Mix Examples - Surface Areas and Volumes Questions in English

(B) The base of a cylinder is a circle.
Given,the radius of the base $r = 7 \, cm$.
The area of the base of a cylinder is given by the formula $\text{Area} = \pi r^2$.
Substituting the value of $r$ and $\pi = \frac{22}{7}$:
$\text{Area} = \frac{22}{7} \times 7 \times 7$
$\text{Area} = 22 \times 7 = 154 \, cm^2$.
Thus,the area of the base is $154 \, cm^2$.

(C) The formula for the curved surface area $(CSA)$ of a cylinder is $2 \pi r h$.
Given:
Radius $(r)$ = $1.4 \, cm = \frac{14}{10} \, cm$
Height $(h)$ = $10 \, cm$
Substituting the values into the formula:
$CSA = 2 \times \frac{22}{7} \times \frac{14}{10} \times 10$
$CSA = 2 \times 22 \times 2$
$CSA = 88 \, cm^2$
Therefore,the curved surface area is $88 \, cm^2$.

(D) The $CSA$ (Curved Surface Area) of a cylinder is given by the formula $2 \pi r h$.
The volume of a cylinder is given by the formula $\pi r^{2} h$.
According to the problem,the $CSA$ and the volume are numerically equal:
$2 \pi r h = \pi r^{2} h$
Dividing both sides by $\pi r h$ (assuming $r \neq 0$ and $h \neq 0$):
$2 = r$
Therefore,the radius of the cylinder is $2$ units.

(B) The base of a cylinder is a circle. The area of a circle is given by the formula $A = \pi r^{2}$.
Given that the area of the base is $616\,cm^{2}$.
So,$616 = \frac{22}{7} \times r^{2}$.
Rearranging the equation to solve for $r^{2}$:
$r^{2} = 616 \times \frac{7}{22}$.
$r^{2} = 28 \times 7$.
$r^{2} = 196$.
Taking the square root of both sides:
$r = \sqrt{196} = 14\,cm$.
Therefore,the radius of the base of the cylinder is $14\,cm$.

(C) For a cone,the slant height $l$ is given by the formula $l = \sqrt{r^2 + h^2}$.
Given,radius $r = 6 \, cm$ and height $h = 8 \, cm$.
Substituting the values into the formula:
$l = \sqrt{6^2 + 8^2}$
$l = \sqrt{36 + 64}$
$l = \sqrt{100}$
$l = 10 \, cm$.
Therefore,the slant height of the cone is $10 \, cm$.

(C) The formula for the $CSA$ (Curved Surface Area) of a cone is given by $CSA = \pi r l$.
Given radius $r = 4 \, cm$ and slant height $l = 6 \, cm$.
Substituting these values into the formula:
$CSA = \pi \times 4 \times 6 = 24 \pi \, cm^{2}$.

(B) The formula for the $CSA$ (Curved Surface Area) of a cone is given by $CSA = \pi r l$.
Given radius $r = 7\, cm$ and slant height $l = 15\, cm$.
Substituting the values into the formula:
$CSA = \frac{22}{7} \times 7 \times 15$
$CSA = 22 \times 15 = 330\, cm^2$.

(D) Given, radius $r = 4\, cm$ and height $h = 3\, cm$.
First, we calculate the slant height $l$ using the formula $l = \sqrt{r^{2} + h^{2}}$.
$l = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5\, cm$.
The Total Surface Area $(TSA)$ of a cone is given by the formula $TSA = \pi r(r + l)$.
Substituting the values, $TSA = \pi \times 4 \times (4 + 5) = \pi \times 4 \times 9 = 36\pi\, cm^{2}$.

(A) For the given cone,the radius $r = 5\,cm$ and the height $h = 12\,cm$.
First,we calculate the slant height $l$ using the formula $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\,cm$.
The $CSA$ (Curved Surface Area) of a cone is given by $\pi r l$.
The area of the base of the cone is $\pi r^2$.
Therefore,the ratio of the $CSA$ to the area of the base is $\frac{\pi r l}{\pi r^2} = \frac{l}{r}$.
Substituting the values,we get $\frac{13}{5}$,which is $13:5$.

