# The base of a triangular pyramid is a triangle with corners at (1 ,2 ), (9 ,5 ), and (4 ,7 ). If the pyramid has a height of 9 , what is the pyramid's volume?

Apr 1, 2018

Volume of a pyramid is $46.5$ cubic.unit.

#### Explanation:

Volume of a pyramid is $\frac{1}{3} \cdot$base area $\cdot$hight.

$\left({x}_{1} , {y}_{1}\right) = \left(1 , 2\right) , \left({x}_{2} , {y}_{2}\right) = \left(9 , 5\right) , \left({x}_{3} , {y}_{3}\right) = \left(4 , 7\right) , h = 9$

Area of Triangle is A_b = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|

A_b = |1/2(1(5−7)+9(7−2)+4(2−5))| or

${A}_{b} = | \frac{1}{2} \left(- 2 + 45 - 12\right) | = | \frac{31}{2} | = \frac{31}{2}$sq.unit

Volume of a pyramid is $\frac{1}{3} \cdot {A}_{b} \cdot h = \frac{1}{3} \cdot \frac{31}{2} \cdot 9 = 46.5$

cubic.unit [Ans]

Apr 1, 2018

$\text{volume } = \frac{93}{2}$

#### Explanation:

$\text{the volume (V) of a pyramid is calculated using}$
$\text{the formula}$

•color(white)(x)V=1/3xx"area of base "xx"height"

$\text{the area of the base (A) is calculated using}$

•color(white)(x)A=1/2|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|

$\text{let } \left({x}_{1} , {y}_{1}\right) = \left(1 , 2\right) , \left({x}_{2} , {y}_{2}\right) = \left(9 , 5\right) , \left({x}_{3} , {y}_{3}\right) = \left(4 , 7\right)$

$A = \frac{1}{2} | 1 \left(5 - 7\right) + 9 \left(7 - 2\right) + 4 \left(2 - 5\right) |$

$\textcolor{w h i t e}{A} = \frac{1}{2} | - 2 + 45 - 12 | = \frac{31}{2}$

$\Rightarrow V = \frac{1}{3} \times \frac{31}{2} \times 9 = \frac{93}{2}$