Question

The radius and slant height of a cone are 12 mm and 25 mm. respectively. Find the volume of the cone.

Answer 1

Answer:-

Your Answer Is 3315.84mm³ approx.

Explanation:-

Given:-

• Radius of cone = 12 mm.

• Slant height of cone = 25 mm.

To Find:-

• Volume of the cone.

Formula Used:-

$\\ \tt\mapsto V = \dfrac{1}{3}\pi\:r^2\:h.$

Where,

• V = Volume of cone.
• r = Radius of cone.
• h = Height of cone.

Solution:-

Let us find out the height of the cone, let the height of cone be x.

From Right Angle Triangle ABC, And Pythagoras Theorem:-

$\\ \tt\mapsto x^2 + r^2 = Slant\:\: Height.$

$\\ \tt\mapsto x^2 + (12\:mm)^2 = (25\:mm)^2.$

$\\ \tt\mapsto x^2 + 144\:m^2 = 625\:mm^2.$

$\\ \tt\mapsto x^2 = 625\:mm^2 - 144\:mm^2.$

$\\ \tt\mapsto x = \sqrt{481\:mm^2}.$

$\\ \tt\mapsto x = 21.9 mm.$

$\\ \large\bf\mapsto x = 22mm (approx)$

Therefore The Height Of Cone is 20 mm.

Therefore,

By using formula,

$\\ \tt\mapsto V = \dfrac{1}{3}\pi\:r^2\:h.$

$\\ \tt\mapsto V = \dfrac{1}{3}\times 3.14\times(12\:mm)^2\times 22\:mm.$

$\\ \tt\mapsto V = \dfrac{1}{3}\times 9947.52\:mm^3.$

$\\ \large\mapsto\boxed{\bf V = 3315.84\:mm^3 (approx).}$

Therefore The Volume Of The Given Cone is 3315.84 mm³.

Answer 2

$\underline{\dag{\underline{\rm{Given}}}}\begin{cases} & \sf{Radius\:(r)\:of\;cone = \bf{12\;mm}} \\ \\ & \sf{Slant\:height\:(l)\;of\;cone = \bf{25\;mm}} \end{cases}$

$\underline{\dag{\underline{\rm{To\:Find}}}:}$

• Volume of the cone = ?

$\underline{\dag{\underline{\rm{Solution}}}:}$

• Firstly, finding height (h) of cone :

$\underline{\dag{\underline{\rm{Using\:Formula}}}:}$

$\qquad\red\bigstar\:{\underline{\boxed{\bf{\green{l^2 = r^2 + h^2}}}}}$

• Putting all known values :

$\longrightarrow\qquad\sf (25)^2 = (12)^2 + h^2$

$\longrightarrow\qquad\sf 625 = 144 + h^2$

$\longrightarrow\qquad\sf 625 - 144 = h^2$

$\longrightarrow\qquad\sf 481 = h^2$

$\longrightarrow\qquad\sf \sqrt{481} = h$

$\longrightarrow\qquad\sf 21.9 = h$

$\longrightarrow\qquad{\boxed{\frak{\pink{h \:\approx\:22\:mm}}}}\:\purple{\bigstar}$

ㅤㅤㅤㅤㅤㅤ━━━━━━━━━━

• Now, finding volume of cone :

$\underline{\dag{\underline{\rm{Using\:Formula}}}:}$

$\qquad\red\bigstar\:{\underline{\boxed{\bf{\green{Volume_{(cone)} = \dfrac{1}{3}\pi r^2 h}}}}}$

• Putting all known values :

$\longrightarrow\qquad\small\sf Volume_{(cone)} = \dfrac{1}{3}\:\times\:\dfrac{22}{7}\:\times\:12\:\times\:12\:\times\:22$

$\longrightarrow\qquad\small\sf Volume_{(cone)} = \dfrac{1}{\cancel{3}}\:\times\:\dfrac{22}{7}\:\times\:\cancel{12}\:\times\:12\:\times\:22$

$\longrightarrow\qquad\small\sf Volume_{(cone)} = \dfrac{22\:\times\:4\:\times\:12\:\times\:22}{7}$

$\longrightarrow\qquad\small\sf Volume_{(cone)} = \dfrac{23232}{7}$

$\longrightarrow\qquad\small\sf Volume_{(cone)} = \dfrac{\cancel{23232}}{\cancel{7}}$

$\longrightarrow\qquad\small\sf Volume_{(cone)} = 3318.8$

$\longrightarrow\qquad\small{\boxed{\frak{\pink{ Volume_{(cone)} \:\approx\:3319\:{mm}^{3}}}}}\:\purple{\bigstar}$

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$\small\therefore\:{\underline{\sf{Volume\:of\:cone\:\approx\:\bf{3319\:{mm}^{3}}\:\sf{respectively.}}}}$

$\underline{\dag{\underline{\rm{More\:to\:know}}}:}$

$\begin{array}{|c|c|c|}\cline{1-3}\bf Shape&\bf Volume\ formula&\bf Surface\ area\ formula\\\cline{1-3}\sf Cube&\tt l^3}&\tt 6l^2\\\cline{1-3}\sf Cuboid&\tt lbh&\tt 2(lb+bh+lh)\\\cline{1-3}\sf Cylinder&\tt {\pi}r^2h&\tt 2\pi{r}(r+h)\\\cline{1-3}\sf Hollow\ cylinder&\tt \pi{h}(R^2-r^2)&\tt 2\pi{rh}+2\pi{Rh}+2\pi(R^2-r^2)\\\cline{1-3}\sf Cone&\tt 1/3\ \pi{r^2}h&\tt \pi{r}(r+s)\\\cline{1-3}\sf Sphere&\tt 4/3\ \pi{r}^3&\tt 4\pi{r}^2\\\cline{1-3}\sf Hemisphere&\tt 2/3\ \pi{r^3}&\tt 3\pi{r}^2\\\cline{1-3}\end{array}$

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