(D) The formula for the surface area of a sphere is $A = 4 \pi r^{2}$.
Given the radius $r = 7\,cm$.
Substituting the value of $r$ into the formula:
$A = 4 \times \frac{22}{7} \times (7)^{2}$
$A = 4 \times \frac{22}{7} \times 49$
$A = 4 \times 22 \times 7$
$A = 88 \times 7 = 616\,cm^{2}$.

(B) The diameter of the hemisphere is $d = 20 \, cm$.
Therefore, the radius $r = d / 2 = 20 / 2 = 10 \, cm$.
The formula for the Curved Surface Area $(CSA)$ of a hemisphere is $2 \pi r^{2}$.
Substituting the value of $r$:
$CSA = 2 \times \pi \times (10)^{2}$
$CSA = 2 \times \pi \times 100$
$CSA = 200 \pi \, cm^{2}$.

(C) The formula for the Total Surface Area $(TSA)$ of a hemisphere is $3 \pi r^{2}$.
Given the radius $r = 10 \, cm$.
Substituting the value of $r$ in the formula:
$TSA = 3 \times \pi \times (10)^{2}$
$TSA = 3 \times \pi \times 100$
$TSA = 300 \pi \, cm^{2}$.

(B) The diameter of the hemisphere is $d = 10 \, cm$.
Therefore, the radius $r = d / 2 = 10 / 2 = 5 \, cm$.
The formula for the $TSA$ of a hemisphere is $3 \pi r^2$.
Substituting the value of $r$ into the formula:
$TSA = 3 \times \pi \times (5)^2$
$TSA = 3 \times \pi \times 25$
$TSA = 75 \pi \, cm^2$.

(D) The area of the circular base of a hemisphere is given by the formula $\pi r^{2}$.
Given,$\pi r^{2} = 154 \, cm^{2}$.
The total surface area $(TSA)$ of a hemisphere is the sum of the curved surface area $(2\pi r^{2})$ and the base area $(\pi r^{2})$,which equals $3\pi r^{2}$.
Therefore,$TSA = 3 \times (\pi r^{2}) = 3 \times 154 = 462 \, cm^{2}$.

(D) Let the radii of the two spheres be $r_{1}$ and $r_{2}$ units.
Given that the ratio of their radii is $\frac{r_{1}}{r_{2}} = \frac{3}{2}$.
The surface area of a sphere is given by the formula $A = 4\pi r^{2}$.
The ratio of the surface areas of the two spheres is $\frac{A_{1}}{A_{2}} = \frac{4\pi r_{1}^{2}}{4\pi r_{2}^{2}}$.
Simplifying this,we get $\frac{A_{1}}{A_{2}} = \left(\frac{r_{1}}{r_{2}}\right)^{2}$.
Substituting the given ratio: $\frac{A_{1}}{A_{2}} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}$.
Therefore,the ratio of their surface areas is $9:4$.

(D) Let the radius of both the sphere and the solid hemisphere be $r$.
The surface area of a sphere is given by the formula $4 \pi r^2$.
The total surface area $(TSA)$ of a solid hemisphere is given by the formula $3 \pi r^2$.
Therefore,the ratio of the surface area of the sphere to the $TSA$ of the hemisphere is $\frac{4 \pi r^2}{3 \pi r^2} = \frac{4}{3}$.
Thus,the required ratio is $4:3$.

(C) The radius of the cylinder $r$ is given by $r = \frac{\text{diameter}}{2} = \frac{14\, cm}{2} = 7\, cm$.
The formula for the volume of a cylinder is $V = \pi r^2 h$.
Substituting the values $r = 7\, cm$ and $h = 10\, cm$ into the formula:
$V = \frac{22}{7} \times (7)^2 \times 10$
$V = \frac{22}{7} \times 49 \times 10$
$V = 22 \times 7 \times 10$
$V = 1540\, cm^3$.
Therefore,the volume of the cylinder is $1540\, cm^3$.

(D) The area of the base of a cylinder is given by the formula $A = \pi r^{2}$.
Given that the area of the base is $20 \, cm^{2}$,we have $\pi r^{2} = 20 \, cm^{2}$.
The volume $V$ of a cylinder is calculated using the formula $V = \pi r^{2} h$.
Substituting the given values,$V = (20 \, cm^{2}) \times (5 \, cm) = 100 \, cm^{3}$.
Therefore,the volume of the cylinder is $100 \, cm^{3}$.

