Question

Find total surface area and volume of a square based pyramid having the slant height 10cm and vertical height 8cm

Answer 1

Find total surface area and volume of a square based pyramid having the slant height 10cm and vertical height 8cm

Answer 2

Given:

A square-based pyramid has a slant height equal to 10cm and a vertical height equal to 8cm.

To find:

The total surface area and volume of a square-based pyramid.

Solution:

The relation between the slant height, base, and height of the square-based pyramid is given by, $l^{2}=h^{2}+(\frac{b}{2})^{2}$.

Substitute $l=10$ and $h=8$ into the above-mentioned equation and solve for b.

$10^2=8^2+(\frac{b}{2})^2\\ 100=64+\frac{b^2}{4}\\ 36=\frac{b^2}{4}\\ 144=b^2\\b=12,-12$

The base of the square is 12 cm.

Use the formula $\rm{TSA}=\frac{1}{2}(\rm{perimeter\ of\ the\ base\times slant\ height})+area\ of\ base$ to find the total surface area of the square-based pyramid.

$\rm{TSA}=\frac{1}{2}(\rm{perimeter\ of\ the\ base\times slant\ height})+area\ of\ base\\=\frac{1}{2}( 4\times12\times10)+12\times12\\=\frac{1}{2}(480)+24\\ =240+24\\=264\ {\rm cm^2}$

To find the volume of a square-based pyramid, use the formula ${\rm{Volume}}=\frac{1}{3} (\rm {base\ area})\times(\rm{height})$ and simplify.

${\rm{Volume}}=\frac{1}{3} (\rm {base\ area})\times(\rm{height})\\=\frac{1}{3}(12^2)\times(8)\\ =2304\ {\rm{cm^3}}$

Thus, the total surface area and volume of the square-based pyramid are 264 square cm and 2304 cubic cm respectively.