Similarity Review

What is Similarity?

Similarity is when two figures have the same shape but different size making them similar. An example of similar shapes would be the images below.



All three triangles are similar because they have the same angle measurements and same shape but they are different in size.



How do we solve similarity problems?

To solve similarity problems you could follow this simple tutorial:


http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=7&sqi=2&ved=0CH4QtwIwBg&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dtm-_6sFdfk8&ei=e4zWT7HYEM746QGi0uGlAw&usg=AFQjCNEGGFNJiYfqx9TKmH2MT_Ku13MgwQ&sig2=5-kVGK027FmUJCbNXJy6Cw

 All you have to do is create a proportion between the original figure and the dilated figure for example, try to find x in the image shown below:

How do we use right triangle similarity?

Right triangle similarity is when an altitude is drawn in a right triangle creating 3 separate triangles.
You can see how this is done in this slideshow:
http://www.slideshare.net/teacherfidel/right-triangle-similarity


To solve right triangle similarity problems is the same as other similarity problems where you first start by creating a proportion.

When are triangles similar?

Two triangles are similar when they have at least two angles of the same measure such as these:

Aim: How do we find the volume of pyramids and cones?

To find the volume of a cone you must use the formula (1/3) times pi times the radius squared times the height.





The volume of this figure would be 6 squared (36) time 15 (540) divided by 3 (180). You could leave this answer in terms of pi which would equal 180pi cm cubed.


To find the volume of a pyramid you have to use the formula (1/3) time base times height.

The volume of this pyramid would be 8 times 8 (36) times 9 (324) divided by 3 (108). The final answer would be 108 cm cubed.









Find the volume of theses two solids:

Aim: How do we find the surface area and volume of the sphere?

To find the volume of a sphere you must use the formula 4 times pi times the radius cubed divided by 3.

For example:

 

The volume of this sphere would be 4 cubed (64) times 4 (254). After this you could just leave the problem in terms of pi which is 256pi and  then divide it by 3. The final answer would be 85.3pi cm cubed.




To find the surface area of a sphere you must use the formula 4pi times radius squared.




 

Using the same sphere the surface area would be 4 squared (16) times 4 (64). After you find this you could just leave the final answer in terms of pi which is 64pi.




Find the surface area and volume of this sphere:

Aim: How do we find surface area and lateral aea of prisms and cylinders?

The formula to find the surface are of a prism is base times height times width

For example in this image the base length is 5, the height is 3  and the length is 10.
5 x 3 x 10 = 150

And to find the lateral are of this figure would would simply subtract the area of the two faces which would be the area of the two squares.
3 x 5 x 2 = 30
Then you would subtract the area of the two faces from the total surface area.
150-30= 120.
Lateral area = 120.

 Find the Surface and lateral area of this prism:


The surface area formula of a cylinder is: 2 π r² + 2 π r h



To find the surface area of his cylinder you would do radius squared (9) times 2 times π.
This will find the area of the two circles. To find the area of the rest of the solid you would have to do
6 x 3 x 2 x π. 
This formula would also solve the lateral area.

But to find the surface area you would have to add the outcome of the first equation and the second equation.


Find the surface and lateral area of this cylinder:


 

Aim: How do we idnetify solids?

Solid- 3- dimensional figures with width, depth and height.

All solids have volume and surface area.
There are two types of solids:
Polyhedra- having all flat surfaces.
Non- Polyhedra- if any surface is not flat.

All solids fall under these two categories.

Figures that would be considered polyhedras are; cubes, pyramids and prisms
Figures that fall under the category of non-polyhedras are figures like; cylinders or spheres.

Aim: How do we find the area of regular polygons?

Area of a regular polygon- The are of a regular polygon is n (number of sides) times a (apothem) times s (side length) divided by 2 (nas 1/2).







Fo example if a (apothem) is 6 cm long and the side (s) is 5 cm long the area of this hexagon would be
5 times 6 times 6, which equals 180 and then you divide 180 by 2. The are of this hexagon would then be 90 cm squared.




What would be the area of a pentagon with an apothem of 7 and a side length of 8?

Aim: How do we find the area of a circle?

Circle Area Formula- The area of a circle is given by the formula: pi times the radius squared.

So for example:




The area of this circle would be the radius (5) squared (25) times pi. You could solve this problem by typing it into a calculator or just leaving it in terms of pi, which would be 25 pi.



Solve the area of this circle: