Volume

Slideshow 50, Mathematics

Mr Richard Sasaki

Room 307

OBJECTIVES

• LEARN HOW TO FIND THE VOLUME OF A PRISM OR CYLINDER

• LEARN HOW TO FIND THE VOLUME OF A SQUARE BASED PYRAMID OR CONE

VOLUME

What do prisms have in common with cylinders?

They both have two congruent faces.

The process of calculating volume for them is similar.

CYLINDERS AND PRISMS

If we know the base, how do we calculate the volume?

If the base is and it has length then its volume is…

𝑉=¿𝑆𝑙Let’s try an example for each.

CYLINDERS AND PRISMS

Example (Cylinder)

A cylinder has radius 2cm and length / height 9cm. Calculate its volume.

𝑆=¿𝜋𝑟 2¿𝜋 ∙22¿ 4𝜋𝑉=¿𝑆𝑙¿ 4𝜋 ∙9¿36𝜋 𝑐𝑚3

CYLINDERS AND PRISMS

Example (Triangular Prism)

A triangular prism properties as shown above. Calculate its volume.

𝑆=¿h𝑏2¿4 ∙32¿6

𝑉=¿𝑆𝑙¿6 ∙6¿36 𝑐𝑚3

336𝑚𝑚3

27𝑚382.5𝑐𝑚3

243𝜋𝑘𝑚3600𝑚3

437.4𝜋𝑚3

2880𝜋𝑐𝑚31260𝑐𝑚3

336𝑚388.2𝜋𝑚3

79.8𝑚3198.276𝑚3

SQUARE BASED PYRAMIDS AND CONES

Have a look at the cube below.

All diagonals meet at the centre and divide the cube into six .

square-based pyramidsIf the cube is cut into half (to form a cuboid),

the apex of a pyramid will touch the top and base will be at the bottom.The pyramid has a third of the volume of the cuboid. This is the coefficient added to the formula about its base and height.

SQUARE BASED PYRAMIDS AND CONES

For a square based pyramid or cone with base and it has height then its volume is…

𝑉=13h𝑆

The method is the same for both but of course calculating for them is different!

SQUARE BASED PYRAMIDS AND CONES

Example 1 Example 2

𝑉=13h𝑆 𝑉=

13h𝑆

𝑉=13∙42∙6 𝑉=

13∙𝜋 ∙22 ∙6

𝑉=32𝑐𝑚3 𝑉=24𝜋3

𝑐𝑚3

Which is bigger?

ANSWERS

5813𝑐𝑚3

16113𝑐𝑚339000𝑚𝑚3

100𝜋𝑐𝑚3

112𝜋𝑐𝑚3713𝜋 𝑘𝑚3

𝑉=13𝜋 r2h

𝑉=13𝑎2h

𝑉=13𝜋 ∙202 ∙60−

13𝜋 ∙102 ∙3 0

𝑉=13𝜋 ∙21000=7000𝜋 cm2