Surface Area of a Right Circular Cone Questions in English

(A) Given: Length of the pipe $(h)$ $= 77 \,cm$.
Outer diameter $(D)$ $= 4.4 \,cm$.
Outer radius $(R)$ $= \frac{D}{2} = \frac{4.4}{2} = 2.2 \,cm$.
The outer curved surface area of a cylinder is given by the formula $2 \pi R h$.
Outer curved surface area $= 2 \times \frac{22}{7} \times 2.2 \times 77 \,cm^{2}$.
$= 2 \times 22 \times 2.2 \times 11 \,cm^{2}$.
$= 44 \times 24.2 \,cm^{2}$.
$= 1064.8 \,cm^{2}$.

(A) The formula for the curved surface area of a right circular cone is given by $CSA = \pi r l$,where $r$ is the base radius and $l$ is the slant height.
Given:
Base radius $r = 7\, cm$
Slant height $l = 10\, cm$
Substituting the values into the formula:
$CSA = \frac{22}{7} \times 7 \times 10\, cm^{2}$
$CSA = 22 \times 10\, cm^{2}$
$CSA = 220\, cm^{2}$

(B) Given: height $h = 16 \, cm$ and radius $r = 12 \, cm$.
First,calculate the slant height $l$ using the formula $l = \sqrt{h^2 + r^2}$.
$l = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20 \, cm$.
Curved surface area of the cone $= \pi rl = 3.14 \times 12 \times 20 = 753.6 \, cm^2$.
Total surface area of the cone $= \pi rl + \pi r^2 = \pi r(l + r) = 3.14 \times 12 \times (20 + 12) = 3.14 \times 12 \times 32 = 1205.76 \, cm^2$.

(C) Since the grains of corn are found only on the curved surface of the corn cob,we need to calculate the curved surface area of the cone.
Given:
Radius $(r)$ = $2.1\, cm$
Height $(h)$ = $20\, cm$
First,we find the slant height $(l)$:
$l = \sqrt{r^2 + h^2} = \sqrt{(2.1)^2 + (20)^2} = \sqrt{4.41 + 400} = \sqrt{404.41} \approx 20.11\, cm$
Curved Surface Area of the cone = $\pi rl$
$= \frac{22}{7} \times 2.1 \times 20.11 = 22 \times 0.3 \times 20.11 = 6.6 \times 20.11 = 132.726\, cm^2$
Given that there are $4$ grains per $1\, cm^2$:
Total number of grains = $132.726 \times 4 = 530.904$
Rounding to the nearest whole number,we get $531$ grains.

(A) Given,diameter of the base of the cone $= 10.5\, cm$.
Radius $(r) = \frac{10.5}{2}\, cm = 5.25\, cm$.
Slant height $(l) = 10\, cm$.
The formula for the curved surface area of a cone is $CSA = \pi rl$.
Substituting the values:
$CSA = \frac{22}{7} \times \frac{10.5}{2} \times 10\, cm^2$.
$CSA = \frac{22}{7} \times 5.25 \times 10\, cm^2$.
$CSA = \frac{22}{7} \times 52.5\, cm^2$.
$CSA = 22 \times 7.5\, cm^2 = 165\, cm^2$.
Therefore,the curved surface area of the cone is $165\, cm^2$.

(B) Given:
Diameter of the base $(d) = 24 \, m$
Radius of the base $(r) = \frac{d}{2} = \frac{24}{2} = 12 \, m$
Slant height $(l) = 21 \, m$
The formula for the total surface area of a cone is given by:
Total Surface Area $= \pi r(r + l)$
Substituting the values:
Total Surface Area $= \frac{22}{7} \times 12 \times (12 + 21) \, m^2$
$= \frac{22}{7} \times 12 \times 33 \, m^2$
$= \frac{8712}{7} \, m^2$
$\approx 1244.57 \, m^2$

(D) Given:
Height of the tent $(h) = 10 \, m$
Radius of the base $(r) = 24 \, m$
The slant height $(l)$ of a cone is given by the formula:
$l = \sqrt{h^2 + r^2}$
Substituting the given values:
$l = \sqrt{10^2 + 24^2} \, m$
$l = \sqrt{100 + 576} \, m$
$l = \sqrt{676} \, m$
$l = 26 \, m$
Thus,the slant height of the tent is $26 \, m$.

(A) Given: Height of the cone $(h) = 10 \, m$,Radius of the base $(r) = 24 \, m$.
First,calculate the slant height $(l)$ of the cone using the formula $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{24^2 + 10^2} = \sqrt{576 + 100} = \sqrt{676} = 26 \, m$.
The curved surface area of the cone is given by $\pi r l$.
Area of the canvas required $= \frac{22}{7} \times 24 \times 26 \, m^2 = \frac{13728}{7} \, m^2$.
The cost of $1 \, m^2$ canvas is $Rs. 70$.
Total cost of the canvas $= \text{Area} \times \text{Rate} = \frac{13728}{7} \times 70 = 13728 \times 10 = Rs. 137280$.

