Topic 4 Scale Factors and Areas of 3-D Shapes Unit 8 Topic 4

Explore Area is a term used to describe a 2D shape. Area is the number of square units inside a 2D shape. Area is measured in units 2. Surface area and volume are terms used to describe a 3D object. Surface area is the number of square units on the surface of a 3D object. Surface area is measured in units 2.

Explore 1. Start by building each 3D object using cube links. To find the surface area, count the number of squares on the outside of the object. Complete the table below. Try this on your own first!!!!

Explore 1. Complete the table below

Explore 2. The original cube has a surface area of 6 units 2. How could you calculate the surface area of the other cubes using the surface area of the original cube, 6 units 2, and the linear scale factors? You should notice that the surface area scale factor is equal to the linear scale factor squared.

Information The relationship between the surface area of a new 3D object and the surface area of the original 3D object can be expressed using the following equation. Surface Area Scale Equation new surface area = old surface area  k 2 where k is the linear scale factor. We can rearrange the equation to isolate the area scale factor, k 2. Area Scale Factor (ASF)

Example 1 Linear Scale Factor and Surface Area a) What is the surface area of a rectangular prism with the dimensions 5 cm by 4 cm by 2 cm? (SA = 2hl + 2lw + 2hw) b) If the rectangular prism is enlarged by a linear scale factor of 3, what are the new dimensions of the prism? Method 1: Using Surface Area Calculation Find the surface area using the dimensions of the new prism. Try this on your own first!!!!

Example 1a: Solution Linear Scale Factor and Surface Area a) What is the surface area of a rectangular prism with the dimensions 5 cm by 4 cm by 2 cm? (SA = 2hl + 2lw + 2hw)

Example 1b: Solution Linear Scale Factor and Surface Area If the rectangular prism is enlarged by a linear scale factor of 3, what are the new dimensions of the prism?

Example 1 Linear Scale Factor and Surface Area Method 1: Using Surface Area Calculation Find the surface area using the dimensions of the new prism.

Example 1 Linear Scale Factor and Surface Area Method 2: Using the Surface Area Scalar Equation Substitute into the surface area scalar equation

Example 2 Surface Area from a Scale Factor a) A baseball has a surface area of cm 2. The diameter of a soccer ball is 2 times the diameter of the baseball. What is the surface area of the soccer ball, to the nearest tenth of a square centimetre? b) The greenhouses at the Muttart Conservatory are each in the shape of a square pyramid. An architect has a model of the largest greenhouses. The model has a glass surface area of 1.15 m 2. A linear scale factor of 40 needs to be applied to the dimensions of the model to produce the largest greenhouse. What is the glass surface area of the largest greenhouse at the Muttart Conservatory? Try this on your own first!!!!

Example 2a: Solution A baseball has a surface area of cm 2. The diameter of a soccer ball is 2 times the diameter of the baseball. What is the surface area of the soccer ball, to the nearest tenth of a square centimetre?

Example 2b: Solution The greenhouses at the Muttart Conservatory are each in the shape of a square pyramid. An architect has a model of the largest greenhouses. The model has a glass surface area of 1.15 m 2. A linear scale factor of 40 needs to be applied to the dimensions of the model to produce the largest greenhouse. What is the glass surface area of the largest greenhouse at the Muttart Conservatory?

Example 3 Linear Scale Factor from Surface Area a) The surface area of a large can of soup is cm 2. The surface area of a small soup can, with the same shape, is cm 2. What linear scale factor needs to be applied to the dimensions of the large can to produce the small can? Answer to the nearest hundredth. b) The amount of canvas needed to make the sides of a small First Nations tipi is 47.1 m 2. The amount of canvas needed to make the sides of a large tipi, with the same shape, is m 2. What linear scale factor needs to be applied to the dimensions of the small tipi to produce the large tipi? Answer to the nearest tenth. Try this on your own first!!!!

Example 3a: Solution What linear scale factor needs to be applied to the dimensions of the large can to produce the small can? Answer to the nearest hundredth.

Example 3b: Solution What linear scale factor needs to be applied to the dimensions of the small tipi to produce the large tipi? Answer to the nearest tenth.

Need to Know: The surface area scale factor, ASF, of a 3D object is The surface area of the original or old shape is multiplied by the surface area scale factor to produce the surface area of the new shape. The surface area scale equation is new surface area = old surface area  k 2, where k is the linear scale factor. You’re ready! Try the homework from this section.