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

1 Answer
Jul 30, 2018

color(green)(V = 1/3 A_b h = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"V=13Abh=(13)7.11717=16.6065, cubic units

Explanation:

Given : A (8,5), B (7,2), C (4,6)A(8,5),B(7,2),C(4,6), h = 7#

Using distance formula we can calculate the lengths of sides a, b, c.

d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)d=(x2x1)2+(y2y1)2

a = sqrt((7-4)^2+(2-6)^2) = 5a=(74)2+(26)2=5

b = sqrt((4-8)^2+(6-5)^2) = sqrt 17 b=(48)2+(65)2=17

c = sqrt((8-7)^2+(5-2)^2) = sqrt 10c=(87)2+(52)2=10

Semi perimeter of base triangle s = (a+b+c)/2s=a+b+c2

s = (5 + sqrt 17 + sqrt 10)/2 = 12.2854/2 ~~ 6.2427s=5+17+102=12.285426.2427

Area of triangle A_t = sqrt(s (s-a) (s-b) (s-c))At=s(sa)(sb)(sc)

A_t = sqrt(6.2427 * (6.2427-5)*(6.2427- sqrt 17)*(6.2427-sqrt 10))At=6.2427(6.24275)(6.242717)(6.242710)

#A_t ~~ 7.1171

Volume of pyramid V = (1/3) * A_t * hV=(13)Ath

color(green)(V = (1/3) * 7.1171 * 7 = 16.6065, " cubic units"V=(13)7.11717=16.6065, cubic units