**Mensuration Formula And Questions for 2D and 3D Shapes**

For any competitive exam, Understanding the Mensuration Formulas is very important. Learn the Mensuration Formula for 2D and 3D Shapes from the article below to ace the exam.

**What are 2D and 3D shapes?**

- 2D Shapes- In geometry, a two-dimensional shape is a flat plane figure or a shape that has only two dimensions i.e. length and width. Two-dimensional or 2-D shapes do not possess any thickness and can be measured in only two faces. Only the area and perimeter of 2D shapes can be calculated.
- 3D Shapes- A three-dimensional shape is where the figure has three dimensions namely length, width, and thickness. We calculate volume, curved surface area, and the total surface area of 3D shapes

**Mensuration Formulas In Detail**

**Mensuration Questions**

**Q1. The radius of a cylinder is 10 cm and the height is 4 cm. The number of centimeters that may be added either to the radius or to the height to get the same increase in the volume of the cylinder is :**

(a) 5

(b) 4

(c) 25

(d) 16

**Ans. (a)**

Sol.

Let ‘a’ cm is added in radius and height

π(10+a)²4 = π (10)² (4 +a)

(10+a)²4 = 10² (4 +a)

⇒ a = 5 cm

**Q2. PQRS is a rectangle. The ratio of the sides PQ and QR is 3: 1. If the length of the diagonal PR is 10 cm, then what is the area (in cm²) of the rectangle?**

(a) 15

(b) 30

(c) 45

(d) 20

**Ans.(b)**

Sol.

PQRS is a rectangle

PR = 10 given

PQ : QR = 3 : 1

In ∆PQR

9x² + x² = 100

10x² = 100

x = √10

Area of rectangle = 3x × 1x

= 3x²

= 3 × 10

= 30

**Q3. The base of a prism is in the shape of an equilateral triangle. If the perimeter of the base is 18 cm and the height of the prism is 20 cm, then what is the volume (in cm³) of the prism?**

(a) 180√3

(b) 30√6

(c) 60√2

(d) 120√3

**Ans. (A)**

Sol.

Given:-

Perimeter = 18 &height=20

Hence ,Side = 6

The volume of prism = area of base × height

= √3/4×6 × 6 × 20

V = 180√3 cm3

**Q4. The height of a cone is 24 cm and the area of the base is 154 cm². What is the curved surface area (in cm²) of the cone?**

(a) 484

(b) 550

(c) 525

(d) 515

**Ans.(B)**

Sol. Area of base = 154

πr² = 154

22/7×r^2=154

r = 7

Height = 24

Radius = 7

Slant height(ℓ) = √(h²+r² )

ℓ =√(24²+7² )

ℓ=25

C.S.A. = πrℓ

= 22/7×7×25

C.S.A. ⇒ 550 cm²

**Q5. A right circular solid cylinder has a radius of the base of 7 cm and height is 28 cm. It is melted to form a cuboid such that the ratio of its side is 2 : 3: 6. What is the total surface area (in cm²) cuboid?**

(a)72 ∛(( (1078)²)/(9²))

(b) ∛(2156/9)

(c) ∛(2148/3)

(d) ∛(2048/3)

**Ans. (A)**

Sol. According to the question, a cylinder is transforming into a cuboid as shown in the figure below

The volume of cylinder = volume of a cuboid

22/7×7×7×28=2x×3x×6x

x^3=1078/9 ⇒x=∛(1078/9)

T. Surface area of cuboid = 2 [ℓb + bh + hℓ] =2×(1078/9)^(2/3) [2×3+3×6+6×2] T.S.A.=72×(1078/9)^(2/3)

**Q6. ABCDEF is a regular hexagon. What is the ratio of the area of triangle ACE and the area of triangle AEF?**

(a) 6: 1

(b) 4: 1

(c) 3: 1

(d) 5: 1

**Ans.(c)**

Sol.

Given is a regular hexagon

In a regular hexagon, there are 6 equilateral triangles as shown by the dotted line

Area of ∆ACE = 1/2×[Area of 6 equilateral triangles] = 3 equilateral triangles.

Area of ∆AEF = 1/2×[Area of 2 equilateral ∆] = 1 equilateral triangle.

Hence ratio of

Area ACE : Area AEF = 3 : 1

Numerically proved

Let the length of the side of the hexagon = 12

Interior angle ∠ABC = ((x–2)×180)/n = ((6–2)×180)/6 = 120

Area of ∆ABC = 1/2×AB×BC×sin120

=1/2×12×12×cos30

Area of ∆ABC = 36√3

Similarly

Area ∆CDE = Area ∆AEF = 36√3

Area of hexagon = 6×√3/4×12×12 = 216√3

Area of ACE = Area of hexagon – [Area of ∆ABC + ∆CDE + ∆AEF] = 216√3 –108√3

= 108√3

∴ Area of ACE: Area of AEF = 108√3 ∶36√3

= 3: 1

**Q7. ABCD is a trapezium. Sides AB and CD are parallel to each other. AB = 6 cm, CD = 18 cm, BC = 8 cm and AD = 12 cm. A line parallel to AB divides the trapezium into two parts of an equal perimeter. This line cuts BC at E and AD at F. If BE/EC = AF/FD, then what is the value of BE/EC?**

(a) 1/2

(b) 2

(c) 4

(d) 1/4

**Ans.(c)**

Sol.

All given values are shown in the diagram

Let BE = x then EC = 8 – x

BE/EC=AF/FD (condition given)

Reverse the given condition & add 1 on both sides

EC/BE+1=FD/AF+1

(EC+BE)/BE=(FD+AF)/AF⇒BC/BE=AD/AF …(i)

Put values in eq. (i) → 8/x=12/AF AF=3x/2 & FD=12–3x/2

Now perimeter FABE = FECD

FA + AB + BE + FE = EC + CD + DF + FE

3x/2+6+x=8 –x+18+(12–3x/2)

5x = 32

x=32/5=BE, hence EC=8 –32/5=8/5