Obj. 57 Effects of Changing Dimensions

The student is able to (I can):

• Predict and calculate how changing one or more dimensions of a shape affects the area of that shape.

What happens to the perimeter and area of a figure when we change dimensions proportionally?

Example: What is the new perimeter and area of the rectangle if the height is doubled?

3

7

6

7

P = 2(3)+2(7)= 20

P = 2(6)+2(7)= 26

A = (3)(7) = 21 A = (6)(7) = 42

Example (cont.)

Now, what would be the effect if the rectangle’s base were doubled?

3

7

3

14

P = 2(3)+2(7)= 20

P = 2(3)+2(14)= 34

A = (3)(7) = 21 A = (3)(14) = 42

Notice that in both cases, doubling one dimension doubles the area. It doesn’t matter whether it is the base or the height.

What happens if we double both the base and the height?

3

7

6

14

P = 20; A = 21

P = 2(6)+2(14)= 40

A = (6)(14)= 84

This time, the perimeter doubled, but the area changed by a factor of 4. Why the difference?

Let’s break down the area on the last example:

2(3)

2(7)

A = (2)(3)(2)(7)= (2)(2)(3)(7)= (4)(21)= 84

Both sides are multiplied by 2, so the perimeter is doubled and the area is multiplied by 22.

P = 2[2(3)]+2[2(7)]= 2[2(3)+2(7)]= 2(20)= 40

Now, let’s look at a circle. What happens to the circumference and area if we triple the radius?

•

•

2

6

C = 2π(2)=4π

A=π(22) = 4π

C = 2π(6) = 12π

A = π(62) = 36π

The circumference increased by 3, and the area increased by 32 or 9.

We can use these ideas to work problems going the other way:

1. If a square’s area is quadrupled (x4), what happens to the perimeter?

Since the area is multiplied by 4, that means that each side was multiplied by the or 2. Thus, the perimeter is doubled.

2. If a circle’s circumference is reduced by half, what happens to the area?

If the circumference is multiplied by ½, then so is the radius. Therefore, the area would be multiplied by or ¼.

4

( )2

12