Question

Find the difference between the lateral and total surface area of a square-based pyramid with side 2.5cm and slant height 8cm.
$A)6.25c{m^2}$
$B)6.75c{m^2}$
$C)6.68c{m^2}$
$D)6.93c{m^2}$

Answer 1

Hint: Here later surface area of a square-based pyramid (LSA) is $\dfrac{1}{2}Pl$ and total surface area (TSA) of a square-based pyramid is $\dfrac{1}{2}Pl + B$ where P is the perimeter of base and l is the slant height. So, now on substituting the required values in the given formula we get LSA and TSA so that we can find the difference between them.

Complete step by step answer:
We know that the general formula for lateral surface area of regular pyramid is (LSA) =$\dfrac{1}{2}Pl$
Here P is the perimeter of square base and l is the slant height.
Here the base of pyramid is square, so the perimeter of base is P=4s
It is given that side of a square-based pyramid is 2.5cm
Therefore
$ \Rightarrow P = 4s$
$ \Rightarrow P = 4 \times 2.5$
$ \Rightarrow P = 10cm$
Hence we know that P=10cm and l=8cm
Now, to get LSA value let us substitute the above value in the formula, where
Lateral surface area of square-based pyramid (LSA) = $\dfrac{1}{2}Pl$
$ \Rightarrow LSA = \dfrac{1}{2} \times 10 \times 8$
$ \Rightarrow LSA = 40c{m^2}$
Therefore we got the value of lateral surface area of square-based pyramid as $40c{m^2}$
We know that the general formula for total surface area of regular pyramid (TSA) is = $\dfrac{1}{2}Pl + B$
Here P is the perimeter of the square base, l is the slant height and B is the base of the area.
Already we know the P and I value from LSA .So, here we have to find the B value where the base is.
Here the given pyramid is a square based pyramid. Then the area of base is ${s^2}$
Therefore
$ \Rightarrow B = {s^2}$
The side of the square based pyramid is 2.5cm.
$ \Rightarrow B = 2.5cm \times 2.5cm$
$ \Rightarrow B = 6.25c{m^2}$
Total surface area of a square based pyramid (TSA) = $\dfrac{1}{2}Pl + B$
$ \Rightarrow TSA = \dfrac{1}{2} \times Pl + B$
$ \Rightarrow TSA = \dfrac{1}{2} \times 80 + 6.25$ $\left[ {\because Pl = 80c{m^2}} \right]$
$ \Rightarrow TSA = 46.25c{m^2}$
The total surface area of square based pyramid is $46.25c{m^2}$
Therefore, total surface area - lateral surface area =$46.25c{m^2} - 40c{m^2} = 6.25c{m^2}$.
Hence the difference between total surface area and lateral surface area of a pyramid is $6.25c{m^2}$.

Thus option (A) is correct.

NOTE: In this problem we have to find the difference between lateral surface area and total surface area of the pyramid. But here the pyramid is a square based so before substituting value in the given formulas we have remembered that the pyramid is square based. We know that the lateral surface area of the pyramid is $\dfrac{1}{2}Pl$ where P is the perimeter and l is the slant height, here the $P = 4s$ since the given pyramid is a square based pyramid. And we also know that Total surface area of a square based pyramid is $\dfrac{1}{2}Pl + B$ where B is the base as we already know it is square based then $B = {s^2}$. We have to make a note that before finding any value we have to check whether the given pyramid is a normal pyramid or any other type pyramid.