Question

The base of a triangular pyramid is a triangle with corners at `(6 ,2 )`, `(3 ,5 )`, and `(4 ,2 )`. If the pyramid has a height of `9`, what is the pyramid's volume?

Answer 1

The volume is `9` cubic units.

Explanation:

This answer is basically in two parts:

Part 1 -- area of the triangular base is half of any side times the height from that side to the opposite vertex.

Part 2 -- vume of the pyramid is one-third the area of the base times the height.

Now to the math.

Part 1 -- it helps to draw out the triangle on graph paper. Note that the first and third vertices are both on the horizontal line `y=2` and the third vertex is at `y=5`. So we have a side that's `2` units long (from `(4,2)` to `(6,2)`) and the height to the opposite vertex is three units (from `y=2` along the entire side to `y=5` on the third vertex). So the area of the triangle is `(1/2)×2×3=3` square units.

Part 2 -- The area of the base is `3` square units and the height is `9` units. Pugthose numbers into the formula given above for the volume of a pyramis: `(1/3)×3×9=9` cubic units.

Answer 2

Volume of pyramid is `9` cubic.unit

Explanation:

Vertices of triangular base are `(6,2) , (3,5) , (4,2)`

The area of the triangular base is `A_b=1/2(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2))` or

`A_t=1/2(6(5-2)+3(2-2)+4(2-5)) = 1/2 (18+0-12)=12*6= 3`sq.unit

Volume of pyramid is `V= 1/3* A_b*h ; h= 9 , A_b=3` or

`V= 1/cancel3*cancel3*9 = 9` cubic.unit [Ans]