MHB [ASK] Find the volume of Pyramid in a Cube

[Monoxdifly]

Given a cube ABCD.EFGH whose side length is 4 cm. If the point I, K, and J is dividing EF, FG, and BG to two equal lengths respectively, determine the volume of pyramid D.IJK!

I think I can work out the pyramid's base area by deriving for the formula of equilateral triangle area. What I can't is determine the pyramid's height. All I knew is that is must be less than 4√3 cm. Anyone willing to help me?

 

[skeeter]

$V = \dfrac{\vec{ID} \cdot (\vec{IK} \times \vec{IJ})}{6}$

$V = \dfrac{(-2,4,-4) \cdot (-4,4,0)}{6} = \dfrac{8 + 16 + 0}{6} = 4$

 

[Monoxdifly]

Given a cube ABCD.EFGH whose side length is 4 cm. If the point I, K, and J is dividing EF, FG, and BG to two equal lengths respectively, determine the volume of pyramid D.IJK!

I am sorry, the bold part was supposed to be BF. Must've been a typo. Also, how to do it without vectors? This is supposed to be for 8 graders, we can only use Pythagorean theorem at most.

 

[skeeter]

Length of diagonal DF = height of pyramid IJKF + height of pyramid IJKD

Height of both pyramids passes through the centroid of equilateral triangle IJK ... recall the ratio of the distance from vertex to centroid : centroid to opposite side is 2:1.
Median length of equilateral triangle IJK should be no problem to determine.

With that information and Pythagoras, you should be able to find the height of pyramid IJKF ... subtract that value from the length of diagonal DF and you have the height of pyramid IJKD.

 

[Monoxdifly]

Let me try:
DJ =$$\sqrt{BD^2+BJ^2}$$=$$\sqrt{(4\sqrt2)^2+2^2}$$=$$\sqrt{16(2)+4}$$=$$\sqrt{32+4}$$=$$\sqrt{36}$$= 6 cm
Height of triangle IJK =$$\sqrt{(2\sqrt2)^2-(\frac12\times2\sqrt2)^2}$$=$$\sqrt{4(2)-(\sqrt2)^2}$$=$$\sqrt{8-2}$$=$$\sqrt6cm$$
Height of pyramid D.IJK =$$\sqrt{DJ^2-(\frac23\times\sqrt6)^2}$$=$$\sqrt{6^2-\frac49(6)}$$=$$\sqrt{36-\frac43(2)}$$=$$\sqrt{\frac{108}{3}-\frac83}$$=$$\sqrt{\frac{100}{3}}$$=$$10\sqrt{\frac13}cm?$$
Because I got $$2\sqrt3cm^2$$ as the base area, so the volume is$$\frac13\times2\sqrt3\times10\sqrt{\frac13}$$=$$\frac{20}{3}$$cc?

 

[skeeter]

yes

 

[Monoxdifly]

Okay, thanks for your help.