Curious about what mensuration is and its significance? Get ready to enhance your knowledge and grasp the fundamental formulas and principles that govern shapes, sizes, and measurements in various geometrical contexts.

What is Mensuration in Maths?

In mathematics, mensuration refers to the branch that measures geometric figures and shapes. It involves calculating various properties of 2D and 3D shapes, such as:

Area: The measure of the space enclosed within a shape.
Perimeter: The total length around a 2D shape.
Volume: The amount of space occupied by a 3D shape.
Surface Area: The total area of all the surfaces of a 3D shape.

Mensuration uses formulas and techniques to determine these measurements, which are essential for solving problems related to geometry and practical applications like construction, design, and spatial analysis.

What are 2D figures in Mensuration?

In mensuration, 2D figures (two-dimensional shapes) refer to flat geometric figures that have only two dimensions: length and width (or height). These shapes do not have thickness or depth, and they lie entirely in a single plane. Studying 2D shapes in mensuration involves calculating various properties such as area, perimeter, and sometimes the length of diagonals.

Mensuration deals with finding the area (the amount of space inside the shape) and the perimeter (the total length around the shape) of 2D shapes. Examples of 2D shapes in mensuration include square, rectangle, triangle, circle, parallelogram, and trapezium.

What are 3D figures in Mensuartion?

In mensuration, 3D figures (three-dimensional shapes) refer to solid geometric figures that have three dimensions: length, width, and height (or depth). Unlike 2D shapes, which are flat and only have area and perimeter, 3D shapes have volume and surface area.

Examples of 3D shapes include cubes, cuboids (Rectangular Prisms), spheres, cylinders, cones, and pyramids. Measuring 3D shapes involves calculating their volume (the amount of space they occupy) and surface area (the total area of all their surfaces).

Basic Terminologies In Mensuration

Let us study the essential terminologies in mensuration:

Terminology	Definition
Area	The amount of space inside a shape or figure
Perimeter	The total distance around the outside of a shape
Volume	The amount of space occupied by a 3D object
Circumference	The distance around the edge of a circle
Surface Area	The total area of all the faces of a 3D object
Diagonal	A line segment connecting two non-adjacent vertices of a polygon
Height	The vertical measurement of an object or figure
Base	The bottom side of a shape or figure
Radius	The distance from the centre to the edge of a circle
Diameter	The distance across a circle through its centre

Basic 2D Formulas in Mensuration

For 2D shapes in mensuration, the area formulas are essential for calculating the space enclosed within the shape. Here are some common area formulas to find the area of 2D shapes:

Shape	Formula
Square	Area = side length×side length
Rectangle	Area = length×width
Triangle	Area = 1/2×base×height
Circle	Area = πr²

Basic 3D Formulas in Mensuration

Calculating the volume of 3D objects involves different formulas based on the shape. Here are some of the important formulas to find the volume of 3D shapes in mensuration:

Shape	Formula
Cube	V = s³
Rectangular Prism	V = l×w×h
Sphere	V = (4/3)πr³
Cylinder	V = πr²h
Cone	V = (1/3)πr²h
Pyramid	V = (1/3)Bh

2D vs 3D in Mensuration

Let us study some of the key features and the differences between 2D & 3D:

Feature	2D Shapes	3D Shapes
Dimensions	Two dimensions: length and width	Three dimensions: length, width, and height
Plane	Flat, existing in a single-plane	Solid, extending through space
Examples	Square, Rectangle, Triangle, Circle	Cube, Cuboid, Sphere, Cylinder, Cone, Pyramid
Area Calculation	Yes, calculates the space within the shape	Not applicable
Perimeter Calculation	Yes, measures the boundary length	Not applicable
Volume Calculation	Not applicable	Yes, measures the space occupied by the shape
Surface Area Calculation	Not applicable	Yes, measures the total area of all surfaces
Representation	Drawn or displayed on a flat surface	Requires 3D models or visualizations

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A comparative table of 2D and 3D figures, including their definitions and formulas for area and volume:

Figure	Definition	2D Formulas	3D Formulas
Square	A quadrilateral with four equal sides and four right angles.	Area: Perimeter:
Rectangle	A quadrilateral with opposite sides equal and four right angles.	Area: Perimeter:
Triangle	A polygon with three sides and three angles.	Area: Perimeter: Sum of all sides
Circle	A round shape where all points are equidistant from the centre.	Area: Circumference:
Cube	A solid with six equal square faces.		Volume: Surface Area:
Cuboid	A solid with six rectangular faces.		Volume: Surface Area:
Sphere	A perfectly round solid where all points on the surface are equidistant from the centre.		Volume: Surface Area:
Cylinder	A solid with two parallel circular bases connected by a curved surface.		Volume: Surface Area:
Cone	A solid with a circular base and a single apex.		Volume: Surface Area:
Pyramid	A solid with a polygonal base and triangular faces meeting at an apex.		Volume: Surface Area: Base Area + Area of triangular faces

Solved Questions With Solutions (MCQs)

Provided below are some of the top selected questions with detailed answers for practice:

Q & A in mensuration

Question 1. When three metal cubes measuring 6 cm, 8 cm, and 10 cm on each side are combined and reshaped into a single cube, determine the length of the new cube's side.

(a) 13 cm

(b) 11 cm

(d) 24 cm

Solution: c) 12 cm

Explanation: The total volume will remain the same. Let the side of the resulting cube be = 'a'. Then, 6³ + 8³ + 10³ = a³ =³√1728 = 12 cm

Question 2. Take the volume of a cube is as 216 cm3. Part of this cube is then melted to form a cylinder of length 8 cm. Find the volume of the cylinder.

(a) 342 cm3

(b) 216 cm3

(d) Data inadequate

Solution: d) Data inadequate

Explanation: Data is inadequate as it’s not mentioned what part of the cube is melted to form a cylinder.

Question 3. The diameters of the two cones are equal. If their slant height is in the ratio 5: 7, find the ratio of their curved surface areas.

(a) 25:7

(b) 25:49

(d) 5:7

Solution: d) 5:7

Explanation: Let the radius of the two cones be = x cm

Let the slant height of 1st cone = 5 cm and

the slant height of 2nd cone = 7 cm

Then the ratio of covered surface area = πx5/πx7 =5:7

Question 4. The curved surface area of a cone is 2376 square cm, and its slant height is 18 cm. Find the diameter.

(a) 6 cm

(b) 18 cm

(d) 12 cm

Solution: c) 84 cm

Explanation: Radius = πrl/πl = 2376/3.14x18 = 42 cm

Diameter = 2 x Radius = 2 x 42 = 84 cm

Question 5. The ratio of the radii of a cylinder to that of a cone is 1: 2. If their heights are equal, find the ratio of their volumes.

(a) 1:3

(b) 2:3

(d) 3:1

Solution: c) 3:4

Explanation: Let the radius of the cylinder = 1(r)

Then the radius of cone be = 2(R)

Then as per question = πr²h/R²h/3= 3πr²h/πR²h=3r²/R²= 3: 4

Question 6. The circumference of a circle is more than its diameter by 16.8 cm. Find the circumference of the circle.

(a) 12.32 cm

(b) 49.28 cm

(d) 24.64 cm

Solution: d) 24.64 cm

Explanation: Let the radius of the circle be = p

Then 2pr – 2r = 16.8

r = 3.92 cm

Then, 2πr = 24.64 cm

Question 7. A spherical shell's outer and inner diameters are 10 cm and 9 cm, respectively. Find the volume of the metal contained in the shell.

(a) 6956 cm³

(b) 141.95 cm³

(d) 478.3 cm³

Solution: b) 141.95 cm³

Explanation: Inner radius(p) = 9 2 = 4.5 cm

Outer radius (R) = 10 2 = 5 cm

Now, the volume of metal contained in the shell = 4πR³- 4πr³= 4π/3 (R³-r³) = 141.95 cm³

Question 8. The radii of two spheres are in the ratio of 1: 2. Find the ratio of their surface areas.

(a) 1:3

(b) 2:3

(d) 3:4

Solution: c) 1:4

Explanation: Let smaller radius (r) = 1

The bigger radius (R) = 2

Then, as per the question

4πr²/4πR² = (r/R)²= (1/2)²=1:4

Question 9. A 7 m wide road surrounds a circular path whose circumference is 352 m. What will be the area of the road?

(a) 2618 m²

(b) 654.5 m²

(d) 5236 m²

Solution: a) 2618 m²

Explanation: Let the inner radius = r

Then 2πr = 352 m. Then r = 56

The outer radius = r + 7 = 63 = R

Now, πR² – πr²= Area of road

π(R²– r²) = 2618 m²

Question 10. Seven equal cubes, each side 5 cm, are joined end to end. Find the surface area of the resulting cuboid.

(a) 750 cm²

(b) 1500 cm²

(d) 700 cm²

Solution: a) 750 cm²

Explanation: Total surface area of 7 cubes = 7 X 6a² = 1050

However, when joining end to end, 12 sides will be covered.

So their area = 12 x a² = 12 x 25 = 300

So the surface area of the resulting figure = 1050 – 300 = 750 cm²

Conclusion

The exploration of mensuration has unveiled a world where mathematical concepts intertwine with practical applications. Understanding the distinctions between 2D and 3D shapes, mastering essential terminologies, and delving into shape formulas are crucial steps in grasping the complexities of mensuration.

One can appreciate how mensuration extends beyond theoretical realms into real-world scenarios by uncovering key concepts and tools necessary for calculations.

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Frequently Asked Questions (FAQs)

1. What is the significance of mensuration in mathematics?

Mensuration in mathematics deals with the measurement of geometric shapes and figures, aiding in calculating areas, volumes, and perimeters involving spatial concepts and measurements.

2. How do 2D and 3D shapes differ in mensuration?

2D shapes are flat figures with length and width, while 3D shapes have depth as well. Mensuration involves different formulas for calculating the areas and volumes of these shapes based on their dimensions.

3. What are some essential terminologies used in mensuration?

Common terminologies in mensuration include area, perimeter, volume, base, height, radius, diameter, circumference, surface area, and lateral area. Understanding these terms is crucial for accurate calculations in geometry.

4. Why is it important to know key mensuration concepts?

Understanding key mensuration concepts enables individuals to solve complex mathematical problems involving measurement and spatial relationships. It provides a foundation for various fields such as engineering, architecture, physics, and more.

5. What practical applications does mensuration have in real life?

Mensuration finds applications in various real-life scenarios such as construction (calculating materials needed), landscaping (determining area for gardening), packaging (estimating box sizes), and sports (measuring playing fields).