Mensuration 3D: Important Questions, Figures & Formulas PDF Download

Mensuration 3D contains volume and surface area of the 3-D (dimension) figures. For example, Cube, Cuboid, Cylinder, Cone, Sphere, Frustum, Pyramid, Prism and Tetrahedron. Volume is the amount of space occupied by the object. Surface area is the surface area of a solid object is measured of the total area that the surface of the object occupies.
In this article, we are going to cover the key concepts of Mensuration 3D along with the various types of questions, and tips and tricks. We have also added a few solved examples, which candidates will find beneficial in their exam preparation. Read the article thoroughly to clear all the doubts regarding the same.

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Mensuration Maths- Definition

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 is a part of mathematics that helps us measure three-dimensional (3D) shapes. These shapes have length, breadth, and height — which means they take up space and have volume.

Cube (like an ice cube)

Cuboid (like a box)

Cylinder (like a can)

Cone (like an ice cream cone)

Sphere (like a ball)

Hemisphere (half of a sphere)

In 3D mensuration, we mainly calculate:

1. Surface Area – the total area covered by the outer surface of the shape.

2. Lateral Surface Area – the area of the sides, not including top and bottom.

3. Volume – the space inside the object.

Mensuration 3D in Maths- Important Terminologies

Let us discover a few more definitions linked to the various geometric shapes.

Let us discover a few more definitions linked to the various geometric shapes.

Terms	Abbreviation	Unit	Definition
Area	A	᠎᠎᠎᠎᠎᠎᠎᠎	The area is the surface that is covered by the closed shape.
Perimeter	P		The measure of the endless line by the boundary of the presented figure is named a Perimeter.
Volume	V		The space utilised by a 3D shape or object is termed a Volume.
Curved Surface Area	CSA	᠎᠎᠎᠎᠎᠎	If there is a curved surface/shape/object, then the total area obtained is called a curved surface area.
Lateral Surface area	LSA		The total area of all the lateral surfaces that encloses the presented figure is termed the lateral surface area.
Total Surface Area	TSA		The aggregate of all the curved and lateral surface areas is designated as the total surface area.
Square Unit	–		The area incorporated by a square of the length of one unit is termed a square unit.
Cube Unit	–		The volume involved by a cube of one side one is termed a cube unit.

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 =
• Surface area =
• 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 =
• 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 =
• 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 =
• Surface area =
• Volume of metal =
• Volume of hollow sphere =
• Here, R = external radius, r = internal radius, a = thinness of wall

Properties of Hemisphere

• Volume =
• Surface area =
• Total surface area =

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

Also read about Volume of a prism, here

Mensuration Formulas For 3D Shapes

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.

Shape	Volume (in cubic units)	Curved Surface Area (CSA) or Lateral Surface Area (LSA) (Square units)	Total Surface Area (TSA) (Square units)
Cube	᠎ Volume=	LSA =	TSA=
Cuboid	Volume=	LSA =	TSA=
Sphere	Volume=	LSA =	TSA=
Hemisphere	Volume=	LSA =	TSA=
Cylinder	Volume=	LSA =	TSA=
Cone	Volume=	LSA =	TSA=

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.

C.S.A=

T.S.A=

T.S.A=

Volume=

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:

᠎Volume of frustum of cone = (OR)

Volume of frustum of cone =

CSA (or) LSA of frustum of cone =

TSA of frustum of cone =

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 =

Curved Surface area =

Total surface area =

Mensuration Formulas for 2D Shapes 

Mensuration is the part of math that deals with measuring the area and perimeter of shapes. Below are the basic formulas used to find the area (the space inside a shape) and perimeter (the distance around a shape) of common 2D (two-dimensional) figures:

Shape	Area (square units)	Perimeter (units)
Square	Side × Side = a²	4 × Side = 4a
Rectangle	Length × Breadth = l × b	2 × (Length + Breadth) = 2(l + b)
Circle	π × Radius² = πr²	2 × π × Radius = 2πr
Scalene Triangle	√[s(s − a)(s − b)(s − c)], where s = (a + b + c) ÷ 2	a + b + c
Isosceles Triangle	½ × Base × Height = ½ × b × h	2 × Equal side + Base = 2a + b
Equilateral Triangle	(√3 ÷ 4) × Side² = (√3/4) × a²	3 × Side = 3a
Right Triangle	½ × Base × Height = ½ × b × h	Base + Height + Hypotenuse
Rhombus	½ × Diagonal1 × Diagonal2 = ½ × d₁ × d₂	4 × Side
Parallelogram	Base × Height = b × h	2 × (Base + Side) = 2(b + l)
Trapezium	½ × Height × (a + c)	a + b + c + d

Differences Between Mensuration 3D and 2D shapes

Check out the difference between the Mensuration 2D and 3D shapes.

