The base of a triangular pyramid is a triangle with corners at #(3 ,8 )#, #(1 ,6 )#, and #(2 ,8 )#. If the pyramid has a height of #4 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(3 ,8  )#, #(1  ,6   )#, and #(2  ,8  )#.  If the pyramid has a height of #4  #, what is the pyramid's volume?

Sadie Allen · Accepted Answer

#\frac{8}{3}# units^2

The area of the base (triangle) can be worked out from the vertices, as Area = #{2 * 2}{2} = 2# Therefore by using the formula #V = {A * h}{3}# we get the volume as #\frac{2*4}{3} = \frac{8}{3}#

Theodore Abad · Answer

#color(blue)("Volume of a pyramid "V_p = 1/3*A_b*h=1/3 *1*4 = 4/3 #

#"Area of triangle knowing three vertices on the coordinate plane is given by "#

#color(crimson)(A_b = |1/2(x_1(y_2−y_3)+x_2(y_3−y_1)+x_3(y_1−y_2))|#

#(x_1,y_1)=(3,8) ,(x_2,y_2)=(1,6),(x_3,y_3)=(2,8) , h=4#

#A_b = |1/2(3(6−8)+1(8−8)+2(8−6))| = 1#

#color(crimson)("Volume of a pyramid " V_p = 1/3* A_b * h#

#color(blue)("Volume of a pyramid "V_p = 1/3*A_b*h=1/3 *1*4 = 4/3 #

Theodore Abad · Answer

undefined To find the volume of a triangular pyramid, you can use the formula: $ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} $.

First, calculate the area of the base triangle using the given coordinates:

$ A = \frac{1}{2} \times \left| (3 \times (6-8)) + (1 \times (8-8)) + (2 \times (8-6)) \right| $.

$ A = \frac{1}{2} \times | (-2) + (0) + (4) | $.

$ A = \frac{1}{2} \times 6 = 3 $.

Then, use the formula for the volume of a triangular pyramid:

$ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 3 \times 4 = 4 $ cubic units.

So, the volume of the triangular pyramid is 4 cubic units.