Volume and Surface Area

3D shapes: cube, cylinder, cone, sphere

104 practice questionsPage updated 27 Mar 2026Review metadata pendingPractice questions

Volume and Surface Area is a crucial topic in mensuration that deals with three-dimensional geometric shapes. This topic covers the calculation of space occupied by solids (volume) and the total area of their surfaces (surface area). Understanding these concepts is essential for solving real-world problems involving containers, packaging, construction, and manufacturing.

Key formulas and rules

Cube: Volume = a^3, TSA = 6a^2, LSA = 4a^2 (where a = side length)

Cuboid: Volume = l x b x h, TSA = 2(lb + bh + hl), LSA = 2h(l + b) (where l = length, b = breadth, h = height)

Cylinder: Volume = pir^2h, TSA = 2pir(r + h), CSA = 2pirh (where r = radius, h = height)

Cone: Volume = (1/3)pir^2h, TSA = pir(r + l), CSA = pirl (where l = sqrt(r^2 + h^2) is slant height)

Sphere: Volume = (4/3)pir^3, TSA = CSA = 4pir^2

Hemisphere: Volume = (2/3)pir^3, TSA = 3pir^2, CSA = 2pir^2

Hollow Cylinder: Volume = pih(R^2 - r^2), TSA = 2pi(R + r)(R - r + h) (where R = outer radius, r = inner radius)

Diagonal of Cuboid = sqrt(l^2 + b^2 + h^2)

Diagonal of Cube = asqrt3

Volume of material in hollow sphere = (4/3)pi(R^3 - r^3)

Key concepts

Total Surface Area vs Curved Surface Area

Total Surface Area (TSA) includes all surfaces of a solid including the base(s) and top. Curved Surface Area (CSA), also called Lateral Surface Area (LSA), includes only the curved or side surfaces, excluding the base and top. For example, for a cylinder, CSA is just the side (2pirh), while TSA includes the two circular bases as well (2pir^2 + 2pirh). Understanding this distinction is crucial as problems often ask specifically for one or the other.

Worked examples

Example 1

A metallic sphere of radius 6 cm is melted and recast into a cone of height 9 cm. Find the radius of the cone.

Solution

Step 1: Calculate the volume of the sphere.
Volume of sphere = (4/3) x pi x r^3 = (4/3) x pi x 6^3 = (4/3) x pi x 216 = 288pi cm^3

Step 2: Since the sphere is recast into a cone, the volume remains the same.
Volume of cone = (1/3) x pi x r^2 x h = 288pi

Step 3: Substitute h = 9 cm and solve for r.
(1/3) x pi x r^2 x 9 = 288pi
3pir^2 = 288pi
r^2 = 288/3 = 96
r = sqrt96 = 4sqrt6 ~= 9.80 cm

Answer: The radius of the cone is 4sqrt6 cm or approximately 9.80 cm.

Example 2

A hollow cylindrical pipe has external diameter 14 cm, internal diameter 10 cm, and length 20 cm. Find the volume of material used to make the pipe and the cost of painting its outer curved surface at 5 per cm^2.

Solution

Step 1: Find radii.
External radius R = 14/2 = 7 cm
Internal radius r = 10/2 = 5 cm

Step 2: Calculate volume of material.
Volume = pih(R^2 - r^2) = pi x 20 x (49 - 25) = pi x 20 x 24 = 480pi ~= 1507.96 cm^3

Step 3: Calculate outer curved surface area.
CSA_outer = 2piRh = 2 x pi x 7 x 20 = 280pi ~= 879.65 cm^2

Step 4: Calculate cost.
Cost = 879.65 x 5 = 4398.25

Answer: Volume of material ~= 1507.96 cm^3, Cost of painting ~= 4398.25

Example 3

A toy is in the form of a hemisphere surmounted by a cone. The diameter of the hemisphere is 12 cm and the total height of the toy is 16 cm. Find the total surface area of the toy.

Solution

Step 1: Determine dimensions.
Radius of hemisphere = 12/2 = 6 cm
Height of cone = Total height - radius = 16 - 6 = 10 cm

Step 2: Find slant height of cone.
l = sqrt(r^2 + h^2) = sqrt(36 + 100) = sqrt136 ~= 11.66 cm

Step 3: Calculate curved surface areas.
CSA of hemisphere = 2pir^2 = 2 x pi x 36 = 72pi cm^2
CSA of cone = pirl = pi x 6 x sqrt136 = 6pisqrt136 ~= 219.42 cm^2

Step 4: Calculate total surface area (note: the base of cone and top of hemisphere are internal).
TSA = CSA of hemisphere + CSA of cone = 72pi + 6pisqrt136 = pi(72 + 6sqrt136) ~= 445.68 cm^2

Answer: Total surface area ~= 445.68 cm^2 or pi(72 + 6sqrt136) cm^2

Example 4

How many spherical lead shots of diameter 2 mm can be obtained from a cuboid of dimensions 8 cm x 6 cm x 4 cm?

Solution

Step 1: Calculate volume of cuboid.
Volume of cuboid = 8 x 6 x 4 = 192 cm^3 = 192000 mm^3

Step 2: Calculate volume of one lead shot.
Radius = 2/2 = 1 mm
Volume of sphere = (4/3) x pi x 1^3 = (4/3)pi mm^3

Step 3: Find number of lead shots.
Number = Volume of cuboid / Volume of one shot
= 192000 / [(4/3) x pi] = 192000 x 3 / (4 x pi)
= 576000 / (4pi) = 144000/pi ~= 45836.68

Answer: Approximately 45,836 complete lead shots can be obtained.

Volume and Surface Area

Key formulas and rules

Key concepts

Total Surface Area vs Curved Surface Area

Hollow Solids

Combining Solids

Conversion and Transformation

Practical Applications

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Key formulas and rules

Key concepts

Total Surface Area vs Curved Surface Area

Hollow Solids

Combining Solids

Conversion and Transformation

Practical Applications

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

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