**LESSON COMING SOON.... (STILL IN PROGRESS)**

**Calculating Surface Area**

Key Terms:

Prism

Pyramid

Net

Polygon

Polyhedron

squared (exponent)

Definition:

Surface area is the measurement of flat space on the outside of a 3-Dimensional shape. These 3D shapes are called polyhedra (the singular form is called a polyhedron).

For more information about calculating area of 2-Dimensional shapes, see the lesson on ***calculating area***(insert hyperlink)

Demonstration:

The formulas for calculating the surface area of many common shapes is provided below in the Basic Rules section. Before these equations are provided, we will explore some simple cases and see how these formulas are obtained.

Our first step in finding the surface area of a 3D shape will be to display the shape as a 2D net: a collection of shapes that can be folded into a closed 3D figure with no overlaps. Each of these shapes was covered in the ***calculating area (insert hyperlink)*** lesson.

For example, the net of a cube can be drawn in multiple ways, as shown here.

***Insert SA image 1***

Any net that folds into a cube has 6 square faces. For a cube of edge length l, the surface area is the total area of all 6 squares. Each square has a surface area of l2, and so the total surface area of a cube is 6l2.

Similarly, a rectangular prism has 3 sets of opposite sides that each have the same area.[Text Wrapping Break]

***Insert SA image 2***

***Insert SA image 3***

where l, w, and h represent length, width and height respectively.

The total surface area is the sum of the areas of each coloured pair. The 2 shows that each colour has two sides.

***Insert SA equation 1***

Using a similar strategy, the surface area of a regular triangular prism (where the ends are equilateral triangles) is the sum of the areas of the two ends, plus the sum of the area of the three rectangular sides.

In this case, we can use pythagorean theorem to find the value of h, even when it’s not given.

***Insert SA image 4***

***Insert SA equation2***

or

***Insert SA equation 3***

The equation for irregular triangular prisms changes only slightly because you require the height, and the three rectangular faces can all have different areas.

[Text Wrapping Break]***Insert SA image 5***

***insert SA equation 4***

Now we will explore a more difficult case. For cones and cylinders, the net doesn’t simply fold up the way we are used to. We actually need to curve one of the faces in order to close the 3D shape.

A cylinder is similar in many ways to a prism. It has a circular top and bottom, for which we already know how to calculate the area. It has one more side, which we will treat as a rectangle. Consider the following net.

***Insert SA image 6***

The height of the cylinder makes up one edge of the rectangle, while the other edge rolls up to make the circumference of the circle.[Text Wrapping Break][Text Wrapping Break]This means that the formula for the area can be given by[Text Wrapping Break][Text Wrapping Break]***Insert SA equation 5***

The last special case we will explore is the cone. This net has an irregular shape that we have not encountered before in our area calculations.

***Insert SA image 7***

We will still use a familiar strategy to calculate its area. In the same way that we approximated a circle’s area in the ***calculating area lesson (insert hyperlink)***, we will take this shape and divide it into smaller pieces, before interlocking them. This forms a shape that looks approximately like a parallelogram. The height of the parallelogram is the length of the sloped face of the cone from tip to base. The base of the parallelogram is half of the circumference of the circle.[Text Wrapping Break]This gives us the formula

***Insert SA equation 6***

Basic Rules:

Most common polyhedra are shown on the table below, along with their nets and the formulas for their surface areas

***insert SA table***.

Examples:

Why do we use this process:

Surface area calculations are used in a wide variety of applications. They are used in construction or building large structures seen in everyday life; they are used to find the size of distant objects like meteors; they are also used on tiny scales when deciding how quickly medicine will be released into your body.[Text Wrapping Break][Text Wrapping Break]Surface area is the amount of space on the outside of a 3D shape, which is usually the only part we see and interact with. Surface area can determine how much paint we need to coat the inside of a room; it can determine how expensive it is to package your favourite drink; it can even determine how brightly a wooden log will burn.

Beyond these uses, surface area calculations are still used in advanced topics of physics and mathematics. There are other, more precise, ways to calculate surface area, but they are usually taught at the second year university level or higher.

Problem Solving:

Some campfires are made so that the logs lean against each other. This allows every side of the log to burn, rather than having half of the log on the ground where it would get no oxygen.[Text Wrapping Break]Two logs are placed into a campfire so that they have their entire surface area exposed. Each log is 20cm long. The first log has a radius of 5cm. The second log has a radius of 8cm. How many times hotter than the first log does the second log burn?[Text Wrapping Break][Text Wrapping Break]***hint 1: Even the circular ends of the log can burn. The amount of heat released is determined by the entire surface area***[Text Wrapping Break][Text Wrapping Break]***hint 2: The answer is a ratio, which we will call heat factor. It can be expressed as a fraction[Text Wrapping Break][Text Wrapping Break]

*Practice Quiz:....*