Is there an easy way to explain to a 14 yr old why the volume of a pyramid equals 1/3 x area of base x height?

Answer 1

This is a tough calculation, but here are some things that may help them get it:

It is fairly clear that the volume is proportional to base area and height. Imagine the effect of doubling the area or the height. Doubling the base area will double the volume of any and every horizontal slice you take throught the pyramid. Doubling the height will double the volume of every vertical slice.

See this pic: http://thinkzone.wlonk.com/Area/VolPyr1.gif

As for why 1/3 is the constant, a good visualization is to think of 6 regular pyramids with bases of 1 meter x 1 meter and heights of 1/2 meter. If you fit all of these together, you could actually make a cube that is 1 meter x 1 meter x 1 meter.

See this pic: http://thinkzone.wlonk.com/Area/VolPyr4.gif

The volume of the whole cube is 1 x 1 x 1

Since the area (a) of the base of the pyramid is 1 x 1 and the height (h) of the pyramid is 1/2, then we can think of the volume of the whole cube as:

a x 2h (i.e. 1 x 1 x (1/2x2))

So the volume of any one pyramid is 1/6th of that which equals

1/3 x a x h

Hope this helps

Answer 2

You can use a model to show him. You can take a cube of any dimension and show him that it can be cut into 6 identical pyramids, each having height half the height of the cube and 1/3 x area of base x height(height of the cube/2) is nothing but one-sixth the volume of the cube. or you can do it the other way round :Starting with six identical pyramids a making it a cube. I hope he will understand it that way.

Answer 3

Not really. To really prove it you would need calculus. You could try explaining why the area of a cube is the way it is, and then rationalize it comparing it to the cube's area. This wouldnt prove it, but it would make the result seem plausable. You could also explain that it can be calculated as 1/3 x area of base x height by submerging pyramids (small ones) in water and measuring the displacement.

Answer 4

Show him a square and calculate area. Then show him the pyramid/triangle and explain that there is not as much volume inside due to the shape. then calculate area. When didviding by three that is subtracting the area not included in the pyramid that IS icluded in the square. You need to elaborate a bit more, sorry if confusing.

Answer 5

Try constructing a Pyramid from a Sheet of PAPER and you will get the Answer with the formula of the Area of a Triangle (its base) in mind and calculation.

Answer 6

Maths is based upon axioms which are called as universal truths. The best way is by some simple practical experiments of comparison like the volume of water displaced etc.

The best way is to study the history of maths and science. You will be surprised to see how much of bickering and politics has happened in accepting certain concepts. One has to understand that we are all trying to explore the universe and our origin. Even great scientists and mathematicians have accepted finally that they feel humble in front of the mighty nature which we call as God.

Unlock the childs imagination and freedom to explore without fear. This is the best education you can provide. Our system kills innovation and free thinking. You have to carefully balance.

Answer 7

Perhaps... give him three pyramids (which could be dismantled by tiny blocks) and have him construct a cube... he'll soon see it takes three pyramids to make a whole cube.

Answer 8

say because I said so. or you can take them to a tutor.

Answer 9

because I said

Answer 10

tell him to find any other answer and sleep well