(B) The formula for the volume of a cylinder is $V = \pi r^2 h$.
Given that the volume $V = 20 \pi \, cm^3$ and the height $h = 5 \, cm$.
Substituting these values into the formula: $20 \pi = \pi r^2 \times 5$.
Dividing both sides by $5 \pi$,we get $r^2 = 4$.
Taking the square root,$r = 2 \, cm$.
The diameter $d$ is twice the radius: $d = 2r = 2 \times 2 = 4 \, cm$.

(D) The formula for the volume of a cylinder is $V = \pi r^{2} h$.
Given that the volume $V = 550 \, cm^{3}$ and the radius $r = 5 \, cm$.
Substituting the values into the formula:
$550 = \frac{22}{7} \times (5)^{2} \times h$
$550 = \frac{22}{7} \times 25 \times h$
$h = \frac{550 \times 7}{22 \times 25}$
$h = \frac{3850}{550}$
$h = 7 \, cm$.
Therefore,the height of the cylinder is $7 \, cm$.

(B) The formula for the volume of a cone is $V = \frac{1}{3} \pi r^{2} h$.
Given radius $r = 5 \, cm$ and height $h = 21 \, cm$.
Substituting the values into the formula:
$V = \frac{1}{3} \times \frac{22}{7} \times (5)^{2} \times 21$
$V = \frac{1}{3} \times \frac{22}{7} \times 25 \times 21$
$V = 22 \times 25 \times \frac{21}{21}$
$V = 22 \times 25 = 550 \, cm^{3}$.

(B) The volume of a cone is given by the formula: $V = \frac{1}{3} \times \text{Area of base} \times \text{height}$.
Given,Area of base $= 60 \, cm^{2}$ and height $= 15 \, cm$.
Substituting these values into the formula:
$V = \frac{1}{3} \times 60 \times 15$
$V = 20 \times 15 = 300 \, cm^{3}$.
Therefore,the volume of the cone is $300 \, cm^{3}$.

(A) Let the radius of both the cone and the cylinder be $r$ and their height be $h$.
The volume of a cone is given by $V_{cone} = \frac{1}{3} \pi r^2 h$.
The volume of a cylinder is given by $V_{cylinder} = \pi r^2 h$.
The ratio of the volume of the cone to the volume of the cylinder is $\frac{V_{cone}}{V_{cylinder}} = \frac{\frac{1}{3} \pi r^2 h}{\pi r^2 h} = \frac{1}{3}$.
Therefore,the ratio is $1: 3$.

(C) The volume of a cylinder is given by the formula $V_{cylinder} = \pi r^{2} h$.
The volume of a cone is given by the formula $V_{cone} = \frac{1}{3} \pi r^{2} h$.
Given that the radii $(r)$ and heights $(h)$ are equal for both shapes,we can compare their volumes:
$V_{cylinder} = \pi r^{2} h$
Since $V_{cone} = \frac{1}{3} \pi r^{2} h$,we can write $\pi r^{2} h = 3 \times (\frac{1}{3} \pi r^{2} h)$.
Therefore,$V_{cylinder} = 3 \times V_{cone}$.

(C) The formula for the volume of a cone is $V = \frac{1}{3} \pi r^{2} h$.
Given that the radius $r = 3 \, cm$ and the height $h = 3 \, cm$.
Substituting these values into the formula:
$V = \frac{1}{3} \times \pi \times (3)^{2} \times 3$
$V = \frac{1}{3} \times \pi \times 9 \times 3$
$V = 9 \pi \, cm^{3}$.

(B) The volume $V$ of a sphere with radius $r$ is given by the formula $V = \frac{4}{3} \pi r^3$.
Let the radii of the two spheres be $r_1$ and $r_2$. Given that $\frac{r_1}{r_2} = \frac{2}{3}$.
The ratio of their volumes is $\frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \left(\frac{r_1}{r_2}\right)^3$.
Substituting the given ratio: $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$.
Thus,the ratio of their volumes is $8:27$.

