Here is the Mensuration Hacks (Part I). We will provide
Mensuration Hacks (Part II) later on. Mensuration is purely formula based topic. So here, we will try to solve
different types of mensuration questions that have appeared in SSC Exams and in this process we will share the
important mensuration concepts / formulas / tricks / hacks.
Q1: The base of a prism is a right angled triangle with two sides 5cm and 12 cm. The height of the prism is 10 cm. The total surface area of the prism is:
A. 360 sq cm
B. 300 sq cm
C. 330 sq cm
D. 325 sq cm
Solution:
Note:
For prism and cylinder [figures with uniform girth]:
1. Lateral surface area = Height*perimeter of the Base.
2. Volume: height*area of the Base.
3. Total surface area: Lateral surface area+area of two Bases.
In above question;
Height of the prism = 10cm.
Perimeter of the base= 5+12+13=30cm by Pythagoras theorem.
So lateral surface area= 10×30=300cm.
Area of the base = 1/2*b*h=1/2×5×12 = 30 sq cm.
So total surface area=300+2*30=360 sq cm.
Answer: (A).
Q2: The height of a cone is 30cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27th of the volume of the given cone, at what height above the base is the section is made?
A. 19cm
B. 20cm
C. 12 cm
D. 15cm
Solution:
Note: In such questions remember one thing similarity
r/R = h/H ........(1)
Here,
r is the radius of small cone.
R is the radius of big cone.
h is the radius of small cone.
H is the height of big cone.
Volume of the cone= 1/3*pi*r^2*h.
Given,
Volume of big cone=27*volume of small cone.
1/3*pi*R^2*H = 1/3*pi*r^2*h
=>27*(r/R)^2 = H/h
Put the value of r/R from eq. ... (1)
27*(h/H)^2 = H/h
Solve it you get h =10 cm.
The question asks height above the base which is (30-h)=20 cm.
Answer: (B).
Q3: The base of a right prism is a quadrilateral ABCD. Given that AB=9CM, BC = 14 CM, CD = 13CM, DA = 12 CM and angle DAB = 90 degree. If the volume of the prism be 2070 cubic cm then the area of the lateral surface is:
A. 720 sq cm
B. 810 sq cm
C. 1260 sq cm
D. 2070 sq cm
Solution:
The base of the prism looks like the fig. below:
BD = 15 cm by Pythagoras theorem.
Area of the base = area of the triangle ABD + area of the triangle BDC.
Area of the triangle ABD = 54 sq cm
Area of the triangle BDC =84 sq cm by Heron's formula.
Area of the base = 138 sq cm.
Volume = height * area of the base
2070 = height * 138
So height of the prism is 15cm.
Lateral Surface Area = Height * Perimeter of the Base
Perimeter of the base = AB + BC + CD + DA = 48 cm
Lateral Surface Area = 48 * 15 = 720 cm^2
Answer: (A).
Q4: The base of a right prism is an equilateral triangle. If the lateral surface area and volume is 120 sq cm and 4*(square root of 3) cm cube respectively. Then side of the base of the prism is:
A. 4cm
B. 5cm
C. 7cm
D. 40 cm
Solution:
Area of the base = √3/4 * a^2,
where a is the side of the equilateral triangle.
Perimeter of the base = 3a.
Volume of the prism = Area of the base * Height =√3/4 * a^2 * h ... (1).
Lateral surface Area of the prism = Perimeter of the base * Height = 3a * h ...(2).
Divide equation (1) by (2)
Volume/Area = (1/4√3) * a
40√3/120 = a/4√3 [Since Volume = 40√3 and Lateral surface Area = 120]
a = 160 * 3/120
a = 4 cm
Answer: (A).
Q5. Base of right pyramid is a square whose area is 324 sq m. If the volume of the pyramid is 1296 cu. m, then area in sq. m of the slant surface is:
A. 1080
B. 360
C. 432
D. 540
Solution:
Note:- This equation is about pyramid. So let me just give you some concepts about pyramid.
Pyramids. SSC can ask questions about two types of Pyramids. Pyramid with triangular base and pyramid with square base. Both these pyramids have different formulas. Look at below fig. and understand the labellings i.e. Slant edge and Slant height.
In the below fig., I have written formulas for both types of pyramids. The formula for volume is same for the both the pyramids.
V = 1/3*A*h
Where A = area of the base (calculation of A will be different for both).
h= height of the pyramid
When lateral surface area is asked you will first calculate the Slant Height. Then with the help of slant height you will find the area of one lateral face (let's call this area M). If the pyramid is having a triangular base then multiply M with 3, to get the lateral surface area of the pyramid. And if the pyramid is having a square base then multiply M with 4.
For a pyramid with triangular base:
e= lateral surface / slant edge
s= slant height
e = square root of (h^2+1/3*a^2)
s = square root of (h^2+1/12*a^2)
Where h is the height and a is the length of a side of the base.
Like all pyramids the volume of triangular pyramid is given by:
V= 1/3*A*h.
For a pyramid with square base:
e = square root of (h^2+1/2*a^2)
s = square root of ( h^2+1/4*a^2)
Now,
The area of the square is 324, hence its side is 18 cm.
Volume of the pyramid = 1/3 * Area of the base * Height.
1296 = 1/3 * 324 * Height.
So Height = 12 cm.
Slant Height of the pyramid with square base = √ (h^2 + a^2/4) = √(12^2 + 18^2/4)
Slant Height = 15cm
Area of the lateral face = 1/2 * Base * Height = 1/2 * 18 * 15 = 135 cm^2.
Pyramid with a square base has 4 lateral faces, so lateral surface area of the pyramid = 4 * 135 = 540 cm^2
Answer: (D).
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