Mensuration Formulas: Mensuration is a branch of mathematics that deals with the area, perimeter, volume, and surface area of various geometrical shapes. It is one of the most important chapters covered in high school Mathematics. Mensuration has immense practical applications in our day-to-day life. It is, for this reason, advanced concepts related to mensuration are covered in higher grades. Knowing the basics of ‘what is mensuration?’ is the key to understand it.

It is also an important and scoring topic for competitive exams, like the Olympiads and NTSE. Mensuration problems are asked in various government job exams as well, like SSC, Banking, Insurance, etc. So it becomes very important for everyone to understand and memorize various mensuration formulas for all 2D and 3D geometrical figures. Keep reading to know more about what is mensuration and mensuration all formula.

Mensuration Formula List: What is Mensuration 2D & 3D Shapes

Mensuration is the branch of mathematics in which we study the surface area, volume, perimeter, length, breadth, and height of geometric shapes. Shapes can be 2D or 3D in nature. Let’s understand what are 2-dimensional and 3-dimensional shapes and what are the differences?

In geometry, a two-dimensional shape or 2D shape is defined as a flat plane figure or a shape that has only two dimensions. These shapes can be represented in a plane with X-axis and Y-axis. Some common examples of 2D shapes are circle, square, rectangle, parallelogram, and rhombus.

What is mensuration formula for a 3D Shape?

A three-dimensional shape or 3D shape is defined as a solid figure or an object that has three dimensions – length, breadth, and height. Three-dimensional shapes can’t be represented on a plane. We need spatial representation for 3D shapes because they have an extra dimension as thickness or depth.

Let’s see the major differences between a 2D and a 3D shape:

2D Shape

3D Shape

A 2D shape is surrounded by 3 or more straight lines that can be represented on a plane surface.

A 3D shape is surrounded by multiple surfaces or planes. They are represented spatially.

2D shapes have only length and breadth, and no height.

3D shapes are solid figures and they have an extra dimension as depth or height.

For 2D shapes, we measure the area and perimeter of figures.

For 3D shapes, we measure their volume, curved surface area, and total surface area.

Mensuration Formulas of 2D Geometric Figures

The table below shows the area and perimeter formulas of common 2-D geometrical figures:

Mensuration Formulas for Different 2D Geometric Shapes

Shape

Area Formula

Perimeter Formula

Figure

Square

\({{a}^{2}}\)

\(4a\)

Rectangle

\(lw\)

\(2(l+w)\)

Right-angled Triangle

\(\frac{1}{2}bh\)

\(b+h+H\) where, H is Hypotenuse

Scalene Triangle

\(\sqrt{s(s-a)(s-b)(s-c)}\), where \(s=\frac{a+b+c}{2}\)

\(a+b+c\)

Isosceles Triangle

\(\frac{1}{2}bh\)

\(2a+b\)

Equilateral Triangle

\(\frac{\sqrt{3}}{4}{{a}^{2}}\)

\(3a\)

Parallelogram

\(bh\)

\(2(a+b)\)

Trapezium

\(\frac{1}{2}h(a+c)\)

\(a+b+c+d\)

Rhombus

\(\frac{1}{2}{{d}_{1}}{{d}_{2}}\)

\(4a\)

Circle

\(\pi{{r}^{2}}\)

\(C=2\pi r\), where C is the circumference

Check the properties of different geometric shapes:

Mensuration Formula Chart: Mensuration Formulas of 3D Geometric Figures

The table below shows the formulas for volume, curved surface area, and total surface area of common 3D geometrical figures:

Mensuration Formulas for Different 3D Geometric Shapes

Shape

Volume Formula

Curved Surface Area Formula

Total Surface Area

Figure

Cube

\({{a}^{3}}\)

\(4{{a}^{2}}\)

\(6{{a}^{2}}\)

Cuboid

\(lbh\)

\(2(l+b)h\)

\(2(lb+bh+hl)\)

