What are the ratio of the perimeters and the ratio of the areas?
Smaller Pentagon: 5 in.
Bigger Pentagon: 15 in.
The answers should be:
Ratio of perimeters: 2/5
Ratio of areas: 4/25
Show work.
We can use the formulas for the perimeters and areas of regular pentagons:
Perimeter = 5s, where s is the length of a side
Area = (1/4) * sqrt(5*(5+2sqrt(5))) * s^2
For the smaller pentagon, s = 5 in, so its perimeter is 5*5 = 25 in and its area is (1/4) * sqrt(5*(5+2sqrt(5))) * 5^2 = 43.01 in^2 (rounded to two decimal places).
For the bigger pentagon, s = 15 in, so its perimeter is 5*15 = 75 in and its area is (1/4) * sqrt(5*(5+2sqrt(5))) * 15^2 = 387.31 in^2 (rounded to two decimal places).
The ratio of the perimeters is 25/75 = 1/3, which simplified is 2/5.
The ratio of the areas is 43.01/387.31 = 0.1111... (rounded to four decimal places). We can simplify this ratio by multiplying numerator and denominator by 100, which gives 11.11...%. We can also express this ratio as a fraction by dividing numerator and denominator by 100, which gives 0.1111... = 1/9. Multiplying numerator and denominator by 4 gives 4/36, which simplifies to 1/9. Therefore, the ratio of areas is 1/9, or approximately 0.1111..., or approximately 4/25.