[ASK] Find the volume of Pyramid in a Cube

In summary, to determine the volume of pyramid D.IJK with a cube ABCD.EFGH whose side length is 4 cm, we can use the formula V = (1/3)Ah, where A is the area of the base and h is the height of the pyramid. To find the height, we can use the Pythagorean theorem and the information that the height passes through the centroid of the equilateral triangle IJK. Once we have the height, we can find the base area using the formula for the area of an equilateral triangle. Plugging in the values, we get a volume of (20/3) cc.
  • #1
Monoxdifly
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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?
 
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  • #2
$V = \dfrac{\vec{ID} \cdot (\vec{IK} \times \vec{IJ})}{6}$

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

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  • #3
Monoxdifly said:
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.
 
  • #4
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.
 
  • #5
Let me try:
DJ =\(\displaystyle \sqrt{BD^2+BJ^2}\)=\(\displaystyle \sqrt{(4\sqrt2)^2+2^2}\)=\(\displaystyle \sqrt{16(2)+4}\)=\(\displaystyle \sqrt{32+4}\)=\(\displaystyle \sqrt{36}\)= 6 cm
Height of triangle IJK =\(\displaystyle \sqrt{(2\sqrt2)^2-(\frac12\times2\sqrt2)^2}\)=\(\displaystyle \sqrt{4(2)-(\sqrt2)^2}\)=\(\displaystyle \sqrt{8-2}\)=\(\displaystyle \sqrt6cm\)
Height of pyramid D.IJK =\(\displaystyle \sqrt{DJ^2-(\frac23\times\sqrt6)^2}\)=\(\displaystyle \sqrt{6^2-\frac49(6)}\)=\(\displaystyle \sqrt{36-\frac43(2)}\)=\(\displaystyle \sqrt{\frac{108}{3}-\frac83}\)=\(\displaystyle \sqrt{\frac{100}{3}}\)=\(\displaystyle 10\sqrt{\frac13}cm?\)
Because I got \(\displaystyle 2\sqrt3cm^2\) as the base area, so the volume is\(\displaystyle \frac13\times2\sqrt3\times10\sqrt{\frac13}\)=\(\displaystyle \frac{20}{3}\)cc?
 
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  • #6
yes
 
  • #7
Okay, thanks for your help.
 

FAQ: [ASK] Find the volume of Pyramid in a Cube

1. What is a pyramid in a cube?

A pyramid in a cube is a three-dimensional shape that has a square base and four triangular sides that meet at a common point, forming a pyramid shape inside a cube.

2. How do you find the volume of a pyramid in a cube?

To find the volume of a pyramid in a cube, you can use the formula V = (1/3)Bh, where B is the area of the base and h is the height of the pyramid.

3. Can the volume of a pyramid in a cube be calculated using the cube's volume?

No, the volume of a pyramid in a cube cannot be calculated using the cube's volume because the pyramid only occupies a portion of the cube's volume.

4. What is the difference between a pyramid in a cube and a regular pyramid?

A pyramid in a cube has a square base and four triangular sides, while a regular pyramid can have a variety of polygonal bases and triangular sides.

5. How is a pyramid in a cube used in real life?

A pyramid in a cube can be seen in architecture, such as the Great Pyramid of Giza, and in packaging, such as a pyramid-shaped tea bag. It can also be used as a visual aid in mathematics and geometry classes.

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