Question

Find the difference between the lateral and total surface area of a square-based pyramid with side $2.5$cm and slant height $8$cm.

• A. $6.25cm$
• B. $6.75cm$
• C. $6.68cm$
• D. $6.93cm$

Answer 1

The correct option is A $6.25cm$

The perimeter of the base is $P=4s$, since it is a square, therefore,

$P=4\times 2.5=10$ cm

The general formula for the lateral surface area of a regular pyramid is $LSA=\frac{1}{2}Pl$ where $P$ represents the perimeter of the base and $l$ is the slant height.

Since the perimeter of the pyramid is $P=10$ cm and the slant height is $l=8$ cm, therefore, the lateral surface area is:

$LSA=\frac{1}{2}Pl=\frac{1}{2}\times 10\times 8=40$ cm${}^2$

Now, the area of the base $B=s^2$ with $s=2.5$ cm is:

$B=s^2={2.5}^2=6.25$cm${}^2$

The general formula for the total surface area of a regular pyramid is $TSA=\frac{1}{2}Pl+B$ where $P$ represents the perimeter of the base, $l$ is the slant height and $B$ is the area of the base.

Since $LSA=\frac{1}{2}Pl=40$cm${}^2$ and area of the base is $B=6.25$ cm${}^2$, therefore, the total surface area is:

$TSA=\frac{1}{2}Pl+B=40+6.25=46.25$cm${}^2$

Therefore, total surface area $-$ lateral surface area $=46.25-40=6.25$cm${}^2$

Hence, the difference between lateral surface area and the total surface area of the pyramid is $6.25$cm${}^2$.