the ratio of the length of the corresponding side of two regular polygons is 3:4. the are of the larger polygon is 320m^2. what is the are of the smaller polygon?
To find the area of the smaller polygon, we need to determine the ratio of their side lengths.
Let's assume that the length of the corresponding side of the larger polygon is 4x, where x is a common multiplier. This means that the length of the corresponding side of the smaller polygon is 3x (since the ratio is given as 3:4).
The area of a regular polygon is given by the formula:
Area = (1/4) * n * s^2 * cot(180/n)
Here, 'n' represents the number of sides and 's' is the length of each side.
Let's denote the number of sides of the larger polygon as 'n1' and the number of sides of the smaller polygon as 'n2'.
Given that the area of the larger polygon is 320m^2, we can set up the following equation:
320 = (1/4) * n1 * (4x)^2 * cot(180/n1)
The area of the smaller polygon is required, so we need to find its number of sides, 'n2'. Since the ratio of their side lengths is given as 3:4, we know that n1:n2 = 4:3.
Thus, n1/n2 = 4/3
We can substitute n1 = 4 and n2 = 3 in the equation above and solve for 'x'.
320 = (1/4) * 4 * (4x)^2 * cot(180/4)
320 = (1/4) * 4 * (16x^2) * cot(45)
320 = x^2 * cot(45)
320 = x^2
x = √(320) = 8
Now that we know the value of 'x', we can calculate the length of the corresponding side of the smaller polygon:
Length of smaller polygon = 3x
= 3 * 8
= 24
Using this length, we can calculate the area of the smaller polygon:
Area = (1/4) * n2 * (24)^2 * cot(180/n2)
Substituting n2 = 3:
Area = (1/4) * 3 * (24)^2 * cot(180/3)
Area = (1/4) * 3 * 576 * cot(60)
Area = 432 * cot(60)
Using the identity cot(60) = √3:
Area = 432 * √3
Therefore, the area of the smaller polygon is 432√3 square units.
To find the area of the smaller polygon, we need to determine the ratio of the areas of the two polygons. Since the ratio of the corresponding sides of the polygons is given as 3:4, the ratio of their areas will be the square of this ratio, which is (3:4)^2 = 9:16.
Given that the area of the larger polygon is 320m^2, we can set up the following equation using the area ratio:
Area of larger polygon / Area of smaller polygon = 9/16
Substituting the known values, we have:
320m^2 / Area of smaller polygon = 9/16
To solve for the area of the smaller polygon, we need to isolate it on one side of the equation.
First, cross-multiply the equation:
320m^2 * 16 = 9 * Area of smaller polygon
(320 * 16)m^2 = (9 * Area of smaller polygon)
Simplify both sides:
5120m^2 = 9 * Area of smaller polygon
Now, isolate the Area of the smaller polygon by dividing both sides by 9:
(5120m^2) / 9 = (9 * Area of smaller polygon) / 9
The 9s cancel out:
(5120m^2) / 9 = Area of smaller polygon
Calculating the result gives us:
Area of smaller polygon ≈ 568.89 m^2
Therefore, the area of the smaller polygon is approximately 568.89 square meters.