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# The height of a cone is 21 cm. Find the area of the base if the slant height is 28 cm.

Last updated date: 17th Jun 2024
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Hint: Here, we will use the formula of slant height, $l = \sqrt {{r^2} + {h^2}}$, where $r$ is the radius of a cone and $h$ is the height of the cone to find the radius of the base of a cone. Then we will substitute the value of radius in the formula of area of base of cone, $\pi {r^2}$ to find the required value.

Given that the height $h$ of a cone is 21 cm and the slant height $l$ of a cone is 28 cm.
Let the radius of a cone is $r$.
We know that the slant height is $l = \sqrt {{r^2} + {h^2}}$, where $r$ is the radius of a cone and $h$ is the height of a cone.
Substituting the values of $r$and $l$ in the above formula of $l$, we get
$\Rightarrow 28 = \sqrt {{r^2} + {{21}^2}} \\ \Rightarrow {28^2} = {r^2} + {21^2} \\ \Rightarrow 784 = {r^2} + 441 \\$
Subtracting the above equation by 441 on each of the sides, we get
$\Rightarrow 784 - 441 = {r^2} + 441 - 441 \\ \Rightarrow 343 = {r^2} \\$
Taking the square root on both sides in the above equation, we get
$\Rightarrow r = \pm \sqrt {343} \\ \Rightarrow r = \pm 7\sqrt 7 {\text{ cm}} \\$
Since the value of the radius can never be negative, so negative value of $r$ is discarded.
Thus, the radius of the base of a cone is $7\sqrt 7$ cm.
We know that the area of the base of a cone is $\pi {r^2}$, where $r$is the radius of the cone.
Substituting the above value of $r$ in $\pi {r^2}$ to find the area of the base of cone, we get
${\text{Area of base of a cone}} = \pi {\left( {7\sqrt 7 } \right)^2} \\ = \pi \left( {49 \times 7} \right) \\$
Using the value of $\pi$ in the above equation, we get
$\dfrac{{22}}{7}\left( {49 \times 7} \right) = 22 \times 49 \\ = 1078{\text{ c}}{{\text{m}}^2} \\$
Thus, the area of the base of a cone is 1078 cm$^2$.

Note: In this question, students should know the formulae of the slant height of cone, $l = \sqrt {{r^2} + {h^2}}$, where $r$ is the radius of a cone and $h$ is height of cone and the area of the cone, area of the base of a cone is $\pi {r^2}$, where $r$is the radius of the cone properly. Also, we are supposed to write the values properly to avoid any miscalculation.