Sphere

\(\frac{4}{3}\pi{{r}^{3}}\)

\(4\pi{{r}^{2}}\)

\(4\pi{{r}^{2}}\)

Hemisphere

\(\frac{2}{3}\pi{{r}^{3}}\)

\(2\pi{{r}^{2}}\)

\(3\pi{{r}^{2}}\)

Cylinder

\(\pi{{r}^{2}}h\)

\(2\pi rh\)

\(2\pi rh+2\pi{{r}^{2}}\)

Cone

\(\frac{1}{3}\pi{{r}^{2}}h\)

\(\pi rl\)

\(\pi r(r+l)\)

Mensuration Formulas PDF: Important Concepts in Mensuration

In mensuration, we come across different terminologies such as area, perimeter, surface area, volume, etc. We have provided definitions for all these terms so that you can be confident about all the mensuration concepts.

Area: Area of a closed 2D geometric shape is defined as the total surface covered by the shape. It is denoted by A. We measure area in m^{2} or cm^{2}. Remember that area is always measured in square units.

Perimeter: We define perimeter of a closed 2D geometric shape as the total length of its boundary. Perimeter is generally denoted by P. It is measured in m or cm.

Volume: Volume of a 3D geometric shape is defined as the total space occupied by the object. It is always measured in cube units. Common measurement units are m^{3} or cm^{3}. We denote volume of a solid figure by V.

Curved Surface Area: Curved surface area is used for curved objects such as sphere. It is defined as the total area covered by the curved part of the object. We denote curved surface area by CSA. Since it is a type of area, CSA is also measured in square units (m^{2} or cm^{2}).

Lateral Surface Area: Lateral surface area is defined as the area occupied by the lateral part of a 3D geometric shape. It is denoted by LSA. We measure lateral surface area in square units (m^{2} or cm^{2}).

Total Surface Area: When we combine the curved surface area and the lateral surface area of a 3D geometric shape, we get its total surface area (TSA). It is also measured in square units.

Some Other Important Mensuration Formulas

Area of Pathway running across the middle of a rectangle = w (l + b – w)

Perimeter of Pathway around a rectangle field = 2 (l + b + 4w)

Area of Pathway around a rectangle field = 2w (l + b + 2w)

Perimeter of Pathway inside a rectangle field = 2 (l + b – 4w)

Area of Pathway inside a rectangle field = 2w (l + b – 2w)

Area of four walls = 2h (l + b)

Solved Problems on Mensuration Formulas

Here we have provided some mensuration problems with solutions.

Question 1: PQRS is a rectangle. The ratio of the sides PQ and QR is 3:1. If the length of the diagonal PR is 10 cm, then what is the area (in cm²) of the rectangle?

Solution: PQRS is a rectangle

PR = 10 given PQ : QR = 3 : 1 In ∆PQR 9x² + x² = 100 10x² = 100 x = √10 Area of rectangle = 3x × 1x = 3x² = 3 × 10 = 30

Question 2: The height of a cone is 24 cm and the area of the base is 154 cm². What is the curved surface area (in cm²) of the cone?

Solution: Area of base = 154 cm² πr² = 154 22/7×r^2 =154 r = 7 Height = 24 Radius = 7 Slant height (ℓ) = √(h²+r² ) ℓ =√(24²+7² ) ℓ=25 C.S.A. = πrℓ = 22/7×7×25 C.S.A. ⇒ 550 cm²

Question 3: ABCD is a trapezium. Sides AB and CD are parallel to each other. AB = 6 cm, CD = 18 cm, BC = 8 cm and AD = 12 cm. A line parallel to AB divides the trapezium in two parts of equal perimeter. This line cuts BC at E and AD at F. If BE/EC = AF/FD, than what is the value of BE/EC?