(D) Given: Diameter of the cylindrical tank $= 2.1 \, m$.
Radius $r = \frac{\text{diameter}}{2} = \frac{2.1}{2} = 1.05 \, m$ or $\frac{21}{20} \, m$.
Height $h = 5 \, m$.
The formula for the Curved Surface Area $(CSA)$ of a cylinder is $2 \pi r h$.
Substituting the values: $CSA = 2 \times \frac{22}{7} \times \frac{21}{20} \times 5$.
$CSA = 2 \times \frac{22}{7} \times \frac{21}{20} \times 5 = 2 \times 22 \times \frac{3}{20} \times 5 = 44 \times \frac{15}{20} = 44 \times 0.75 = 33 \, m^2$.
Therefore,the $CSA$ is $33 \, m^2$.

(B) The volume of a cylinder is given by the formula $V = \pi r^2 h$.
Given radius $r = 1.4 \, m$ and height $h = 4 \, m$.
Substituting the values,$V = \frac{22}{7} \times (1.4)^2 \times 4$.
$V = \frac{22}{7} \times 1.96 \times 4 = 22 \times 0.28 \times 4 = 24.64 \, m^3$.
Since $1 \, m^3 = 1000 \, \text{litres}$,the capacity in litres is $24.64 \times 1000 = 24640 \, \text{litres}$.

(B) Given: Height $(h) = 3\,m$,Diameter $= 8\,m$.
Radius $(r) = \frac{\text{diameter}}{2} = \frac{8}{2} = 4\,m$.
For a cone,the slant height $(l)$ is given by $l = \sqrt{r^{2} + h^{2}}$.
$l = \sqrt{4^{2} + 3^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5\,m$.
$CSA$ of the conical tank $= \pi r l$.
$CSA = 3.14 \times 4 \times 5 = 3.14 \times 20 = 62.8\,m^{2}$.

(A) The surface area of a sphere is given by the formula $A = 4 \pi r^{2}$.
Given $A = \frac{763}{3} \, m^{2}$ and $\pi = 3.14$.
Substituting the values into the formula:
$\frac{763}{3} = 4 \times 3.14 \times r^{2}$
$\frac{763}{3} = 12.56 \times r^{2}$
$r^{2} = \frac{763}{3 \times 12.56}$
$r^{2} = \frac{763}{37.68}$
$r^{2} \approx 20.25$
Taking the square root of both sides:
$r = \sqrt{20.25} = 4.5 \, m$.
Thus,the radius of the sphere is $4.5 \, m$.

(B) The formula for the volume of a cone is $V = \frac{1}{3} \pi r^{2} h$.
Given radius $r = 6 \, cm$ and height $h = 21 \, cm$.
Substituting the values,we get $V = \frac{1}{3} \times \frac{22}{7} \times (6)^{2} \times 21$.
$V = \frac{1}{3} \times \frac{22}{7} \times 36 \times 21$.
$V = 22 \times 36 = 792 \, cm^{3}$.

(C) The formula for the volume of a cone is $V = \frac{1}{3} \pi r^{2} h$.
Given: $V = 9504\,cm^{3}$,$r = 18\,cm$,and $\pi = \frac{22}{7}$.
Substituting the values into the formula:
$9504 = \frac{1}{3} \times \frac{22}{7} \times (18)^{2} \times h$
$9504 = \frac{1}{3} \times \frac{22}{7} \times 324 \times h$
$9504 = 22 \times 108 \times \frac{h}{7}$
$9504 = 2376 \times \frac{h}{7}$
$h = \frac{9504 \times 7}{2376}$
$h = 4 \times 7 = 28\,cm$.
Thus,the height of the cone is $28\,cm$.

(C) The formula for the surface area of a sphere is $A = 4\pi r^{2}$.
Given that the surface area $A = 1256\,cm^{2}$ and $\pi = 3.14$.
Substituting the values into the formula:
$1256 = 4 \times 3.14 \times r^{2}$
$1256 = 12.56 \times r^{2}$
$r^{2} = \frac{1256}{12.56}$
$r^{2} = 100$
Taking the square root on both sides,we get $r = 10\,cm$.

