a The perimeter increases by a factor. Does the area increase by a factor too?
B
b Say that the length of the side of the smaller square is l units. What area does it get? How long is the length of the side of the larger square?
A
a A_s=4 square units and A_l=16 square units. The area is quadrupled.
B
b Yes, see solution.
Practice makes perfect
a Let's first draw the two squares in a diagram.
The area of a square we find using the formula
A=s^2where s is the side of the square. Let's calculate the length of the side of the smaller square, s_s. Two of its sides coincide with the coordinate axes. Therefore we can calculate the length of the side using the Ruler Postulate.
The length of the side is 2 units. The square's area is therefore
A_s=s_s^2=2^2=4 square units.
We will now calculate the length of the side of the larger square, s_l. Here two of the square's sides also coincide with both the x- and the y-axes and we can therefore calculate the side's length using the Ruler Postulate.
The length of the side of the larger square is 4 units. This square's area is therefore
A_l=s_l^2=4^2=16 square units.
We find that the larger square's area is 16 square units. That is 4 times larger than the smaller square's area, which is 4 square units. When we went from the smaller to the larger square we doubled the length of the sides and therefore we also doubled the perimeter. When we doubled the length of the perimeter we quadrupled the area.
b Let's now look at two other squares. We have one smaller square, which side is l units long, and one larger, which side is twice as long, 2l units.
The length of the side is doubled when we go from the smaller to the larger square. Let's see what the area of each square is. Let's begin by determining the smaller square's area, A_s.
