Intermediate Test Prep (Grades7-8) Using Mathematical Sentences to Solve Problems

The Cone:
The surface which is created by a straight line which passes through a fixed point and travels in a circular path.

The Pyramid:
A 3-dimensional figure which has a polygonal base and triangular faces which meet at a common point.

The Cone:
We have all seen all kinds of examples of this shape out in the real world....
ice cream cones, funnels, party hats, megaphones, all have the cone as it's basic shape.

If you will let your imagination go for a minute you should be able to visualize that a cone could be dropped into a cylinder which has the same sized circular base, and which has have the same height, and although it would be a perfect fit at the very bottom, there would be lots of empty space as the cone gradually builds to a point. I'm hoping that you see that the volume of the cone, although related to that of a cylinder, must be quite a bit less. In fact the volume of a cone is exactly 1/3 the volume of a cylinder which has the same radius and height.
Therefore the formula for the volume of a cone is:
V = 1/3Π r²h
Let's look at an example:
Find the volume of a cone which
has a radius of 4 cm, and a height of 9 cm.
V = 1/3Π r²h
V = 1/3(3.14)(4²)(9)
V = 1/3(3.14)(16)(9)
V = 1/3(452.16)
V = 150.72 cubic cm!

The Pyramid:
Although the variety of examples for the shape of a pyramid seems to be more difficult to list, everyone immediately pictures a desert scene with the pyramid rising up out of the sand as if dropped there by the wave of a magic wand. It is simply incredible to understand the amount of time and work it must have taken for these ancient buildings to have been created.

If we think along the same lines we used to develop the volume formula for a cone it should make sense that the volume of a pyramid, just like the cone, is equal to 1/3 the volume of the prism which has the exact same base. You should be reminded that although we normally think of a pyramid as having a square or rectangular base, pyramids can have a base in the shape of any polygon....triangles, pentagons, hexagons, etc...are all examples of possible bases for a pyramid.
Let's look at an example:
V = 1/3 Bh (where B = area of the polygonal base)
Find the volume of the pyramid having a
square base where each side of the square is 7 cm,
and the pyramid is 12 cm high.
V = 1/3Bh
B = the area of a square which = s²
V = 1/3 (7²)(12)
V = 1/3(49(12)
V = 1/3(588)
V = 196 cubic cm!

For practice on your own....
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