The scale factor of two regular octagons is 5 : 2. Find the ratio of their perimeters and the ratio of their areas.
Let the side length of the smaller octagon be 2x, then the side length of the larger octagon is 5x.
The perimeter of the smaller octagon is 8(2x) = 16x.
The perimeter of the larger octagon is 8(5x) = 40x.
Therefore, the ratio of their perimeters is:
(Perimeter of larger octagon) : (Perimeter of smaller octagon) = 40x : 16x = 5 : 2.
The area of the smaller octagon is (1 + sqrt(2)) * (2x)^2 ≈ 9.66x^2.
The area of the larger octagon is (1 + sqrt(2)) * (5x)^2 ≈ 60.66x^2.
Therefore, the ratio of their areas is:
(Area of larger octagon) : (Area of smaller octagon) = 60.66x^2 : 9.66x^2 = 6.27 : 1.
the ratio of their perimeter, which is a linear measurement = 5 : 2
the ratio of their areas = 5^2 : 2^2 = 25 : 4
Sorry about that, you are correct. The ratio of their areas should be (5x)^2 : (2x)^2 = 25 : 4. Thank you for pointing that out!
To find the ratio of the perimeters, we can simply divide the lengths of the sides of the larger octagon by the lengths of the sides of the smaller octagon.
The ratio of the perimeters is 5:2.
To find the ratio of the areas, we need to consider that the area of a regular polygon is proportional to the square of its side length. This means that if we multiply the side length by a certain factor, the area will increase by the square of that factor.
In this case, the scale factor is 5:2. Therefore, the ratio of the areas will be the square of this scale factor.
The ratio of the areas is (5:2)^2, which simplifies to 25:4.
So, the ratio of the perimeters is 5:2, and the ratio of the areas is 25:4.