(B) Given: Base radius $(r) = 6\, m$,Height $(h) = 8\, m$.
Slant height $(l) = \sqrt{r^2 + h^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\, m$.
Curved surface area of the tent $= \pi rl = 3.14 \times 6 \times 10 = 188.4\, m^2$.
Let the length of the tarpaulin be $L$. Since the width is $3\, m$,the area is $L \times 3 = 188.4\, m^2$.
$L = \frac{188.4}{3} = 62.8\, m$.
Extra length for margins and wastage $= 20\, cm = 0.2\, m$.
Total length required $= 62.8\, m + 0.2\, m = 63.0\, m$.

(A) Given: Slant height $(l) = 25 \, m$ and base diameter $(d) = 14 \, m$.
First,calculate the radius $(r)$ of the base:
$r = \frac{d}{2} = \frac{14}{2} = 7 \, m$.
Next,calculate the curved surface area $(CSA)$ of the conical tomb:
$CSA = \pi r l = \frac{22}{7} \times 7 \times 25 = 22 \times 25 = 550 \, m^2$.
The rate of white-washing is $Rs. 210$ per $100 \, m^2$.
Therefore,the total cost of white-washing is:
$\text{Cost} = \left( \frac{210}{100} \right) \times 550 = 2.1 \times 550 = Rs. 1155$.

(B) Given: Radius of the base $(r) = 7\, cm$ and height $(h) = 24\, cm$.
First,calculate the slant height $(l)$ of the cone using the formula $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25\, cm$.
The lateral surface area of one cone is given by $\pi rl$.
Lateral surface area $= \frac{22}{7} \times 7 \times 25 = 550\, cm^2$.
To find the area of the sheet required for $10$ such caps,multiply the area of one cap by $10$.
Total area $= 10 \times 550 = 5500\, cm^2$.

(C) Given:
Diameter of the base $= 40 \, cm$
Radius $(r) = \frac{40}{2} \, cm = 20 \, cm = 0.2 \, m$
Height $(h) = 1 \, m$
Slant height $(l) = \sqrt{r^2 + h^2} = \sqrt{(0.2)^2 + (1)^2} \, m = \sqrt{0.04 + 1} \, m = \sqrt{1.04} \, m = 1.02 \, m$
Curved surface area of one cone $= \pi r l = 3.14 \times 0.2 \times 1.02 \, m^2$
Curved surface area of $50$ cones $= 50 \times 3.14 \times 0.2 \times 1.02 \, m^2$
$= 50 \times 0.2 \times 3.14 \times 1.02 \, m^2 = 10 \times 3.14 \times 1.02 \, m^2 = 32.028 \, m^2$
Cost of painting $= \text{Total Area} \times \text{Rate} = 32.028 \times 12 = Rs. 384.336$
Rounding to two decimal places,the cost is $Rs. 384.34$.

(B) The total surface area of a cone is given by the formula: $\text{Total Surface Area} = \text{Area of the base} + \text{Curved surface area}$.
Given,radius $R = \frac{r}{2}$ and slant height $L = 2l$.
$\text{Total Surface Area} = \pi R^2 + \pi RL$
Substituting the values:
$= \pi \left(\frac{r}{2}\right)^2 + \pi \left(\frac{r}{2}\right) \times (2l)$
$= \pi \left(\frac{r^2}{4}\right) + \pi rl$
$= \pi r \left(\frac{r}{4} + l\right)$
$= \pi r \left(l + \frac{r}{4}\right)$.

(A) The area of canvas required for a conical tent is equal to its curved surface area,which is given by the formula $\pi r l$.
First,we calculate the slant height $(l)$ using the formula $l = \sqrt{r^{2} + h^{2}}$,where $r = 12 \, m$ and $h = 3.5 \, m$.
$l = \sqrt{(12)^{2} + (3.5)^{2}} = \sqrt{144 + 12.25} = \sqrt{156.25} = 12.5 \, m$.
Now,calculate the curved surface area:
Area $= \pi r l = \frac{22}{7} \times 12 \times 12.5$.
Area $= \frac{22 \times 150}{7} \approx 471.428 \, m^{2}$.
Rounding to two decimal places,the area required is $471.42 \, m^{2}$.

