Question

The area of a regular octagon is 35 cm 2 what is the area of a regular octagon with sides three times as large

Answer 1

The area of a polygon = n*side^2 / (4 * tan(180/n))
where "n" is the number of sides

Let's calculate area for side = 2
area = 8*2^2 / (4 * tan(180/8))
area = 32 / (4 * tan(22.5))
area = 32 / 4*0.41421
area = 19.3138746047

Now, let's calculate area for side = 6
area = 8*6^2 / (4 * tan(180/8))
area = 288 / 1.65684
area = 173.824871442

173.824871442 / 19.3138746047 =  9

So, the area would be 9 times larger.

ALSO, looking at the formula
n*side^2 / (4 * tan(180/n))
we can see that the side length appears just once in the formula and we are to square it in the calculation.  So, if we increase the side length is increased by 3, then the area increases by 3^2 or 9.