The areas of two similar octagons are 112 in.² and 63 in.². What is the ratio (larger to smaller) of their perimeters?
The ratio of the areas of two similar figures is the square of the ratio of their corresponding side lengths. Thus, we can find the ratio of the lengths of corresponding sides of the two octagons by taking the square root of the ratio of their areas:
sqrt(112/63) = 1.5
This means that the corresponding sides of the larger octagon are 1.5 times longer than those of the smaller octagon. Therefore, the ratio of their perimeters is also 1.5:
Perimeter of larger octagon / Perimeter of smaller octagon = 1.5
Answer: The ratio of the larger octagon's perimeter to the smaller octagon's perimeter is 1.5.
To find the ratio of perimeters of two similar octagons, we need to first find the ratio of their corresponding sides.
The ratio of the areas of two similar shapes is equal to the square of the ratio of their corresponding sides. Therefore, we have:
(area of larger octagon) / (area of smaller octagon) = (side length of larger octagon)² / (side length of smaller octagon)²
Let's denote the side length of the larger octagon as 'a' and the side length of the smaller octagon as 'b'. We can now set up the equation using the given areas:
112 in.² / 63 in.² = a² / b²
To find the ratio of the perimeters, we need to take the square root of this equation:
√(112 in.² / 63 in.²) = √(a² / b²)
Simplifying further:
√(1.7778) = a / b
Now, we can find the ratio of the perimeters by multiplying both sides of the equation by 8 (as there are 8 sides in an octagon):
8 * (√(1.7778)) = 8 * (a / b)
This gives us the ratio of the perimeters as:
8√(1.7778) : 8
Simplifying this expression:
8 * 1.3333 : 8
10.6664 : 8
Therefore, the ratio of the perimeters (larger to smaller) is approximately 10.6664 : 8, or 1.3333 : 1.
To find the ratio of the perimeters of two similar octagons, we need to know the relationship between their areas. Let's start by finding the ratio of the areas of the two octagons.
Given:
Area of the larger octagon = 112 in.²
Area of the smaller octagon = 63 in.²
To determine the relationship between the areas, we can use the formula for the ratio of areas:
(Ratio of areas)² = (Area of larger shape) / (Area of smaller shape)
Substituting the given values:
(Ratio of areas)² = 112 in.² / 63 in.²
Simplifying:
(Ratio of areas)² = 1.7778
Taking the square root of both sides:
Ratio of areas = √1.7778 ≈ 1.3333
So, the ratio of the areas between the larger and smaller octagon is approximately 1.3333.
Since the octagons are similar, the ratio of corresponding sides should also be 1.3333. As an octagon has eight sides, the ratio of their perimeters will be:
(Ratio of perimeters) = (Ratio of sides) * 8
Substituting the value we found:
(Ratio of perimeters) = 1.3333 * 8
Calculating:
(Ratio of perimeters) ≈ 10.6666
Therefore, the ratio of the perimeters (larger to smaller) of the two similar octagons is approximately 10.6666.