Important Mensuration (3D) Formulas

It is very important to have an understanding of different 3D Mensuration formulas to comfortably attempt Maths questions which covers a good portion of the Quant Section of Competitive Exams. Here we are providing you formulas and shortcuts on how to solve mensuration questions.

Important Mensuration (3D) Formulas

Cube

• s = side
• Volume: V = s^3
• Lateral surface area = 4a²
• Surface Area: S = 6s^2
• Diagonal (d) = s√3

Cuboid

• Volume of cuboid: length x breadth x width
• Total surface area = 2 ( lb + bh + hl)

Right  Circular  Cylinder

• Volume of Cylinder = π r^2 h
• Lateral Surface Area (LSA or CSA) = 2π r h
• Total Surface Area = TSA = 2 π r (r + h)

Hollow-Cylinder

r₁ = outer radius

r₂ = inner radius

* Volume of Hollow Cylinder = π(pie) h(r_1(Square) – r_2(Square))

Right Circular Cone

• l^2 = r^2 + h^2
• Volume of cone = 1/3 π r^2 h
• Curved surface area: CSA=  π r l
• Total surface area = TSA = πr(r + l )

Important relation between radius, height and slant height of similar cone.

Frustum of a Cone

• r = top radius, R = base radius,
• h = height, s = slant height
• Volume: V = π/ 3 (r^2 + rR + R^2)h
• Surface Area: S = πs(R + r) + πr^2 + πR^2

Sphere

• r = radius
• Volume: V = 4/3 πr^3
• Surface Area: S = 4π^2

Hemisphere

• Volume-Hemisphere = 2/3 π r^3
• Curved surface area(CSA) = 2 π r^2
• Total surface area = TSA = 3 π r^2

Quarter-Sphere

Let ‘r’ is the radius of given diagram. You have to imagine this diagram, this is 1/4^th part of Sphere.

 

Prism

• Volume = Base area x height
• Lateral Surface area = perimeter of the base x height

Pyramid

• Volume of a right pyramid = (1/3) × area of the base × height.
• Area of the lateral faces of a right pyramid = (1/2) × perimeter of the base x slant height.
• Area of whole surface of a right pyramid = area of the lateral faces + area of the base.

Important:
1.From a solid cylinder no. of maximum solid cone of same height and radius as cylider are 3.
2.From a solid sphere, no. of maximum solid cone having height and radius equal can be made are 4.
3.From a solid hemisphere, no. of maximum solid cone having height and radius equal can be made are 2.

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