गए सबसे बड़े बेलन का व्यास = 8 सेमी (त्रिज्या r = 4 सेमी) और ऊँचाई h = 8 सेमी होगी। वक्रपृष्ठ का क्षेत्रफल = 2πrh = 2 × π × 4 × 8 = 64π सेमी²। 128π पाने के लिए त्रिज्या 8 मान लेना गलती है।

Question 117

The side of the cube made by melting cubes of sides 3 cm, 4 cm, 5 cm will be

  1. A12 सेमी
  2. B6 सेमी
  3. C216 सेमी
  4. D50 सेमी

Explanation

पिघलाकर नया घन बनाने पर कुल आयतन समान रहता है: 3³ + 4³ + 5³ = 27 + 64 + 125 = 216 सेमी³। नए घन की भुजा = ∛216 = 6 सेमी। 216 आयतन है, भुजा नहीं; 12 सेमी भुजाओं का योग है।

Practice questions

Question 1

The total surface area of a cube is 294 sq cm. Find its side.

  1. A6 सेमी
  2. B7 सेमी
  3. C8 सेमी
  4. D9 सेमी

Explanation

6a² = 294 ⇒ a² = 49 ⇒ a = 7 cm.

Question 2

A cuboid has length 10 cm, breadth 5 cm and height 4 cm. Find its volume.

  1. A150 सेमी³
  2. B200 सेमी³
  3. C180 सेमी³
  4. D220 सेमी³

Explanation

Volume = l × b × h = 10 × 5 × 4 = 200 cm³.

Question 3

Find the curved surface area of a cylinder with radius 7 cm and height 10 cm (take π = 22/7).

  1. A440 सेमी²
  2. B220 सेमी²
  3. C400 सेमी²
  4. D880 सेमी²

Explanation

CSA = 2πrh = 2 × 22/7 × 7 × 10 = 440 cm².

Question 4

Find the volume of a cone with radius 6 cm and height 7 cm (take π = 22/7).

  1. A264 सेमी³
  2. B252 सेमी³
  3. C308 सेमी³
  4. D176 सेमी³

Explanation

Volume = (1/3)πr²h = (1/3) × 22/7 × 36 × 7 = (1/3) × 22 × 36 = 264 cm³.

Question 5

The volume of a cube is 125 cm³. Find the length of its diagonal.

  1. A5√3 सेमी
  2. B5√2 सेमी
  3. C25 सेमी
  4. D15 सेमी

Explanation

a³ = 125 ⇒ a = 5 cm. Diagonal = a√3 = 5√3 cm.

Question 6

Three cubes of sides 6 cm, 8 cm and 10 cm are melted to form a new cube. Find the side of the new cube.

  1. A12 सेमी
  2. B24 सेमी
  3. C18 सेमी
  4. D1728 सेमी

Explanation

Total volume = 6³ + 8³ + 10³ = 216 + 512 + 1000 = 1728 cm³. The new cube's side = ∛1728 = 12 cm.

Question 7

Find the curved surface area of the largest cylinder that can be cut from a cube of side 14 cm (take π = 22/7).

  1. A616 सेमी²
  2. B308 सेमी²
  3. C1232 सेमी²
  4. D196 सेमी²

Explanation

The largest cylinder has radius = 14/2 = 7 cm and height = 14 cm. CSA = 2πrh = 2 × 22/7 × 7 × 14 = 616 cm².

Question 8

Two cylinders of equal volume have their radii in the ratio 2:1. Find the ratio of their heights.

  1. A1:4
  2. B4:1
  3. C1:2
  4. D2:1

Explanation

When volumes are equal, r₁²h₁ = r₂²h₂. So h₁/h₂ = r₂²/r₁² = 1²/2² = 1/4, i.e. h₁:h₂ = 1:4.

Question 9

In which solid are the curved surface area (CSA) and the total surface area (TSA) equal?

  1. Aबेलन
  2. Bशंकु
  3. Cगोला
  4. Dघनाभ

Explanation

A sphere has no flat base, so its entire surface is curved surface — CSA = TSA = 4πr².

Question 10

Which is the most suitable activity to help students understand 'conservation of volume' (volume does not change when shape changes)?

  1. Aसूत्र बोर्ड पर लिखकर याद कराना
  2. Bएक निश्चित मात्रा का पानी अलग-अलग आकार के बर्तनों में डालकर दिखाना कि आयतन (मात्रा) समान रहता है
  3. Cकेवल पाठ्यपुस्तक के उदाहरण हल कराना
  4. Dविषय को छोड़ देना

Explanation

Pouring a fixed quantity of water into containers of different shapes and showing that the volume (quantity) stays the same gives children direct, hands-on proof — far more lasting than memorising a formula.

Frequently asked questions

How many questions come from surface area and volume in REET?+

REET 2024 had 4 questions from this topic. These are numerical questions, so along with memorising the formulas, regular calculation practice is essential.

How do you find a cone's slant height?+

A cone's slant height l = √(r² + h²), where r is the radius and h is the cone's height — this is based on the Pythagoras theorem and is used in both the TSA and CSA formulas of a cone.

What's the difference between the surface areas of a sphere and a hemisphere?+

A sphere's total surface area is 4πr², while a hemisphere's curved surface is 2πr² and total surface (including the base) is 3πr² — the difference is exactly the area of the base circle, πr².

What stays unchanged when two solids are melted to form a third?+

The total volume doesn't change when the shape changes through melting — this is the principle of volume conservation, and questions based on it are solved by setting the volumes of the two solids equal.

What should you watch for when finding the surface area of a composite solid (like a cone + hemisphere)?+

A surface that gets hidden where it joins another solid (like a cone's base joined to a hemisphere) is not counted in the total surface area — only the surfaces visible from outside are added, while volume adds up the volumes of both parts directly.

Read next

Perimeter and Area

Revise the perimeter and area formulas for 2D shapes.

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Next topic — reading data in tables and graphs.

REET Syllabus

See the complete REET Level 1 and Level 2 syllabus in one place.

REET 2024 Solved Papers

Read all of REET 2024's solved questions, subject by subject.

Change Log

  • 17 September 2026 — English page published — mirrors the fact-checked Hindi page, independently bilingual-accuracy checked