SSC CGL Mensuration Notes: 2D & 3D Formulas, Examples & Practice

Mensuration is the most formula-heavy chapter in SSC CGL Quant — and one of the most predictable. The shapes never change, the formulas never change, and the questions are usually a direct application of one formula. If you can recall the formulas under pressure, the marks are practically free.

The official syllabus lists everything from triangles and circles through cubes, cuboids, cylinders, cones, spheres, hemispheres and right pyramids. This chapter covers the full 2D and 3D list in the order you should learn them — 2D first, because every 3D formula is built on a 2D base.

No derivations, no theory fluff — just the formulas, how to choose between them (CSA vs TSA is where most marks die), and worked examples showing the exact steps SSC-style questions expect.

Key Concepts & Formulas

Triangle — perimeter and area

Perimeter is the length of the boundary; area is the surface covered. A triangle's area is half the product of base and height, where the height must be perpendicular to that base. For a triangle whose three sides are known but no height is given, Heron's formula takes over: compute the semi-perimeter s, then the area is the square root of s(s-a)(s-b)(s-c).

Formulas:

• Perimeter of triangle = a + b + c
• Area of triangle = (1/2) x base x height
• Heron: s = (a+b+c)/2; Area = sqrt(s(s-a)(s-b)(s-c))

Rectangle, square, parallelogram, trapezium

The rectangle and square are the easiest shapes to score on: area is just length times breadth (or side squared), and the perimeter is the round-trip distance. Two facts people forget: the diagonal of a square is side x sqrt(2), and the area of a rhombus equals half the product of its diagonals.

Formulas:

• Rectangle: Area = length x breadth; Perimeter = 2(length + breadth)
• Square: Area = side^2; Perimeter = 4 x side; Diagonal = side x sqrt(2)
• Parallelogram: Area = base x height
• Rhombus: Area = (1/2) x d1 x d2
• Trapezium: Area = (1/2) x (sum of parallel sides) x height

Circle — area and circumference

For a circle, the radius is the star of the show — nearly every formula uses it, not the diameter. Area is pi x r^2 and the circumference (perimeter) is 2 x pi x r. In SSC, use pi = 22/7 when the radius is a multiple of 7 and pi = 3.14 otherwise. For a sector of angle theta, take theta/360 of the full circle's area or circumference.

Formulas:

• Area of circle = pi x r^2
• Circumference = 2 x pi x r
• Diameter = 2 x radius
• Sector area = (theta/360) x pi x r^2; Arc length = (theta/360) x 2 x pi x r

Cube and cuboid

A cube is a box with all edges equal; a cuboid has three different edges. Total surface area counts every face, volume counts the space inside. For the cuboid, the surface formula 2(lb + bh + hl) comes from the three distinct face pairs. The space diagonal of a cuboid — useful when the question involves the longest rod that fits inside — is sqrt(l^2 + b^2 + h^2).

Formulas:

• Cube: TSA = 6 x a^2; Volume = a^3; Diagonal = a x sqrt(3)
• Cuboid: TSA = 2(lb + bh + hl); Volume = l x b x h; Diagonal = sqrt(l^2 + b^2 + h^2)

Cylinder and cone

A cylinder is a circle stretched along a height: the curved (lateral) surface is 2 x pi x r x h, and adding the two circular caps gives the total surface area. A cone is one-third of the cylinder with the same base and height — its curved surface uses the slant height l = sqrt(r^2 + h^2), not the vertical height. That one substitution, h vs l, is where most cone marks are lost.

Formulas:

• Cylinder: CSA = 2 x pi x r x h; TSA = 2 x pi x r x (r + h); Volume = pi x r^2 x h
• Cone: slant height l = sqrt(r^2 + h^2); CSA = pi x r x l; TSA = pi x r x (r + l); Volume = (1/3) x pi x r^2 x h

Sphere and hemisphere

A sphere's surface area is exactly four times the area of its great circle: 4 x pi x r^2. Its volume is (4/3) x pi x r^3. A hemisphere (half a sphere) has a curved surface of 2 x pi x r^2 — but its TOTAL surface area adds the flat circular base, making 3 x pi x r^2. If the question says 'total surface area of a hemisphere' and you answer 2 x pi x r^2, you have missed the base.

Formulas:

• Sphere: SA = 4 x pi x r^2; Volume = (4/3) x pi x r^3
• Hemisphere: CSA = 2 x pi x r^2; TSA = 3 x pi x r^2; Volume = (2/3) x pi x r^3

Solved Examples

Practice Questions

Common Traps (galtiyan jo marks katwati hain)

• Radius vs diameter: most circle formulas need the RADIUS. If the question gives the diameter, halve it first — SSC options always include the unhalved answer as a trap.
• Cone slant height: surface-area formulas use l = sqrt(r^2 + h^2), NOT the vertical height h. This single mix-up ruins more cone questions than any other mistake.
• CSA vs TSA: curved surface area excludes the flat faces; total surface area includes them. A hemisphere's TSA (3 x pi x r^2) includes the circular base its CSA (2 x pi x r^2) leaves out.
• Unit discipline: areas are square units (sq cm), volumes are cubic units (cubic cm). If your units are wrong, your formula choice was wrong.
• pi choice: use 22/7 when the radius is a multiple of 7, 3.14 otherwise. Mismatching gives answers that match no option.
• Heron's formula: s is the SEMI-perimeter (a+b+c)/2, not the full perimeter. Using the full perimeter inside the square root is a classic error.

FAQs