# The base of a triangular pyramid is a triangle with corners at (2 ,5 ), (6 ,5 ), and (3 ,8 ). If the pyramid has a height of 15 , what is the pyramid's volume?

Jun 22, 2016

Volume of pyramid is $30.015$ cubic units.

#### Explanation:

Volume of such a pyramid is one third of base of its area multiplied by its height. While height has been given, we have to find area of triangular base, which can bee found using Heron's formula, which gives are $\Delta = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$, where $s = \frac{1}{2} \left(a + b + c\right)$ and $a$, $b$ and $c$ are the three sides of the base triangle.

For this find the sides of triangle formed by $\left(2 , 5\right)$, $\left(6 , 5\right)$ and $\left(3 , 8\right)$ by using distance formula $\sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

The distance between pair of points will be

$a = \sqrt{{\left(6 - 2\right)}^{2} + {\left(5 - 5\right)}^{2}} = \sqrt{16 + 0} = \sqrt{16} = 4$

$b = \sqrt{{\left(3 - 6\right)}^{2} + {\left(8 - 5\right)}^{2}} = \sqrt{9 + 9} = \sqrt{18} = 4.2426$ and

$c = \sqrt{{\left(3 - 2\right)}^{2} + {\left(8 - 5\right)}^{2}} = \sqrt{1 + 9} = \sqrt{10} = 3.1623$

Hence, $s = \frac{1}{2} \times \left(4 + 4.2426 + 3.1623\right) = \frac{11.4049}{2} = 5.7025$

And area of triangle $\Delta = \sqrt{5.7025 \left(5.7025 - 4\right) \left(5.7025 - 4.2426\right) \left(5.7025 - 3.1623\right)}$

= $\sqrt{5.7025 \times 1.7025 \times 1.4599 \times 2.5402} = \sqrt{36.0034} = 6.0003$

Hence volume of pyramid is $\frac{1}{3} \times 6.0003 \times 15 = 30.015$