Mensuration – Basics, Formulas, Area and Volume, Solved Examples, Explanation (Geometric Figures - Triangle, Square, Rectangle, Circle, Cube, Cuboid, etc.)

Formulas for Areas and Volumes (Mensuration)

Circle:

1. Diameter, D = 2R
2. Area = πR² sq. units
3. Circumference = 2πR units

Square:

1. Area = a² sq. units
2. Perimeter = 4a units
3. Diagonal, d = √2 a units

Rectangle:

1. Area = L*B sq. units
2. Perimeter = 2(L+B) units
3. Diagonal, d = √L²+B² units

Right Angled Triangle:

1. Area = (½)bxh sq. units
2. Perimeter = b + h + hypotenuse
3. Hypotenuse = √b²+h² units

Equilateral Triangle:

1. Area = √4 a² sq. units
2. Perimeter = 3a units, where a = side of the triangle

Scalene Triangle:

1. Area: √s(s-a)(s-b)(s-c) sq. units; s = (a+b+c)/2
2. Perimeter = (a+b+c) units

Isosceles Triangle:

1. Area = b/4 √4a²-B² sq units
2. Perimeter = 2a + b units, where b = base length; a = equal side length

Cube:

1. Volume = a³ cubic units
2. Lateral Surface Area (LSA) = 4a² sq. units
3. Total surface area (TSA) = 6a² sq. units
4. Length of diagonal = a√3 units

Cuboid:

1. Volume = (Cross section area * height) = L * B * H cubic units
2. Lateral Surface Area (LSA) = 2[(L+B)H] sq. units
3. Total surface area (TSA) = 2(LB+BH+HL) sq. units
4. Length of the diagonals = √L²+B²+H² units

Sphere:

1. Volume = (4/3) πR³ cubic units
2. Surface Area = 4πR² sq. units
3. If R and r are the external and internal radii of a spherical shell, then its Volume = (4/3) [R³-r³] cubic units

Hemisphere:

1. Volume = (2/3) πR³ cubic units
2. TSA = 3πR² sq. units

Cylinder:

1. Volume = πr²h cubic units
2. Curved surface Area (CSA) (excludes the areas of the top and bottom circular regions) = 2πRh sq. units
3. TSA = Curved Surface Area + Areas of the top and bottom circular regions = 2πRh + 2πR² = 2πR[R+h] sq. units

Cone:

1. Volume = (1/3) πR²h cubic units
2. Slant Height of cone, L = √R²+H² units
3. CSA = πRL sq. units

Mensuration and Geometric Problems (Areas and Volumes Questions)

Q.1: A right circular cone is placed over a cylinder of the same radius. Now the combined structure is painted on all sides. Then they are separated now the ratio of area painted on Cylinder to Cone is 3:1. What is the height of Cylinder if the height of Cone is 4 m and radius is 3 m?
A. 5m
B. 6m
C. 8m
D. 10m
Answer: B. 6m
Explanation:
Cylinder painted area = 2πrh+πr²
Cone painted area = πrl
2h+r/√ (r² +h1²) = 3:1
h = 6

Q.2: The diameter of Road Roller is 84 cm and its length is 150 cm. It takes 600 revolutions to level once on a particular road. Then what is the area of that road in m²?
A. 2376
B. 2476
C. 2496
D. 2516
Answer: A. 2376
Area: 600*2*22/7*42/100*150/100 = 2376


Q.3: A smaller triangle is having three sides. Another big triangle is having sides exactly double the sides of the smaller triangle. Then what is the ratio of Area of Smaller triangle to Area of the bigger triangle?
A. 1:2
B. 2:1
C. 1:4
D. 4:1
Answer: C. 1:4
Explanation:
Smaller triangle sides = a, b, c
Area = √s(s-a) (s-b) (s-c);
s = a+b+c/2
= √(a+b+c)(b+c-a)(a+c- b)(a+b-c)/4
Bigger triangle =2a, 2b, 2c
Area = √(a+b+c)(b+c-a)(a+c- b)(a+b-c)
Ratio = 1:4

Q.4: ABCD is a square of 20 m. What is the area of the least-sized square that can be inscribed in it with its vertices on the sides of ABCD?
A. 100 m²
B. 120 m²
C. 200 m²
D. 250 m²
Answer: C. 200 m²
Explanation:
It touches on midpoints on the sides of the square ABCD
Side = √ (10² +10²) = √200
Area = 200 m²

Q.5: A hemispherical bowl of diameter 16cm is full of ice cream. Each student in a class is served exactly 4 scoops of ice cream. If the hemispherical scoop is having a radius of 2cm, then ice cream is served to how many students?
A. 16
B. 32
C. 64
D. 128
Answer: A. 16
Explanation:
2/3*π*8³ = n*4*2/3*π*2³
n = 16

Q.6: A hollow cylindrical tube is made of plastic is 4 cm thick. If the external diameter is 18 cm and length of the tube is 59cm, then find the volume of the plastic?
A. 10380 cm³
B. 10384 cm³
C. 10440 cm³
D. 10444 cm³
Answer: B. 10384 cm³
Explanation:
R = 9, r = 5
V = 22/7*59(9² – 5²)
= 22/7*59(81 – 25)
= 10384

Q.7: What is the radius of the circle whose area is equal to the sum of the areas of two circles whose radii are 20 cm and 21 cm?
A. 27m
B. 28m
C. 29m
D. 30m
Answer: C. 29m
Explanation:
πR² = πr1² + πr2²
πR² = π(r1² + r2²)
R² = (400 + 441)
R = 29

Q.8: A well with 14 m diameter is dug up to 49 m deep. Now the soil taken out during dug is made into cubical blocks of 3.5m side each. Then how many such blocks were made?
A. 22
B. 44
C. 88
D. 176
Answer: D. 176
Explanation:
22/7*7²*49 = n*(7/2)³
n = 176


Q.9: If the ratio of radius two Cylinders A and B are in the ratio of 2:1 and their heights are in the ratio of 2:1 respectively. The ratio of their total surface areas of Cylinder A to B is?
A. 1:2
B. 2:1
C. 1:4
D. 4:1
Answer: D. 4:1
Explanation:
Cylinder A: 2πr1 (r1 + h1)
Cylinder B: 2πr2 (r2 + h2)
r1/r2 = 2:1; h1/h2 = 2:1
TA/TB = 2πr1 (r1 + h1)/2πr2 (r2 + h2)

Q.10: The area of the Circular garden is 88704 m². Outside the garden a road of 7m width laid around it. What would be the cost of laying road at Rs. 2/m².
A. Rs. 7,546
B. Rs. 10,036
C. Rs. 11,092
D. Rs. 15,092
Answer: D. Rs. 15,092
Explanation:
88704 = 22/7*r²
r = 168
Outer radius = 168+7 = 175
Outer area = 22/7*175² = 96250
Road area = 96250 – 88704 = 7546
Cost = 7546*2 = 15092