3D Mensuration (original) (raw)

Last Updated : 3 Dec, 2025

3D Mensuration is the branch of mathematics that deals with the measurement of three-dimensional geometric shapes, including their surface area, volume, lateral surface area, and other related parameters. Unlike 2D shapes (flat figures), 3D objects have length, width, and height (or depth), making them solid structures.

3D Shapes: Cube, Cuboid, Sphere, Cylinder, Cone, Pyramid, Prism, etc.

Mensuration Terminologies

Here is the list of terms you will come across in mensuration class. We have provided the term, it's abbreviation, unit, and definition for easy understanding.

Terms Abbreviation Unit Definition
**Volume V cm3 or m3 A 3D shape's space is referred to as its volume.
**Curved Surface Area CSA m2 or cm2 The overall area is known as a Curved surface area if there is a curved surface. Example: Sphere.
**Lateral Surface area LSA m2 or cm2 The term "Lateral Surface area" refers to the combined area of all lateral surfaces that encircle the provided figure.
**Total Surface Area TSA m2 or cm2 The total surface area is the total of all the curved and lateral surface areas.
**Square Unit - m2 or cm2 A square unit is the area that a square of side one unit covers.
**Cube Unit – m3 or cm3 The space taken up by a cube with a single side.

Mensuration Formula for 3D Shapes

The following table provides a list of all mensuration formulas for 3D shapes:

**Shape **Volume **Curved Surface Area or Lateral Surface Area **Total Surface Area **Figure
**Cube a3 LSA = 4 a2 6a2 Shapes-11cube dimensions
**Cuboids l × b × h LSA = 2h(l + b) 2(lb +bh +hl) Shapes-12cuboid dimensions
**Sphere (4/3)πr3 4πr2 4πr2 Shapes-13sphere dimensions
**Hemisphere (⅔)πr3 2πr2 3πr2 Shapes-14hemisphere dimensions
**Cylinder πr2h 2πrh 2πrh + 2πr2 Shapes-15cylinder dimensions
**Cone (⅓)πr2h πrl πr(r + l) Shapes-16cone dimensions

Mensuration 3D - Questions and Answers

**Q1: Find the length of the largest rod that can be kept in a cuboidal room of dimensions 10 x 15 x 6 m.

**Solution :

_Largest rod would lie along the diagonal.
_=> Length of largest rod = Length of diagonal of the room = (L 2 + B 2 _+ H 2 )1/2
_=> Length of the largest rod = (102 + 152 + 62)1/2 = (100 + 225 + 36)1/2 = (361)1/2
_=> Length of the largest rod = 19 m

**Q2: Find the number of bricks of dimension 24 x 12 x 8 cm each that would be required to make a wall 24 m long, 8 m high and 60 cm thick.

**Solution :

_Volume of 1 brick = 24 x 12 x 8 = 2304 cm _3
_Volume of wall = 2400 x 800 x 60 = 115200000 cm 3
_Therefore, number of bricks required = 115200000 / 2304 = 50000

**Q3: A rectangular sheet of paper measuring 22 cm x 7 cm is rolled along the longer side to make a cylinder. Find the volume of the cylinder formed.

**Solution:

_Let the radius of the cylinder be ‘R’.
_The sheet is rolled along the longer side.
_=> 2 π R = 22
_=> R = 3.5 cm
_Also, height = 7 cm
_Therefore, volume of the cylinder = π R2 H = π (3.5)2 7 = 269.5 cm 3

**Q4: If each edge of a cube is increased by 10 %, what would be the percentage increase in volume?

**Solution:

_Let the original edge length be ‘a’
_=> Original volume = a 3
_Now, new edge length = 1.1 a
_=> New volume = (1.1 a) 3 _= 1.331 a 3
_=> Increase in volume = 1.331 a 3 – 1 a 3 = 0.331 a 3
_Therefore, percentage increase in volume = (0.331 a 3 / a 3) x 100 = 33.1 %

**Q5: Three metal cubes of edge lengths 3 cm, 4 cm, and 5 cm are melted to form a single cube. Find the edge length of such a cube.

**Solution:

_Volume of new cube = Volume of metal generated on melting the cubes = Sum of volumes of the three cubes
_=> Volume of new cube = 3 _3 + 4 _3 _+ 5 _3 = 216
_=> Edge length of new cube = (216) 1/3 = 6 cm