(D) When a right triangle is revolved about one of its sides (containing the right angle),a cone is formed.
Here,the side about which the triangle is revolved becomes the height $(h)$ of the cone,and the other side becomes the radius $(r)$ of the base.
Given:
Height $(h) = 8 \, cm$
Radius $(r) = 6 \, cm$
Slant height $(l) = 10 \, cm$
The curved surface area of a cone is given by the formula: $\text{CSA} = \pi r l$
Substituting the values:
$\text{CSA} = \frac{22}{7} \times 6 \times 10$
$\text{CSA} = \frac{1320}{7} \approx 188.57 \, cm^2$
Thus,the curved surface area of the solid formed is $188.57 \, cm^2$.

(N/A) For a cone:
Radius $(r) = 15 \, cm$ and slant height $(l) = 25 \, cm$.
Curved surface area of a cone:
$= \pi r l$
$= 3.14 \times 15 \times 25 \, cm^2$
$= 1177.5 \, cm^2$
Total surface area of a cone:
$= \pi r(l + r)$
$= 3.14 \times 15 \times (25 + 15) \, cm^2$
$= 3.14 \times 15 \times 40 \, cm^2$
$= 1884 \, cm^2$

(A) Given:
Diameter of the cone,$d = 6\, cm$.
Radius of the cone,$r = d / 2 = 6 / 2 = 3\, cm$.
Slant height of the cone,$l = 11\, cm$.
The formula for the total surface area of a cone is $TSA = \pi r(r + l)$.
Substituting the values:
$TSA = \pi \times 3 \times (3 + 11)$
$TSA = \pi \times 3 \times 14$
$TSA = 42\pi\, cm^2$.
Using $\pi \approx 22/7$:
$TSA = 42 \times (22 / 7) = 6 \times 22 = 132\, cm^2$.

(N/A) Given: Radius $(r) = 3.5 \, cm$,Height $(h) = 12 \, cm$.
First,find the slant height $(l)$ using the formula $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{(3.5)^2 + (12)^2} = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5 \, cm$.
Curved Surface Area $(CSA) = \pi rl = \frac{22}{7} \times 3.5 \times 12.5 = 22 \times 0.5 \times 12.5 = 137.5 \, cm^2$.
Total Surface Area $(TSA) = \pi r(r + l) = \frac{22}{7} \times 3.5 \times (3.5 + 12.5) = 22 \times 0.5 \times 16 = 11 \times 16 = 176 \, cm^2$.

(N/A) $1$. When revolved about $AB$: The height $h = AB = 28 \, cm$ and the radius $r = BC = 21 \, cm$. The slant height $l = \sqrt{r^2 + h^2} = \sqrt{21^2 + 28^2} = \sqrt{441 + 784} = \sqrt{1225} = 35 \, cm$. The curved surface area $CSA = \pi r l = \frac{22}{7} \times 21 \times 35 = 22 \times 3 \times 35 = 2310 \, cm^2$.
$2$. When revolved about $BC$: The height $h = BC = 21 \, cm$ and the radius $r = AB = 28 \, cm$. The slant height $l = \sqrt{r^2 + h^2} = \sqrt{28^2 + 21^2} = 35 \, cm$. The curved surface area $CSA = \pi r l = \frac{22}{7} \times 28 \times 35 = 22 \times 4 \times 35 = 3080 \, cm^2$.

(B) Given: Diameter of the cone $d = 42 \, m$,so radius $r = 21 \, m$. Height $h = 20 \, m$.
First,calculate the slant height $l$ of the cone using the formula $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{21^2 + 20^2} = \sqrt{441 + 400} = \sqrt{841} = 29 \, m$.
The area of the canvas required is the curved surface area of the cone,given by $CSA = \pi r l$.
$CSA = \frac{22}{7} \times 21 \times 29 = 22 \times 3 \times 29 = 1914 \, m^2$.
The cost of the canvas is $1914 \, m^2 \times ₹ 75/m^2 = ₹ 143550$.

(N/A) Let the radius $r = 2x$ and the slant height $l = 7x$.
The curved surface area of a cone is given by $CSA = \pi rl$.
Given $CSA = 704 \, cm^2$ and taking $\pi = \frac{22}{7}$:
$704 = \frac{22}{7} \times (2x) \times (7x)$
$704 = 22 \times 2 \times x^2$
$704 = 44x^2$
$x^2 = \frac{704}{44} = 16$
$x = 4$.
Thus,radius $r = 2 \times 4 = 8 \, cm$ and slant height $l = 7 \times 4 = 28 \, cm$.
The height $h$ is given by $h = \sqrt{l^2 - r^2} = \sqrt{28^2 - 8^2} = \sqrt{(28-8)(28+8)} = \sqrt{20 \times 36} = \sqrt{720} = 12\sqrt{5} \, cm$.