(C) For the frustum of a cone, the slant height $l$ is given by the formula:
$l = \sqrt{h^2 + (r_1 - r_2)^2}$
Given $h = 4\,cm$, $r_1 = 10\,cm$, and $r_2 = 7\,cm$:
$l = \sqrt{4^2 + (10 - 7)^2} = \sqrt{16 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\,cm$
The curved surface area $(CSA)$ of a frustum of a cone is given by:
$CSA = \pi l(r_1 + r_2)$
Substituting the values:
$CSA = \pi \times 5 \times (10 + 7) = \pi \times 5 \times 17 = 85\pi\,cm^2$

(B) The volume of a frustum of a cone is given by the formula: $V = \frac{1}{3} \pi h (r_{1}^{2} + r_{2}^{2} + r_{1} r_{2})$.
Given: $r_{1} = 3\,cm$,$r_{2} = 2\,cm$,and $h = 21\,cm$.
Substituting the values into the formula:
$V = \frac{1}{3} \times \frac{22}{7} \times 21 \times (3^{2} + 2^{2} + 3 \times 2)$
$V = 22 \times (9 + 4 + 6)$
$V = 22 \times 19$
$V = 418\,cm^{3}$.
Thus,the capacity of the glass is $418\,cm^{3}$.

(B) The volume of a cylinder is given by the formula $V = \pi r^2 h$.
Let the radii of the two cylinders be $r_1$ and $r_2$,and their heights be $h_1$ and $h_2$.
Given that $\frac{r_1}{r_2} = \frac{3}{4}$ and $\frac{h_1}{h_2} = \frac{4}{5}$.
The ratio of their volumes is $\frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right)$.
Substituting the given values: $\frac{V_1}{V_2} = \left(\frac{3}{4}\right)^2 \times \left(\frac{4}{5}\right)$.
$\frac{V_1}{V_2} = \frac{9}{16} \times \frac{4}{5} = \frac{9 \times 4}{16 \times 5} = \frac{9}{4 \times 5} = \frac{9}{20}$.
Thus,the ratio of their volumes is $9:20$.

(C) The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^{3}$.
Given the radius $r = 1.5\,cm$.
Substituting the value of $r$ into the formula:
$V = \frac{4}{3} \times \pi \times (1.5)^{3}$
$V = \frac{4}{3} \times \pi \times 1.5 \times 1.5 \times 1.5$
$V = 4 \times \pi \times 0.5 \times 1.5 \times 1.5$
$V = 2 \times \pi \times 2.25$
$V = 4.5 \pi\,cm^{3}$.

(A) The volume $V$ of a sphere with radius $r$ is given by the formula $V = \frac{4}{3} \pi r^3$.
If the radius is doubled,the new radius $r' = 2r$.
The new volume $V'$ is given by $V' = \frac{4}{3} \pi (r')^3$.
Substituting $r' = 2r$ into the formula,we get $V' = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi (8r^3) = 8 \times (\frac{4}{3} \pi r^3)$.
Therefore,$V' = 8V$.
Thus,the volume becomes $8$ times the original volume.

(B) The Total Surface Area $(TSA)$ of a hemisphere is given by the formula: $TSA = 3\pi r^{2}$.
Given,$TSA = 462\,cm^{2}$.
So,$3\pi r^{2} = 462$.
Using $\pi = \frac{22}{7}$,we have $3 \times \frac{22}{7} \times r^{2} = 462$.
$r^{2} = 462 \times \frac{7}{3 \times 22}$.
$r^{2} = 462 \times \frac{7}{66}$.
$r^{2} = 7 \times 7 = 49$.
$r = \sqrt{49} = 7\,cm$.
The diameter $d = 2r = 2 \times 7 = 14\,cm$.

(A) The volume of a cone with radius $r$ and height $h$ is given by the formula $V = \frac{1}{3} \pi r^{2} h$.
This formula represents one-third of the volume of a cylinder with the same base radius and height.

(D) Let the radii of the two spheres be $r_1$ and $r_2$. Given that $r_1 : r_2 = 3 : 4$.
The volume of a sphere is given by the formula $V = \frac{4}{3} \pi r^3$.
Therefore,the ratio of the volumes of the two spheres is $\frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \left( \frac{r_1}{r_2} \right)^3$.
Substituting the given ratio,we get $\frac{V_1}{V_2} = \left( \frac{3}{4} \right)^3 = \frac{3^3}{4^3} = \frac{27}{64}$.
Thus,the ratio of their volumes is $27:64$.