2D Shape (Mensuration)	3D Shape (Mensuration)
If a shape or pattern is enclosed by three or added straight lines in a plane, then such a shape is a 2D shape.	If a shape or pattern is enclosed by several surfaces or planes then it is a 3D shape.
Two-dimensional shapes hold no depth/height.	Three-dimensional shapes are also termed solid shapes and when compared to 2D they possess height or depth.
2D shapes as the name suggests have only two dimensions, that is the length and breadth.	The three-dimensional shapes as compared to 2D have three dimensions namely depth/height, breadth and length.
In two dimensions, we can measure the area and perimeter of shapes.	In three dimensions, we can measure the volume, CSA(Curved Surface area), LSA( Lateral Surface area.) or TSA(Total Surface area.) of the given shapes.

How to Solve Question Based on Mensuration 3D – Tips and Tricks

Candidates can find different tips and tricks from below for solving the questions related to Mensuration 3D.

• Tip # 1: Make sure you remember all the properties and formulas mentioned above to solve the questions related to this section quickly.
• Tip # 2:Attempt mock and quizzes to brush up your concepts of Mensuration 3D.

Mensuration Solved Problems

Question 1: If the volume of a cube is 4913 cm3, then find the total surface area.

Solution 1: Volume =

⇒ = 4913

∴a = 17 cm

⇒Total surface area =

∴Total surface area = 6 × 172 = 1734

Question 2: If the sum of three sides is 45 cm and the length of diagonal is 21 cm in a cuboid, find the total surface area of this cuboid.

Solution 2: l + b + h = 45————— (1)

⇒Diagonal =

⇒ = 21

⇒ = 441———— (2)

∴From (1),

⇒ = 452

⇒ + 2(lb + bh + hl) = 2025 441 + 2(lb + bh + hl) = 2025

⇒2(lb + bh + hl) = 2025 – 441 = 1584

∴Surface area of this cuboid is 1584

Question 3: Find the total surface area and volume of a cylinder which has a height of 21m and a base of diagonal is 12m.

Solution 3: Diagonal = 2 × radius Radius = 12/2 = 6 m Total surface area = 2πrh

⇒Total surface area = 2 × 22/7 × 6 × 21 = 792

⇒Volume =

∴Volume = 22/7 ×62×21=2376

Question 4: The base of a solid right prism is a triangle whose sides are 9 cm, 12 cm and 15 cm. The slant height of the prism is 5 cm, Find the total surface area of the prism.

Solution 4: Perimeter of base = 9 + 12 + 15 = 36 cm Area of base = ½ × 12 × 9 = 54

⇒Total surface area = Perimeter of base × slant height + 2 × Area of base Total surface area = (36 × 5) + 2 × 54

∴Total surface area =288

Exams where Mensuration 3D is Part of Syllabus

Questions based on Mensuration 3D come up often in various prestigious government exams some of them are as follows.

• SBI PO, SBI Clerk, IBPS PO, IBPS Clerk
• SSC CGL, SSC CHSL, SSC MTS
• LIC AAO, LIC ADO
• RRB NTPC, RRB ALP
• UPSC
• MPSC
• KPSC
• BPSC
• WBPSC
• Other State Level Recruitment Examinations

If you are checking Mensuration 3D article, also check the related maths articles in the table below:
Mensuration 2D	Parabola, Ellipse and Hyperbola
Height and Distance	Pie Diagram
Data Interpretation	Linear Equation In Two Variables

We hope you found this article regarding Mensuration 3D was informative and helpful, and please do not hesitate to contact us for any doubts or queries regarding the same. You can also download the Testbook App, which is absolutely free and start preparing for any government competitive examination by taking the mock tests before the examination to boost your preparation.For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam:

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