Mensuration – Complete Notes for Competitive Exams

1. Introduction

Mensuration deals with the measurement of geometrical figures and solids, including area, surface area, and volume.

It is a vital topic in Quantitative Aptitude sections of SSC, Banking, Railways, and other competitive exams.

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2. Important Formulas

2.1 Plane Figures (2D Shapes)

Shape	Area Formula	Perimeter Formula
Square	$A = a^2$A=a2	$P = 4a$P=4a
Rectangle	$A = l \cdot b$A=l⋅b	$P = 2(l + b)$P=2(l+b)
Triangle	$A = \frac{1}{2} \cdot b \cdot h$A=21​⋅b⋅h	$P = a + b + c$P=a+b+c
Circle	$A = \pi r^2$A=πr2	$C = 2\pi r$C=2πr
Parallelogram	$A = b \cdot h$A=b⋅h	$P = 2(a + b)$P=2(a+b)
Trapezium	$A = \frac{1}{2}(a+b)h$A=21​(a+b)h	$P = a + b + c + d$P=a+b+c+d
Rhombus	$A = \frac{1}{2} \cdot d_1 \cdot d_2$A=21​⋅d1​⋅d2​	$P = 4a$P=4a

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2.2 Solid Figures (3D Shapes)

Shape	Surface Area (SA)	Volume (V)
Cube	$6a^2$6a2	$a^3$a3
Cuboid	$2(lb + bh + hl)$2(lb+bh+hl)	$l \cdot b \cdot h$l⋅b⋅h
Sphere	$4\pi r^2$4πr2	$\frac{4}{3}\pi r^3$34​πr3
Hemisphere	$3\pi r^2$3πr2	$\frac{2}{3}\pi r^3$32​πr3
Cylinder	$2\pi r(h+r)$2πr(h+r)	$\pi r^2 h$πr2h
Cone	$\pi r(l + r)$πr(l+r) (l = slant height)	$\frac{1}{3}\pi r^2 h$31​πr2h
Frustum of cone	$\pi (r_1 + r_2) l + \pi (r_1^2 + r_2^2)$π(r1​+r2​)l+π(r12​+r22​)	$\frac{1}{3}\pi h(r_1^2 + r_1r_2 + r_2^2)$31​πh(r12​+r1​r2​+r22​)

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3. Important Tips

• Always identify the shape first
• Memorize formulas for cube, cuboid, sphere, cylinder, and cone
• Use π=3.1416\pi = 3.1416π=3.1416 or 227\frac{22}{7}722​ depending on exam preference
• For composite figures, break into simpler shapes

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Top 25 Practice Questions – Mensuration

Plane Figures (2D)

Q1. Area of a square with side 6 cm
Q2. Perimeter of a rectangle with length 10 cm and breadth 5 cm
Q3. Area of a triangle with base 8 cm and height 5 cm
Q4. Circumference of a circle with radius 7 cm
Q5. Area of a parallelogram with base 12 cm and height 5 cm
Q6. Area of a rhombus with diagonals 10 cm and 8 cm
Q7. Area of a trapezium with bases 10 cm and 6 cm, height 5 cm
Q8. Perimeter of an equilateral triangle with side 9 cm
Q9. Area of a circle with diameter 14 cm
Q10. Area of rectangle with length 15 cm, width 10 cm

Solid Figures (3D)

Q11. Surface area of a cube with side 4 cm
Q12. Volume of a cuboid with dimensions 5 cm × 4 cm × 3 cm
Q13. Surface area of a sphere with radius 7 cm
Q14. Volume of a cylinder with radius 3 cm and height 10 cm
Q15. Curved surface area of a cone with radius 5 cm and slant height 13 cm
Q16. Volume of a cone with radius 3 cm and height 4 cm
Q17. Total surface area of a hemisphere with radius 7 cm
Q18. Volume of a hemisphere with radius 7 cm
Q19. Lateral surface area of a cylinder with radius 7 cm and height 10 cm
Q20. Total surface area of a cuboid with dimensions 3 cm × 4 cm × 5 cm
Q21. Volume of a cube with side 6 cm
Q22. Total surface area of a cone with radius 3 cm and slant height 5 cm
Q23. Volume of a frustum of a cone with $r_1 = 5$r1​=5, $r_2 = 3$r2​=3, $h = 6$h=6
Q24. Curved surface area of a frustum of a cone with $r_1 = 7$r1​=7, $r_2 = 4$r2​=4, $l = 6$l=6
Q25. Total surface area of a sphere with radius 14 cm

Answer

Answers – Mensuration

Plane Figures (2D)

Q1. 36 cm²
Q2. 30 cm
Q3. 20 cm²
Q4. 44 cm
Q5. 60 cm²
Q6. 40 cm²
Q7. 40 cm²
Q8. 27 cm
Q9. 154 cm²
Q10. 150 cm²

Solid Figures (3D)

Q11. 96 cm²
Q12. 60 cm³
Q13. 615.44 cm²
Q14. 282.74 cm³
Q15. 282.74 cm²
Q16. 37.70 cm³
Q17. 923.63 cm²
Q18. 1436.76 cm³
Q19. 439.82 cm²
Q20. 94 cm²
Q21. 216 cm³
Q22. 75.36 cm²
Q23. 226.19 cm³
Q24. 330.19 cm²
Q25. 2463.01 cm²