Mensuration Formulas: All 2D and 3D Shapes with Examples

Mensuration is the branch of geometry that measures shapes: perimeter and area for 2D figures, and surface area and volume for 3D solids. The whole topic comes down to one formula table for flat shapes, one for solids, and a reliable habit of checking units.

2D mensuration formulas (perimeter and area)

Shape	Perimeter	Area
Square (side $a$ )	$4a$	$a^2$
Rectangle ( $l \times b$ )	$2(l + b)$	$lb$
Triangle (sides $a,b,c$ )	$a + b + c$	$\tfrac{1}{2} \times \text{base} \times \text{height}$
Equilateral triangle (side $a$ )	$3a$	$\tfrac{\sqrt{3}}{4} a^2$
Parallelogram	$2(a + b)$	$b \times h$
Rhombus (side $a$ )	$4a$	$\tfrac{1}{2} d_1 d_2$
Trapezium	sum of sides	$\tfrac{1}{2}(a + b)h$
Circle (radius $r$ )	$2\pi r$	$\pi r^2$
Semicircle	$\pi r + 2r$	$\tfrac{1}{2}\pi r^2$

For a triangle with three known sides, Heron's formula avoids the height entirely: with $s = \tfrac{a+b+c}{2}$ ,

A = \sqrt{s(s-a)(s-b)(s-c)}

3D mensuration formulas (CSA, TSA, volume)

Exams distinguish curved (lateral) surface area, the wrap-around surface only, from total surface area, which adds the flat tops and bottoms.

Solid	CSA / LSA	TSA	Volume
Cube (edge $a$ )	$4a^2$	$6a^2$	$a^3$
Cuboid ( $l,b,h$ )	$2h(l+b)$	$2(lb + bh + lh)$	$lbh$
Cylinder ( $r,h$ )	$2\pi r h$	$2\pi r(r + h)$	$\pi r^2 h$
Cone ( $r,h$ , slant $l$ )	$\pi r l$	$\pi r(r + l)$	$\tfrac{1}{3}\pi r^2 h$
Sphere ( $r$ )	$4\pi r^2$	$4\pi r^2$	$\tfrac{4}{3}\pi r^3$
Hemisphere ( $r$ )	$2\pi r^2$	$3\pi r^2$	$\tfrac{2}{3}\pi r^3$

The cone's slant height connects to its radius and height by the Pythagorean relation:

l = \sqrt{r^2 + h^2}

Notice the pattern: the cone's volume is exactly one third of the cylinder with the same base and height, and a hemisphere's volume is half a sphere's. Patterns like these cut memorization roughly in half.

Worked example 1: cylinder surface area and volume

A closed cylinder has radius $7$ cm and height $10$ cm. Find its total surface area and volume (use $\pi = \tfrac{22}{7}$ ).

Total surface area:

\text{TSA} = 2\pi r(r + h) = 2 \times \frac{22}{7} \times 7 \times (7 + 10)

\text{TSA} = 44 \times 17 = 748\ \text{cm}^2

Volume:

V = \pi r^2 h = \frac{22}{7} \times 7 \times 7 \times 10 = 1540\ \text{cm}^3

Choosing $r = 7$ with $\pi = \tfrac{22}{7}$ makes the $7$ cancel — a deliberate setup that appears constantly in CBSE and ICSE papers.

Worked example 2: combination solid (cone on a hemisphere)

A toy is a cone of height $4$ cm mounted on a hemisphere, both of radius $3$ cm. Find the total volume and the total surface area.

Volume is the sum of both parts:

V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3 = \frac{1}{3}\pi (9)(4) + \frac{2}{3}\pi (27) = 12\pi + 18\pi = 30\pi \approx 94.2\ \text{cm}^3

Surface area is where most marks are lost. The flat circle where the cone meets the hemisphere is inside the toy, so the exposed surface is the cone's CSA plus the hemisphere's CSA — not any TSA.

Slant height of the cone:

l = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\ \text{cm}

Total exposed surface:

S = \pi r l + 2\pi r^2 = \pi(3)(5) + 2\pi(9) = 15\pi + 18\pi = 33\pi \approx 103.7\ \text{cm}^2

Common mistakes in mensuration problems

Using diameter where the formula needs radius — halve the diameter first.
Adding TSA of both solids in a combination problem instead of only the exposed surfaces.
Mixing units, such as a radius in cm with a height in m; convert before substituting.
Reporting area in cm or volume in cm² — area is always square units, volume always cubic units.
Using vertical height in $\pi r l$ for a cone instead of the slant height $l$ .
Dropping the $\tfrac{1}{3}$ in cone and pyramid volumes.

A revision strategy that actually works for exams

Rather than rereading the tables, rebuild them from memory once a day in the week before the exam: blank page, two tables, then check against this list. Combination-solid questions (Class 10 Surface Areas and Volumes) reward one habit above all — sketch the solid, shade the exposed surfaces, and only then pick formulas. Conversion questions (melting a sphere into wires or cones) always reduce to "volume stays equal," so set the two volume expressions equal and solve for the unknown dimension.

Frequently Asked Questions

What is mensuration in maths?: Mensuration is the branch of geometry concerned with measuring shapes. For 2D figures it covers perimeter and area, and for 3D solids it covers curved surface area, total surface area, and volume. It is a core chapter in Class 8, 9, and 10 and a standard scoring topic in board and competitive exams.
What is the difference between CSA and TSA?: Curved surface area, or CSA, measures only the wrap-around lateral surface of a solid, such as the side wall of a cylinder. Total surface area, or TSA, adds the flat faces too, like the circular top and bottom. For a sphere there is no flat face, so CSA and TSA are identical.
What are the most important mensuration formulas for Class 10?: The Class 10 essentials are cylinder, cone, sphere, and hemisphere: their curved surface areas, total surface areas, and volumes, plus the slant height relation for a cone. Combination-solid problems, where a cone sits on a hemisphere or a cylinder, build entirely from these six solids.
How do you solve combination solid problems?: Sketch the solid and identify which surfaces are actually exposed. Volumes simply add together, but surface areas do not: the joining faces are hidden inside, so you add only the exposed curved or flat areas of each part. Most lost marks come from adding total surface areas instead of exposed ones.
Is mensuration 2D or 3D?: Both. 2D mensuration deals with flat figures such as triangles, quadrilaterals, and circles, where you compute perimeter and area. 3D mensuration deals with solids such as cubes, cylinders, cones, and spheres, where you compute surface area and volume. Exam syllabi usually split the topic along exactly this line.

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