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Surface Area of a Frustum

Lateral surfa...

Question

Lateral surface area of the frustum of a cone is

π (r_{2} - r_{1}) h

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π (r_{1} + r_{2}) h

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π (r_{1} - r_{2}) l

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π (r_{1} + r_{2}) l

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Solution

The correct option is D $π (r_{1} + r_{2}) l$
By ratio and proportion:
$\frac{l_{1}}{r_{1}} = \frac{l}{r_{1} - r_{2}}$
$\Rightarrow l_{1} = \frac{l r_{1}}{r_{1} - r_{2}}$

And from the figure we observe that:
$l_{2} = l_{1} - l$
$= \frac{l r_{1}}{r_{1} - r_{2}} - l$
$= \frac{l r_{1} - l (r_{1} - r_{2})}{r_{1} - r_{2}}$
$= \frac{l r_{2}}{r_{1} - r_{2}}$

The length of arc is the circumference:
$s_{1} = 2 π r_{1}, s_{2} = 2 π r_{2}$

The lateral surface area of a cone is given by $\frac{1}{2} s l,$ where $s$ is the circumference of the base and $l$ is the slant height.

Hence, by using the above formula,
$\begin{matrix} A & = \frac{1}{2} s_{1} l_{1} - \frac{1}{2} s_{2} l_{2} = \frac{1}{2} (2 π r_{1}) (\frac{l r_{1}}{r_{1} - r_{2}}) - \frac{1}{2} (2 π r_{2}) (\frac{l r_{2}}{r_{1} - r_{2}}) = \frac{π l r_{1}^{2}}{r_{1} - r_{2}} - \frac{π l r_{2}^{2}}{r_{1} - r_{2}} = \frac{π l (r_{1}^{2} - r_{2}^{2})}{r_{1} - r_{2}} = \frac{π l (r_{1} + r_{2}) (r_{1} - r_{2})}{r_{1} - r_{2}} = π (r_{1} + r_{2}) l \end{matrix}$

Hence, option $D$ is the correct answer.

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Similar questions

Lateral surface area of frustum of cone =

π (r_{1} + r_{2}) l

Q. Question 7
The curved surface area of a frustum of a cone is

π (r_{1} + r_{2}) l

, where

l = \sqrt{h^{2} + (r_{1} + r_{2})^{2}}

r_{1} a n d r_{2}

are the radii of the two ends of the frustum and h is the vertical height.

Q. The circumferences of circular faces of a frustum are 132 cm and 88 cm and its height is 24 cm. To find the curved surface area of the frustum complete the following activity.(

π = \frac{22}{7}

)

circumference₁ = 2

π

r₁= 132

r₁=

\frac{132}{π} = 134

circumference₂ =2

π

r₂= 88

r₂=

\frac{88}{2 π} = 134

slant height of frustum, l =

\sqrt{h^{2} + {(r_{1} - r_{2})}^{2}} \sqrt{1223^{2} + 1234^{2}} = 1234 cm

curved surface area of the frustum =

π

(r₁+ r₂ ) l

=

π \times 123 \times 132 = 123 sq . cm .

Q. The total surface area of a cone of radius

\frac{r}{2}

and length 2l, is
(a) 2

π r (1 + r)

(b)

πr (1 + \frac{r}{4})

(c)

π r (1 + r)

(d) 2

π r l

Question 5
The total surface area of a cone whose radius is $\frac{r}{2}$ and slant height 2l is:

A) $2 π r (l + r)$
B) $π r (l + \frac{r}{4})$
C) $π r (l + r)$
D) $2 π r l$

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