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

2 Answers

Volume of pyramid V = color(red)(5V=5 cubic units

Explanation:

enter image source here

First to find the area of the triangular base.

If three sides are known, area of the triangle is given by the formula

A = sqrt((s )(s-a) (s-b) (s-c))A=(s)(sa)(sb)(sc)

where s is the semi perimeter of the triangular base, a,b and c the sides of the base.

Using distance formula we can find the sides.

c = sqrt((4-3)^2 + (2-7)^2) = color(brown)(5.099c=(43)2+(27)2=5.099

a = sqrt(5-3)^2 + (3-7)^2) = color (brown)(4.4721a=532+(37)2)=4.4721

b = sqrt((5-4)^2+(3-2)^2) = color(brown)(1.4142b=(54)2+(32)2=1.4142

Semi perimeter p = (a + b + c)/2 = (5.099 + 4.4721 + 1.4142)/2 = color(purple)(5.4919p=a+b+c2=5.099+4.4721+1.41422=5.4919

Area of triangular base A = sqrt(5.4919 * (5.4919-4.4721) * (5.4919-5.099) * (5.4919-1.4142)) ~~ color(green)(3A=5.4919(5.49194.4721)(5.49195.099)(5.49191.4142)3

Volume of pyramid V = (1/3) * A * h = (1/cancel3) * cancel3 * 5= color(red)(5 cu. units

Feb 18, 2018

5

Explanation:

"the volume (V) of a pyramid is calculated using the formula"

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

"the area of the base (A) can be found 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)|

"let "(x_1,y_1)=(4,2),(x_2,y_2)=(3,7),(x_3,y_3)=(5,3)

A=1/2|4(7-3)+3(3-2)+5(2-7)|

color(white)(A)=1/2|16+3-25|=3

rArrV=1/3xx3xx5=5