Mensuration 3D with Volume and Surface Area Formulas

Mensuration 3D deals with the measurement of solid figures such as cuboids, cubes, cylinders, cones, spheres and hemispheres. Questions generally require volume, curved or lateral surface area, total surface area, capacity, or a missing dimension. This page covers mensuration 3D formulas, unit conversions, standard properties and calculation methods used in aptitude questions.

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What is Mensuration 3D?

Mensuration 3D is the study of the surface area and volume of three-dimensional solids. Surface area is measured in square units, while volume and capacity are measured in cubic units.

A solid has length, breadth or width, and height. Surface area measures the exposed outer surface, and volume measures the space occupied by the solid. Use consistent units before calculation: 1 m = 100 cm, 1 m² = 10,000 cm², and 1 m³ = 1,000,000 cm³. Also, 1 litre = 1,000 cm³ and 1 millilitre = 1 cm³.

Mensuration 3D Video Lessons

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8 Lessons
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Combined Solids: Volume and Surface Area

Learn how to calculate the volume and surface area of combined three-dimensional solids by identifying component shapes and applying the appropriate mensuration formulas.

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Mensuration 3D Formula & Tricks

Important Formulas

Cuboid volume
V = l × b × h

For a cuboid with length l, breadth b and height h, volume is the product of its three dimensions.

Cuboid surface areas
LSA = 2h(l + b), TSA = 2(lb + bh + hl)

LSA is the area of the four side faces. TSA includes all six faces.

Cube
V = a³, LSA = 4a², TSA = 6a²

Here, a is the edge of the cube. All six faces are equal squares.

Cylinder
V = πr²h, CSA = 2πrh, TSA = 2πr(h + r)

Here, r is the radius and h is the height. CSA excludes the two circular bases.

Cone
l = √(r² + h²), V = (1/3)πr²h, CSA = πrl, TSA = πr(l + r)

For a right circular cone, l is the slant height, r is the base radius and h is the perpendicular height.

Sphere
V = (4/3)πr³, Surface area = 4πr²

A sphere has one continuous curved surface and no separate base.

Hemisphere
V = (2/3)πr³, CSA = 2πr², TSA = 3πr²

The total surface area includes the circular base, while curved surface area excludes it.

Prism
V = base area × height

The height is the perpendicular distance between the two congruent parallel bases.

Quick Tricks

Use diameter carefully

If a question gives diameter d, first use r = d/2 in formulas containing r² or r³. Do not substitute the diameter directly for radius.

Example: For a cylinder with d = 14 cm, r = 7 cm. Its base area is π × 7², not π × 14².
Compare volumes by dimensions

When solids have the same height and radius or base dimensions, common factors can be cancelled before calculation.

Example: A cone and cylinder with the same r and h have volumes in the ratio (1/3)πr²h : πr²h = 1 : 3.
Check the required surface

Use CSA or LSA when bases are not exposed or are excluded. Use TSA when every outer face, including bases, is counted.

Example: Painting the curved wall of an open cylinder requires CSA, 2πrh. Painting a closed cylinder requires TSA, 2πr(h + r).
Convert capacity after finding volume

Calculate volume in cubic units first, then convert it to capacity. Use 1 litre = 1,000 cm³ and 1 m³ = 1,000 litres.

Example: A tank with volume 2.5 m³ has capacity 2,500 litres.

Mensuration 3D Concepts

Cuboid and cube

A cuboid has three possibly different dimensions, whereas a cube has all twelve edges equal.

For a cuboid, volume is lbh and total surface area is 2(lb + bh + hl). For a cube of edge a, volume is a³ and total surface area is 6a². The diagonal of a cuboid is √(l² + b² + h²), while the diagonal of a cube is a√3.

Example: A cuboid measuring 5 cm × 4 cm × 3 cm has volume 5 × 4 × 3 = 60 cm³ and total surface area 2(20 + 12 + 15) = 94 cm².

Cylinder and its exposed surfaces

A cylinder has two equal circular bases joined by a curved surface.

Its curved surface area is 2πrh, and its total surface area is 2πr(h + r). The volume is πr²h. For an open cylinder, exclude the area of the missing base: one-base area is 2πrh + πr², and no-base area is 2πrh.