(B) The total surface area of a cone is given by the formula: $TSA = \pi r(r + l)$,where $r$ is the radius and $l$ is the slant height.
Given: $r = 10\, cm$,$TSA = 880\, cm^2$,and $\pi = 22/7$.
Substituting the values into the formula:
$880 = (22/7) \times 10 \times (10 + l)$
$880 \times 7 = 220 \times (10 + l)$
$6160 = 220 \times (10 + l)$
$10 + l = 6160 / 220$
$10 + l = 28$
$l = 28 - 10$
$l = 18\, cm$.
Thus,the slant height of the cone is $18\, cm$.

(C) The total surface area $(TSA)$ of a cone is given by the formula: $TSA = \pi r(r + l) = \pi rl + \pi r^{2}$.
Given,$TSA = 3696 \, cm^{2}$ and the curved surface area $(CSA)$ is $\pi rl = 2310 \, cm^{2}$.
Substituting the value of $CSA$ into the $TSA$ formula:
$3696 = 2310 + \pi r^{2}$.
$\pi r^{2} = 3696 - 2310 = 1386$.
Using $\pi = \frac{22}{7}$,we get $\frac{22}{7} r^{2} = 1386$.
$r^{2} = \frac{1386 \times 7}{22} = 63 \times 7 = 441$.
$r = \sqrt{441} = 21 \, cm$.
Now,use the $CSA$ formula to find the slant height $(l)$:
$2310 = \pi rl = \frac{22}{7} \times 21 \times l$.
$2310 = 22 \times 3 \times l = 66l$.
$l = \frac{2310}{66} = 35 \, cm$.
The ratio of the radius $(r)$ to the slant height $(l)$ is $r:l = 21:35$.
Dividing both by $7$,we get $3:5$.

(B) Let the radius $r = 4x$ and the slant height $l = 7x$.
The curved surface area $(CSA)$ of a cone is given by $\pi rl$.
Substituting the values,$CSA = \pi(4x)(7x) = 28\pi x^2$.
The total surface area $(TSA)$ of a cone is given by $\pi r(r + l)$.
Substituting the values,$TSA = \pi(4x)(4x + 7x) = \pi(4x)(11x) = 44\pi x^2$.
The ratio of the total surface area to the curved surface area is $\frac{TSA}{CSA} = \frac{44\pi x^2}{28\pi x^2} = \frac{44}{28} = \frac{11}{7}$.
Thus,the ratio is $11:7$.

(D) The total surface area of a cone is given by the formula: $TSA = \pi r(r + l)$,where $r$ is the radius and $l$ is the slant height.
Given: $r = 5 \, cm$ and $l = 9 \, cm$.
Substituting the values into the formula:
$TSA = \pi \times 5 \times (5 + 9)$
$TSA = \pi \times 5 \times 14$
$TSA = 70 \pi \, cm^2$.
Using $\pi \approx \frac{22}{7}$:
$TSA = 70 \times \frac{22}{7} = 10 \times 22 = 220 \, cm^2$.
Therefore,the total surface area is $220 \, cm^2$.

(C) The total surface area of a cone consists of the area of the circular base and the curved surface area of the cone.
$1$. The area of the circular base is $\pi r^2$.
$2$. The curved surface area of the cone is $\pi r l$,where $r$ is the radius and $l$ is the slant height.
$3$. Total Surface Area = $\text{Base Area} + \text{Curved Surface Area} = \pi r^2 + \pi r l$.
$4$. Factoring out $\pi r$,we get $\pi r(r + l)$ or $\pi r(l + r)$.

(A) The curved surface area of a cone is given by the formula $\pi r l$,where $r$ is the radius of the base and $l$ is the slant height of the cone.

(C) In a right circular cone,the slant height $(l)$,the radius of the base $(r)$,and the vertical height $(h)$ form a right-angled triangle.
According to the Pythagorean theorem,the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Therefore,$l^{2} = r^{2} + h^{2}$.

(C) The total surface area of a cone is given by the formula: $TSA = \pi r(r + l)$,where $r$ is the radius and $l$ is the slant height.
Given: $r = 10 \, cm$,$l = 25 \, cm$.
Using $\pi \approx \frac{22}{7}$ is not ideal here,let us check if $\pi \approx 3.14$ is intended or if the result is a multiple of $\pi$.
Calculation: $TSA = \pi \times 10 \times (10 + 25) = \pi \times 10 \times 35 = 350\pi$.
If we take $\pi \approx 3.14$,$TSA = 350 \times 3.14 = 1099 \approx 1100 \, cm^2$.
Thus,the correct option is $C$.

(B) Given: Radius $(r)$ = $7 \, cm$,Height $(h)$ = $24 \, cm$.
First,find the slant height $(l)$ using the formula: $l = \sqrt{r^2 + h^2}$.
$l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, cm$.
The curved surface area $(CSA)$ of a cone is given by the formula: $CSA = \pi rl$.
$CSA = \frac{22}{7} \times 7 \times 25$.
$CSA = 22 \times 25 = 550 \, cm^2$.