Archive for the ‘Lesson 11: Volume of Solids’ Category

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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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Lesson 11: Volume of Solids (I)

May 31, 2012

Vocabulary on Volume of Solids

Lesson 11: Volume of Solids 

Volume Worksheet

Volume Word Problems

 I Have to Know by the End of this Lesson

 

1. Volume of a solid

1.1. Volume units

Volume is length by length by length, so the basic unit of volume is a cube with edge length one metre. Its volume is 1 metre × 1 metre × 1 metre, which is written m3 (cubic metre).

The basic unit of volume is the cubic metre, which is written m3.

Multiples and submultiples of cubic metre are the following:

  unit it is the volume of a it equals symbol
multiples cubic kilometre cube whose edges measure one kilometre 1000000000 m3 km3
cubic hectometre cube whose edges measure one hectometre 1000000 m3 hm3
cubic decametre cube whose edges measure one decametre 1000 m3 dam3
base unit cubic metre cube whose edges measure one metre   m3
submultiples cubic decimetre cube whose edges measure one decimetre 0.001 m3 dm3
cubic centimetre cube whose edges measure one centimetre 0.000001 m3 cm3
cubic millimetre  cube whose edges measure one millimetre 0.000000001 m3 mm3

 You have to bear in mind that volume units are cubes and three-dimensional. The conversion of a unit into another one is done by dividing or multiplying both the length, the height and the width, that is dividing and multiplying by ten three times; in other words, dividing and multiplying by 1000.

The volume units go up by a factor of 1000 at the time.

Each jump to a smaller unit is equivalent to multiply by 1000, you have to move the decimal point three places to the right. Each jump to a larger unit is equivalent to divide by 1000, you have to move the decimal point three places to the left. Here you are a sketch:

 

Now you can practice on the following links:

1 – Volume unit conversion. (with hints)

2 – Volume unit conversion.

3 – Match the equal volumes.

 

 1.2 Volume of an object

 Volume of an object is the amount of space it occupies.

 We will begin by a basic solid, the cube.

2. Relationships among volume, capacity and mass

2.1 Volume and capacity

The capacity of a container is known as the volume of the liquid or gas that it can hold.
Capacity and volume have equivalent meanings. Establishing a special unit to measure volume isn’t necessary, so using the cubic metre would be enough, but for practical use the litre was established as a unit. If you pour one litre in a cube with an edge of 1 dm it will fit in the cube exactly.

The litre is identical to the cubic decimetre (dm³), although it wasn’t always so . Recall

The litre is the capacity of a cubic decimetre.

We know 1 dm is 0·1 m or 10 cm, so 1 dm³ is 10×10×10 cm:

1 dm3, 10 cm along each side, 1000 cm3

This, of course, means that there are 1,000 cm³ in a litre, or that 1 cm³ is equal to 1 mL.

The following table shows the equivalence between volume units and capacity units.

Volume units m3     dm3     cm3
Capacity units kl hl dal l dl cl ml


2.2. Volume, mass and capacity

A recipient contains a litre of pure water, which occupies 1 dm3. We weight it and it weights 1 kilogram exactly.

One kilogram is the weight of 1 dm3  of pure water.

If we weigh a container with 1 ml of pure water with, which occupies 1 cm3, it weights 1 gram.

One gram is the weight of 1 cm3 of water

 The following table shows the equivalence among volume units, capacity units and mass units for pure water.

Volume units m3     dm3     cm3
Capacity units kl hl dal l dl cl ml
Mass units t q mag kg dag hg g

1 l =1 dm3 = 1 kg of pure water

 NOTE: If we have other substance different from pure water one litre doesn’t weight one kilo.

In the following video you can find a long explanations about these topics: