The base of a triangular pyramid is a triangle with corners at (4 ,2 )(4,2), (3 ,6 )(3,6), and (7 ,5 )(7,5). If the pyramid has a height of 8 8, what is the pyramid's volume?

2 Answers
Oct 18, 2017

2020

Explanation:

The triangle with vertices (4, 2)(4,2), (3, 6)(3,6) and (7, 5)(7,5) can be drawn inside a 4 xx 44×4 square with vertices (3, 2)(3,2), (7, 2)(7,2), (7, 6)(7,6) and (3, 6)(3,6), dividing it into 44 triangles...

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The two smaller triangles are right angled triangles with legs of lengths 11 and 44, so their total area is 44.

The larger isosceles right angled triangle has area 1/2 * 3 * 3 = 9/21233=92

So the total area of the given triangle is:

4^2 - 4 - 9/2 = 15/242492=152

The pyramid has volume:

1/3 * "base" * "height" = 1/3 * 15/2 * 8 = 2013baseheight=131528=20

Oct 18, 2017

2020

Explanation:

The area of a triangle with vertices (x_1, y_1)(x1,y1), (x_2, y_2)(x2,y2), (x_3, y_3)(x3,y3) is given by the formula:

"Area" = 1/2 abs(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)Area=12|x1y2+x2y3+x3y1x1y3x2y1x3y2|

(see https://socratic.org/s/aK6h7PjW)

So putting:

{ ((x_1, y_1) = (4, 2)), ((x_2, y_2) = (3, 6)), ((x_3, y_3) = (7, 5)) :}

we find that the base of our pyramid has area:

1/2 abs((4)(6)+(3)(5)+(7)(2)-(4)(5)-(3)(2)-(7)(6))

=1/2 abs(24+15+14-20-6-42)

=1/2 abs(-15)

=15/2

Then the volume of the pyramid is:

1/3 xx "base" xx "height" = 1/3 * 15/2 * 8 = 20