Volume of Basic Shapes

So today we looked at finding the volume of basic prisms. When I say basic prisms, what I mean are prisms that have a constant cross section. This means that the face you see on the shape is the same, no matter where you cut through the shape. For example;

The prism above has a pentagon as a face. If we were to cut a cross section at any point through this shape, it would have the same pentagon face no matter what. This works for a variety of shapes. For example; 

If a prism has this constant cross section, it is then known as a basic prism. 

To find the volume of these basic prisms we use the following formula; 

Volume = Area of Face x Height (or depth)

The face we find the area of is the face that has the constant cross section throughout the prism. For example, with the cylinder above we would find the area of the circular face, and then multiply it by the height or depth of the whole prism. 

Using this information, you should be able to make a start on the "Volume" questions on the first work sheet, which can be found in the "Work Sheets" tab. 

If you are still stuck on this topic, the following website is handy; 

http://www.mathsisfun.com/geometry/prisms.html

If you are still stuck, feel free to comment on this post, or email or chat to me in person. 

Surface Area of Cylinders

Today we looked at the total surface area of a cylinder. A cylinder looks like this:

The two pieces of information we need to find the total surface area of a cylinder are the radius (marked with an "r" on the diagram above) and the height of the cylinder (marked with a "h" on the diagram above). Remember, the radius is only half way across the circle, so if you are told the full distance across the circle you will need to half this. 

To find the surface area, imagine this cylinder all folded out as a flat 2D shape. We call this a net. The net of a cylinder looks like this: 

From this we can see that a cylinder is simply 2 circles and a rectangle. We know how to find the area of both of these. We know the area of a circle is: 

We have two of these circles, so the total area of the two circles is two lots of this, or:

Now we need to find the area of the rectangle part. For the area of a rectangle, we need the length and the width. We already know the length. This is the height of the cylinder (marked "h" on our cylinder and the net of the cylinder). We now need to find the width of this rectangle. If you look carefully you will see that the width is actually the same as the distance around the outside of the circle. This is known as the circumference of a circle. The formula for finding this is simply: 

So this is our width, and we know that the height of our rectangle is its length, so the total area of the rectangle can be found by multiplying these two together. We get:

So finally, to get the total surface area of the cylinder we combine the area of this rectangle by the area of the two circular pieces. This gives us a final formula of;

We can use this now to easily find the surface area of a cylinder. All we need to do is substitute the length of the radius (r) and the height of our cylinder (h) into this formula and use your calculator to get an answer for the total surface area. 

You should now be able to finish off most of the work from the surface area section of the worksheet. You can find a copy of this worksheet in the "Worksheets" tab. It is the first worksheet on surface area and volume. 

If you have any questions about this feel free to comment on this post, or email or chat to me in person. 

Pythagoras' Theorem

Last lesson we looked at using Pythagoras' Theorem to find unknown sides on our prisms. This is a really important skill to know and use. I did notice that a lot of students were struggling with this, or had simply forgot how to apply Pythagoras' Theorem. For today's lesson I want to look through the following document:

https://tasedu-my.sharepoint.com/personal/adam_van_sant_education_tas_gov_au/_layouts/15/WopiFrame.aspx?sourcedoc={C47B44FA-187B-45E1-95A1-F3DA9FB1A297}&file=Pythagoras%20recap.docx&action=default

Work through this today. The first two pages contain some information and some instructions. The second two pages contain some questions to work through.

If you have any questions on this topic feel free to comment on this post, or email or chat to me in person.

Surface Area of Prisms - Using Pythagoras' Theorem

So we have been looking at finding the surface area of prisms. Often the diagram will give us all of the information we need to solve these problems straight away, but sometimes we need to find some for ourselves first. An example of this can be seen with the following shape:

If we are asked to find the total surface area of this shape, we can easily do this as it is made up of basic triangles and rectangles. If we look carefully though, we will see that the top rectangular face doesn't have a length for the edge touching the triangle. 

Without this length, we cannot find the area of this face, and so we cannot find the total surface area of the prism. Fortunately we do have a way to find this side length, and that is using Pythagoras' Theorem on right angled triangles. This should have been covered in grade 8 or 9, but if you need a refresher the following link is handy: 

http://www.mathsisfun.com/pythagoras.html

Now looking at this prism, we will just look at the triangular face for now:

Using Pythagoras' Theorem we can find the length of this unknown side quite easily:

Now  that we know the unknown side length, our prism now looks like this:

We now have all of the information we need to find the total surface area of this whole prism. 

If you work this out, you should get an answer of 75 square centimeters.

We may also have a prism that looks like this:

If we look carefully at this one, we can see that we don't actually know the height of the triangular face, so we cannot find the area of this face. We will again use Pythagoras' Theorem to find this height. Again, I will just focus on the triangular face:

Now Pythagoras' Theorem will only work with right angled triangles, and this triangle above is clearly not right angled. We can, however cut this triangle in half and turn it into a right angled triangle. It would then look like so:

You will notice that the base length of the triangle is now 2cm, since we have cut it in half. We can now use Pythagoras' Theorem to find our unknown side like so: 

We now know that our height of our triangular face is 3.46cm. Our original prism now looks like so:

We can now easily find the total surface area of this prism. If you work it out you should get an answer of 73.84 square centimeters. 

I have uploaded a worksheet for this topic into the "Work Sheets" tab. You should be able to complete the first question. There are 6 different prisms for you to find the surface area of. 

As always, if you have any questions on this topic feel free to comment on this post, or email or chat to me in person. 

Welcome Back + New Unit

Firstly, welcome back from your time off. I hope you all had a great few weeks off and are feeling rested and ready to go for your last ever term of high school.

We have now started a new unit on space and measurement. This will involve finding the surface area and volume of shapes and prisms. We will start with more basic shapes and then move up to harder problems involving more complex shapes.

The first thing we need to understand is how to find the area of basic shapes. The details for this can be found in the following table:

This is a handy table that gives formulas for finding surface areas of basic shapes. We can use these when we find surface areas of more complex 3D shapes. For example, if we were asked to find the surface area of this shape:

Finding the surface area involves simply finding the area of each face, then adding all of these together to find the total surface area. We can easily find the surface area of each of the faces of this prism, as they are all rectangles. If we look at the table above, we know that the area of a rectangle is:

A = l x w 

We can apply this formula to all of the faces of this shape, then simply add all of the faces together. 

We can also apply the same ideas to different prisms, such as this one: 

This is a prism made of two triangular faces and 3 rectangular faces. If we use the formulas from the table above, we can easily find the areas of all of these faces. We can simply add all of these faces together and get the total surface area of this prism. 

This is a brief introduction to the unit, and more specifically how we find surface area of shapes. I have put a word document that has the table above in it in the "In Class Information" tab. 

If you have any questions from this feel free to comment on this post, or email or chat to me in person,