MENSURATION 3D

Mensuration, within the realm of mathematics, delves into the exploration of measuring geometric shapes and their various attributes such as length, volume, shape, surface area, lateral surface area, and more. Acquaint yourself with mensuration concepts in fundamental mathematical studies

Mensuration in Maths- Important Terminologies and Formulas

Terminology	Formula	Description
Area of Square	A = s²	s is the side length
Area of Rectangle	A = l × w	l is the length and w is the width
Area of Triangle	A = ½ × b × h	b is the base and h is the height
Area of Circle	A = πr²	π is a constant (≈ 3.14) and r is the radius
Volume of Cube	V = s³	s is the side length
Volume of Cuboid	V = l × w × h	l is the length, w is the width, and h is the height
Volume of Cylinder	V = πr²h	π is a constant (≈ 3.14), r is the radius, and h is the height
Volume of Sphere	V = (4/3)πr³	π is a constant (≈ 3.14) and r is the radius
Perimeter of Square	P = 4s	s is the side length
Perimeter of Rectangle	P = 2(l + w)	l is the length and w is the width
Perimeter of Triangle	P = a + b + c	a, b, and c are the side lengths
Perimeter of Circle	P = 2πr	π is a constant (≈ 3.14) and r is the radius
Area of Trapezoid	A = ½ × (b1 + b2) × h	b1 and b2 are the parallel bases and h is the height
Volume of Cone	V = (1/3)πr²h	π is a constant (≈ 3.14), r is the radius, and h is the height
Surface Area of Sphere	S = 4πr²	π is a constant (≈ 3.14) and r is the radius

$l$ represents length, $w$ represents width, $b$ represents base, $h$ represents height, and $r$ represents the radius.
$π$ is the mathematical constant approximately equal to 3.14159.
Surface area and lateral surface area formulas are not included in this table for brevity, but they can be derived based on the figures

Mensuration Formulas For 2D Shapes

Note:

$π$ represents the mathematical constant approximately equal to 3.14159.
For triangles, the base and height must be perpendicular to each other.
The perimeter and circumference formulas for these shapes are not included in this table for brevity

Mensuration Formulas for 3D Shapes

Shape	Surface Area	Volume
Cube	6s²	s³ (where s is the side length)
Cuboid	2(lb + bh + hl) (where l, b, and h are length, breadth, and height, respectively)	l × b × h
Sphere	4πr² (where r is the radius)	(4/3)πr³
Cylinder	2πr² + 2πrh	πr²h
Cone	πr² + πrl	(1/3)πr²h
Pyramid	(1/2)lb + bh + ch (where l is the base area, b and h are the base width and height, and c is the slant height)	(1/3)lbh
Hemisphere	2πr² + πr²	(1/2)(4/3)πr³

Solved Problems of Mensuration

Problem 1:

Find the area of a rectangle with a length of 8 cm and width of 5 cm.

Solution: The formula for the area ( $A$ ) of a rectangle is given by $A = length \times width$ .

Given the length ( $l$ ) is 8 cm and width ( $w$ ) is 5 cm: $A = 8 cm \times 5 cm = 40 cm^{2}$

Therefore, the area of the rectangle is

Problem 2:

Determine the volume of a cube with a side length of 3 cm.

Solution: The formula for the volume ( $V$ ) of a cube is given by $V = side^{3}$ .

Given the side length ( $s$ ) is 3 cm: $V = 3 cm \times 3 cm \times 3 cm = 27 cm^{3}$

Therefore, the volume of the cube is

Problem 3:

Calculate the surface area of a cylinder with a radius of 4 cm and a height of 6 cm. (Take $π \approx 3.14$ )

Solution: The formula for the surface area ( $S A$ ) of a cylinder is given by $S A = 2 π r^{2} + 2 π r h$ .

Given the radius ( $r$ ) is 4 cm and height ( $h$ ) is 6 cm:

$S A = 2 \times 3.14 \times 4^{2} + 2 \times 3.14 \times 4 \times 6$

$S A = 2 \times 3.14 \times 16 + 2 \times 3.14 \times 24$

$S A = 100.48 + 150.72$

$S A = 251.20 cm^{2}$

Therefore, the surface area of the cylinder is

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MENSURATION 3D

MENSURATION 3D

Mensuration Formulas for 3D Shapes

Problem 1:

Problem 2:

Problem 3:

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