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Mensuration

A section of mathematics that communicates about the length, volume, or area of various geometric shapes is termed Mensuration. These shapes or patterns exist in two dimensions or three dimensions.

Mensuration 3D in Maths- Important Terminologies

Let us discover a few more definitions linked to the various geometric shapes.

Overview: Mensuration | Quantitative Techniques for CLAT

Properties of Cube


Overview: Mensuration | Quantitative Techniques for CLAT


  • All sides are equal.
  • The Cube has all six faces in a square shape.
  • Each of the faces meets the other four faces.
  • The edges opposite to each other are parallel.
  • Side = a

Formula for a Cube:

  • Volume = a3
  • Surface area = 6a2
  • Diagonal = Overview: Mensuration | Quantitative Techniques for CLAT

Properties of Cuboid

Overview: Mensuration | Quantitative Techniques for CLAT

  • Three-dimensional shape with length, breadth, and height
  • Each of the faces meets the other four faces.
  • The edges opposite to each other are parallel.
  • Height = h, Length = l and Breadth = b

Formula for a cuboid:

  • Volume = l × b × h
  • Volume = Area of base × height
  • Surface area = 2(lb + bh + hl)
  • Diagonal = l2+b2+h2
  • Area of four walls = Perimeter of base × height = 2(l + b)× h
  • Volume of metal = External volume – internal volume

Properties of Cylinder

Overview: Mensuration | Quantitative Techniques for CLAT

  • A cylinder has one curved side.
  • A cylinder has two vertical flat ends in the circular shape.

Formula for a cylinder:

  • Volume = πr2h
  • Surface area = 2πrh
  • Total surface area = 2πr(h + r)

Properties of Cone

Overview: Mensuration | Quantitative Techniques for CLAT

  • Height = h, Slant height = l and Radius = r
  • It has only one vertex point.
  • It’s base is circular.

Formula for a cone:

  • Volume = (1/3)πr2h
  • l = Overview: Mensuration | Quantitative Techniques for CLAT
  • Surface area = πrl
  • Total surface area = πr(l + r)

Properties of Sphere

Overview: Mensuration | Quantitative Techniques for CLAT

  • Volume = 4/3πR3
  • Surface area =4πR3
  • Volume of metal = 4/3π(R3−r3)
  • Volume of hollow sphere = 4/3π(R−a)3
  • Here, R = external radius, r = internal radius, a = thinness of wall

Properties of Hemisphere

Overview: Mensuration | Quantitative Techniques for CLAT

  • Volume = 2/3πr3
  • Surface area = 2πr2
  • Total surface area = 3πr2

Properties of Prism

Overview: Mensuration | Quantitative Techniques for CLAT

  • Volume = Area of base×height
  • Lateral surface area = Perimeter of base × slant height
  • Total surface area = Perimeter of base × slant height + 2 × Area of base

Mensuration 3D Formulas


With the knowledge of terms, properties and formulas of 3D shapes let us summarise some of the important formulas in the below table, these mensuration formulas are very useful while solving mensuration problems.
Overview: Mensuration | Quantitative Techniques for CLAT

Here’s some more 3D shapes with their formulas and images.

Quarter Sphere

Overview: Mensuration | Quantitative Techniques for CLAT

A quarter sphere is specifically one-fourth share of a full sphere. i.e., if we break a sphere into four equal sections, each portion is termed a quarter sphere. Therefore, the volume of a quarter sphere is one-fourth of the volume of a sphere.

Overview: Mensuration | Quantitative Techniques for CLAT

Frustum of a Cone

Overview: Mensuration | Quantitative Techniques for CLAT

The frustum of a cone is the section of the cone without vertex when the given cone is split into two pieces with a plane that is parallel to the bottom of the cone.

