Overview: Mensuration | Quantitative Techniques for CLAT PDF Download

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.

Properties of Cube

• 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 =

Properties of Cuboid

• 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

• 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

• 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 =
• Surface area = πrl
• Total surface area = πr(l + r)

Properties of Sphere

• 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

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

Properties of Prism

• 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.

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

Quarter Sphere

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.

Frustum of a Cone

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:

Hollow Cylinder (Hollow Right Circular Cylinder)

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(R²−r²)
Curved Surface area = 2πRh+2πrh=2πh(R+r)
Total surface area = 2πH(R+r)+2π(R²−r²)

Differences Between Mensuration 3D and 2D shapes