what happens to the area of a shape when it doubles and why?

Question

I know what happens, but I can't explain why .

I mean a quadrilateral

Answer 1

If you define "doubling a shape" as increasing a shape's size until it's length / width have doubled, then the answer to your question is that the area will be increased by a factor of 4.

This is easy to think about with a square. Double the square and you have doubled the length and doubled the width of the square.

Before the area was length times width, and now it is doubled length times doubled width. Write this as an equation and you'll see

L * W = "Area of shape"

and

2 L * 2 W = "Area of doubled shape"

or

4 * ( L * W ) = "Area of doubled shape"

and since we said L * W = "Area of shape" then we substitute that into this equation and get

4 * "Area of Shape" = "Area of Double Shape"

Try this with other area formulas, it always works to show the Area of the doubled shape is equal to 4 times the area of the un-doubled shape

Answer 2

What shape? And what measure of the shape is doubling?

Take a circle of radius r. It's area is PI*r^2.

Now let the radius double to 2r. The area increases to PI*(2r)^2 or 4*PI*r^2. It has increased by a factor of 4 due to the squaring of the radius.