(C) The volume of a sphere is given by the formula $V = \frac{4}{3} \pi r^3$.
Given that $V = 4.5 \pi \, cm^3$,we have:
$4.5 \pi = \frac{4}{3} \pi r^3$
Dividing both sides by $\pi$:
$4.5 = \frac{4}{3} r^3$
$r^3 = 4.5 \times \frac{3}{4} = \frac{45}{10} \times \frac{3}{4} = \frac{9}{2} \times \frac{3}{4} = \frac{27}{8}$
Taking the cube root of both sides:
$r = \sqrt[3]{\frac{27}{8}} = \frac{3}{2} = 1.5 \, cm$
The diameter $d$ is $2r$:
$d = 2 \times 1.5 = 3 \, cm$.

(C) Area is defined as the amount of two-dimensional space that an object occupies. The standard unit of area is square units (e.g.,$m^2$,$cm^2$,$km^2$).
$cm^3$ and $m^3$ are units of volume.
$cm$ is a unit of length.
Therefore,$cm^2$ is a unit of area.

(B) Volume is defined as the amount of space occupied by a three-dimensional object.
In the International System of Units $(SI)$,the base unit of length is the meter $(m)$.
Since volume is calculated as the product of three dimensions (length $\times$ width $\times$ height),its unit is the cube of the unit of length.
Therefore,the unit of volume is $m \times m \times m = m^3$.
- $m$ is a unit of length.
- $m^2$ is a unit of area.
- $cm^2$ is also a unit of area.
Thus,the correct option is $B$.

(B) The volume of a cubical tank is given by the formula $V = \text{side}^3$.
Given the side length is $1\, m$,the volume $V = (1\, m)^3 = 1\, m^3$.
We know that $1\, m^3 = 1000\, \text{litres}$.
Therefore,the maximum amount of water that can be stored in the tank is $1000\, \text{litres}$.

(D) The formula for the total surface area of a cube with side length $a$ is given by $6a^{2}$.
Given the side length $a = 4 \, cm$.
Substituting the value of $a$ into the formula:
Total Surface Area $= 6 \times (4 \, cm)^{2}$
Total Surface Area $= 6 \times 16 \, cm^{2}$
Total Surface Area $= 96 \, cm^{2}$.
Therefore,the correct option is $D$.

(A) The formula for the total surface area of a cuboid is $2(lb + bh + hl)$,where $l$,$b$,and $h$ are the length,breadth,and height of the cuboid respectively.
Given dimensions are $l = 15\,cm$,$b = 12\,cm$,and $h = 10\,cm$.
Substituting these values into the formula:
Total surface area $= 2(15 \times 12 + 12 \times 10 + 10 \times 15)$
$= 2(180 + 120 + 150)$
$= 2(450)$
$= 900\,cm^{2}$.

(B) For the given hemisphere,the radius $r = \frac{\text{diameter}}{2} = \frac{2}{2} = 1 \, cm$.
The total surface area of a hemisphere is given by the formula $3 \pi r^{2}$.
Substituting the value of $r = 1 \, cm$ into the formula:
Total surface area $= 3 \times \pi \times (1)^{2} = 3 \pi \, cm^{2}$.

(C) The formula for the curved surface area of a cylinder is $2 \pi r h$.
Given that the curved surface area is $132 \, cm^2$ and the height $h = 3 \, cm$.
Substituting the values into the formula:
$132 = 2 \times \frac{22}{7} \times r \times 3$
$132 = \frac{132}{7} \times r$
Multiplying both sides by $\frac{7}{132}$:
$r = 132 \times \frac{7}{132} = 7 \, cm$.
The diameter $d$ is twice the radius:
$d = 2r = 2 \times 7 = 14 \, cm$.

(D) The formula for the total surface area of a cylinder is $2 \pi r(h + r)$.
Given,radius $r = 7 \, cm$ and height $h = 13 \, cm$.
Substituting the values into the formula:
Total surface area $= 2 \times \frac{22}{7} \times 7 \times (13 + 7)$
$= 2 \times 22 \times 20$
$= 44 \times 20 = 880 \, cm^{2}$.