Example: For r = 3 cm and h = 10 cm, using π = 22/7, the volume is π × 9 × 10 = 1980/7 cm³.

Cone and slant height

The slant height of a right circular cone is l = √(r² + h²), and it is different from the perpendicular height h.

The radius, perpendicular height and slant height form a right triangle. The cone volume is one-third of the volume of a cylinder having the same base radius and height. Curved surface area uses πrl, while total surface area adds the base area πr².

Example: If r = 5 cm and h = 12 cm, then l = √(25 + 144) = 13 cm. The curved surface area is 65π cm².

Sphere and hemisphere

A sphere has no flat face, while a hemisphere is half of a sphere and has one circular base.

A sphere of radius r has volume 4πr³/3 and surface area 4πr². A hemisphere has volume 2πr³/3, curved surface area 2πr² and total surface area 3πr². The base is included only in the total surface area of a hemisphere.

Example: For a hemisphere with r = 3 cm, its total surface area is 3π × 9 = 27π cm².

Combined solids and volume conservation

For a solid made by joining or removing shapes, calculate each relevant volume separately and add or subtract them.

When a solid is melted and recast without loss, its volume remains constant. Thus, volume of the original solid equals the sum of the volumes of the recast solids. For a cavity, subtract the cavity volume from the outer solid volume.

Example: A solid cylinder of volume 500 cm³ is recast into two equal cones. Each cone has volume 250 cm³.

Mensuration 3D Visual Guide & Data

Use the table and chart below to understand the data better.

Standard 3D mensuration formulas

Common volume and surface-area formulas for solids used in aptitude questions.

Solid Volume Curved or lateral surface area Total surface area
Cuboid lbh 2h(l + b) 2(lb + bh + hl)
Cube a³ 4a² 6a²
Cylinder πr²h 2πrh 2πr(h + r)
Cone (1/3)πr²h πrl πr(l + r)
Sphere (4/3)πr³ 4πr² 4πr²
Hemisphere (2/3)πr³ 2πr² 3πr²

Note: In the cone, l is slant height. For a sphere, the entire surface is curved, so curved and total surface areas are equal.

Cuboid with length, breadth and height

Visual guide: Cuboid with length, breadth and height
l: 6 cm b: 4 cm h: 3 cm
A cuboid has volume l × b × h and total surface area 2(lb + bh + hl).

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Practice Mensuration 3D Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Mensuration 3D revision points

Recall the standard solid formulas, surface-area rules and unit conversions.

  • Volume is measured in cubic units; surface area is measured in square units.
  • Cuboid: V = lbh and TSA = 2(lb + bh + hl).
  • Cylinder: V = πr²h, CSA = 2πrh and TSA = 2πr(h + r).
  • Cone: l = √(r² + h²), V = (1/3)πr²h and CSA = πrl.
  • Sphere: V = 4πr³/3 and surface area = 4πr².
  • Hemisphere: V = 2πr³/3, CSA = 2πr² and TSA = 3πr².
  • Use r = d/2 when diameter is given.
  • 1 litre = 1,000 cm³; 1 m³ = 1,000 litres.

Mensuration 3D FAQs

What is the formula for the total surface area of a cuboid?

The total surface area is 2(lb + bh + hl), where l, b and h are its length, breadth and height.

What is the difference between CSA and TSA of a cylinder?

Cylinder CSA is 2πrh and includes only the curved surface. TSA is 2πrh + 2πr² = 2πr(h + r), so it also includes both circular bases.

How is the slant height of a cone calculated?

For a right circular cone, l = √(r² + h²). If r = 5 cm and h = 12 cm, l = √169 = 13 cm.

What is the total surface area of a hemisphere?

The total surface area is 3πr². It equals curved surface area 2πr² plus the circular base area πr².

How do you convert cubic metres into litres?

Multiply by 1,000. Therefore, 1 m³ = 1,000 litres and 2.5 m³ = 2,500 litres.

What happens to volume when a solid is melted and recast?

The volume remains unchanged if there is no loss of material. The original volume equals the total volume of the recast pieces.

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