Solution: Let BE = x then EC = 8 – x

BE/EC = AF/FD (Given) Reverse the given condition & add 1 both side EC/BE + 1 = FD/AF + 1 (EC+BE)/BE = (FD+AF)/AF ⇒ BC/BE = AD/AF … (i) Put values in eq. (i) → 8/x = 12/AF AF = 3x/2 Similarly, FD = 12–3x/2 Now perimeter FABE = FECD FA + AB + BE + FE = EC + CD + DF + FE 3x/2 + 6 + x = 8 – x + 18 + (12–3x/2) 5x = 32 x = 32/5 =BE Hence EC = 8 –32/5 = 8/5 ∴ BE/EC = (32/5)/(8/5) =4

Question 4: Find the area and perimeter of a square whose side is 10 cm.

Solution: Given: Side = a = 10 cm Area of a square = a^{2 }square units Substitute the value of “a” in the formula, we get Area of a square = 10^{2} A = 10 x 10 = 100 Therefore, the area of a square = 100 cm^{2} The perimeter of a square = 4a units P = 4 x 10 =40 Therefore, the perimeter of a square = 40 cm.

Question 5: Find out the height of a cylinder with a circular base of radius 70 cm and volume 154000 cubic cm.

Solution: Given, r= 70 cm V= 154000 cm^{3} Since formula is, V = πr^{2}h h = V/πr^{2} = 154000/15400 =10 Hence, height of the cylinder is 10 cm.

Question 6: If the sides of a triangle are 26 cm, 24 cm, and 10 cm, what is its area?

Solution: The triangle with sides 26 cm, 24 cm and 10 cm is right-angled, where the hypotenuse is 26 cm. Area of the triangle = 1/2 * 24 * 10 = 120 cm^{2}

Question 7: Find the area of a trapezium whose parallel sides are 20 cm and 18 cm long, and the distance between them is 15 cm.

Solution: Area of a trapezium = 1/2 (sum of parallel sides) * (perpendicular distance between them) = 1/2 (20 + 18) * (15) = 285 cm^{2}

Question 8: Find the area of a parallelogram with a base of 24 cm and a height of 16 cm.

Solution: Area of a parallelogram = base * height = 24 * 16 = 384 cm2

At Embibe, you can solve mensuration practice questions for:

Frequently Asked Questions on Mensuration Formulas

Students can find some general FAQs on the topic down below:

Q.1: What is the formula for mensuration? Ans: Mensuration is commonly referred to as the study of geometry and the formulas that come under it involving the calculation of Area, Perimeter, Volume, and Surface Area of different types of 2D and 3D figures. For the full list of formulas, you can refer to this article.

Q.2: How can we remember mensuration formulas? Ans: The best way to remember mensuration formulas would be by understanding area and perimeter concepts and then using the formula tables provided in this article. You can either take a printout of the page or bookmark it whenever you need it.

Q.3: Which is the easiest way of learning mensuration formulas? Ans: The easiest way of learning mensuration formulas will be by taking the printout of the formulas provided in this article and sticking them near your study table so that you can revise them whenever you want or you can bookmark this page and visit for revision.

Q.4: Is there any difference between mensuration and geometry? Ans: Mensuration deals with the calculation of perimeter, area, volume, and other parameters for 2D or 3D geometric shapes. Geometry is concerned with the properties and relations of points and lines of various shapes.

Q.5: What are 2D and 3D mensuration? Ans: 2D mensuration deals with the area, perimeter, volume, and other parameters related to 2D geometric shapes such as square, rectangle, rhombus, circle, etc. On the other hand, 3D mensuration is concerned with the calculation of volume, curved surface area, lateral surface area, and total surface area of 3D geometric shapes such as a sphere, cylinder, cone, etc.

Now you are provided with all the necessary information regarding different mensuration formulas. Practice more questions and master this concept. Students can make use of NCERT Solutions for Maths provided by Embibe for their exam preparation.

We hope this detailed list of mensuration formulas help you. If you have any queries, feel to ask in the comment section below. We will get back to you at the earliest.