The frustum of a cone is also called a truncated cone. Similar to any other 3D shape or object, the frustum of a cone also holds surface area and volume. The formula for the same are as follows:

Overview: Mensuration | Quantitative Techniques for CLAT

Overview: Mensuration | Quantitative Techniques for CLAT

Overview: Mensuration | Quantitative Techniques for CLAT

Overview: Mensuration | Quantitative Techniques for CLAT

Hollow Cylinder (Hollow Right Circular Cylinder)

Overview: Mensuration | Quantitative Techniques for CLAT

The volume of a hollow cylinder is defined as the 3D space contained by it. For instance, the volume of glass indicates the available area inside it. In other words, we can say that volume represents the maximum space that can be filled by water if the water flows into the glass.
Here the inner radius of the base is ‘r’, the outer radius of the base is ‘R’ and the height of the hollow right circular cylinder is ‘h’.
Volume = πh(R2r2)
Curved Surface area = 2πRh+2πrh=2πh(R+r)
Total surface area = 2πH(R+r)+2π(R2r2)

Differences Between Mensuration 3D and 2D shapes


Check out the difference between the Mensuration 2D and 3D shapes.
Overview: Mensuration | Quantitative Techniques for CLAT

How to Solve Question Based on Mensuration 3D – Tips and Tricks

Candidates can find different tips and tricks from below for solving the questions related to Mensuration 3D.

  • Tip # 1: Make sure you remember all the properties and formulas mentioned above to solve the questions related to this section quickly.
  • Tip # 2:Attempt mock and quizzes to brush up your concepts of Mensuration 3D.

Mensuration 3D Sample Questions

Question 1: If the volume of a cube is 4913 cm3, then find the total surface area.
Solution 1: 
Volume = a3
a3= 4913
∴a = 17 cm
⇒Total surface area = 6a2
∴Total surface area = 6 × 172 = 1734 cm2

Question 2: If the sum of three sides is 45 cm and the length of diagonal is 21 cm in a cuboid, find the total surface area of this cuboid.
Solution 2:
 l + b + h = 45————— (1)
⇒Diagonal = l2+b2+h2
l2+b2+h= 21
l2+b2+h2= 441———— (2)
∴From (1),
(l+b+h)2 = 452
l2+b2+h2 + 2(lb + bh + hl) = 2025 441 + 2(lb + bh + hl) = 2025
⇒2(lb + bh + hl) = 2025 – 441 = 1584 cm2
∴Surface area of this cuboid is 1584 cm2

Question 3: Find the total surface area and volume of a cylinder which has a height of 21m and a base of diagonal is 12m.
Solution 3: 
Diagonal = 2 × radius Radius = 12/2 = 6 m Total surface area = 2πrh
⇒Total surface area = 2 × 22/7 × 6 × 21 = 792 m2
⇒Volume = πr2h
∴Volume = 22/7 ×62×21=2376 m3

Question 4: The base of a solid right prism is a triangle whose sides are 9 cm, 12 cm and 15 cm. The slant height of the prism is 5 cm, Find the total surface area of the prism.
Solution 4:
 Perimeter of base = 9 + 12 + 15 = 36 cm Area of base = ½ × 12 × 9 = 54 cm2
⇒Total surface area = Perimeter of base × slant height + 2 × Area of base Total surface area = (36 × 5) + 2 × 54
∴Total surface area =288 cm2.

The document Overview: Mensuration | Quantitative Techniques for CLAT is a part of the CLAT Course Quantitative Techniques for CLAT.
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FAQs on Overview: Mensuration - Quantitative Techniques for CLAT

1. What is mensuration?
Ans. Mensuration is a branch of mathematics that deals with the measurement of geometric figures and their parameters such as length, area, volume, and angles.
2. How is mensuration used in real life?
Ans. Mensuration has practical applications in various fields such as engineering, architecture, construction, and surveying. It is used to calculate the area of land, volume of materials required for construction, and dimensions of objects.
3. What are the basic formulas used in mensuration?
Ans. Some of the basic formulas used in mensuration include the formula for the area of a rectangle (length x width), the formula for the area of a circle (π x radius^2), and the formula for the volume of a cylinder (π x radius^2 x height).
4. How can I calculate the area of an irregular shape?
Ans. To calculate the area of an irregular shape, you can divide it into smaller regular shapes for which you know the formulas. Then, calculate the area of each smaller shape and add them together to get the total area of the irregular shape.
5. What is the difference between perimeter and area?
Ans. Perimeter is the distance around the outside of a two-dimensional shape, while area is the measure of the space enclosed by the shape. Perimeter is measured in units of length (e.g., centimeters or inches), while area is measured in square units (e.g., square centimeters or square inches).
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