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Lesson 11: Volume of Solids (II)

June 6, 2012

3. Volume of a cuboid

Volume of a cuboid whose side lengths are l, w and h is:

Click below and you  will practice how to calculate the volume of a cuboid.

 3.1. Cavallieri’s  principle

In English we name cross sections to the shape made when a solid is cut through parallel to the base.

Note: don’t draw the rest of the object, just the shape made when you cut through.

cross-section of a pyramid

Example:

The cross-section of a rectangular pyramid is a rectangle

cross-section of a cylinder

Example:

The cross-section of a circular cylinder is a circle

Compare the figures below.

That the stacks have equal volume is made clearer with a simple transformation of the stack on the right.If we cut both figures before through parallel to the base, the cross sections are the same.

The theorem bellow is known as Cavallieri’s Principle.

If, in two solids of equal altitude, the areas of cross sections at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.

Cavalieri’s principle reveals its power when applied to more complicated shapes (or when the rectangles are allowed to become infinitely thin). We will apply it to find many volumes of solids we studied in lesson 10.

4. Volume of prisms and cylinders

 4.1. Volume of a prism

Applying Cavallieri’s Principle, the volume of a cuboid and the volume of a prism with the same height and whose bases have the same area, are the same. If we take into account that , where  is the base area we can conclude:

The volume of a prism with height h and base area AB is

V = AB · h

For instance, if we have a triangular prism AB is the area of a triangle (base · times height of the triangle divides by two).

Click below if you want to see a video on this topic:

You can practice below:

Volume of prisms and pyramids

4.2. Volume of a cylinder

 Notice that any cross-section of a cylinder is equal to the base. Applying Cavallieri’s Principle again, the volume of a cuboid and the volume of a cylinder with the same height and whose bases have the same area are the same.

The volume of a cylinderwith height and radius  is V = AB · h =πr2 · h

 Click below if you want to see a video on this topic:

5. Volume of pyramids and cones

5.1. Volume of a pyramid

 

The prism and the pyramid on the right have the same height and the same base.

 

 

If we fill up the pyramid with sand, we can see we need three times the sand contained in the pyramid to fill up the prism. So, we can deduce that the volume of the pyramid is a third of the volume of the prism.

The volume of a pyramid with height and base area is

V = 1/3 · (AB · h)  

Note:If we have a cone and a pyramid with the same height  and the same base area , by applying Cavallieri’s principle we can conclude that they have the same volume.

 Click below if you want to see a video on this topic:

 You can practice on the following link:

Volume of prisms and cylinders

5.2. Volume of a cone

The cylinder and the cone below have the same height and the same base. 

If we fill up the cone with sand, we can see we need three times the sand contained in the cone to fill up the cylinder. So we can deduce that the volume of the cone is a third of the volume of the cylinder.

The volume of a cone with height  and radius is

V = 1/3 · (πr2 · h) 

Click beñow if you want to see a video on this topic:

You can practice on the link below:

Volume of pyramids and cones

6. Volume of a sphere

 Consider the cylinder circumscribed to a sphere with radius . This cylinder has height 2r and radius r.

 If we fill up the semi sphere with sand, we need three times the sand that contains the semi sphere to fill up the cylinder. 

 

So the volume of the sphere, that is twice the volume of the semi sphere, is

V =  4/3 π r³. 

See a video on this topic:

You can practice on the following link:

Volume and surface area of spheres

 

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