The area of a regular decagon is 50cm2. What is the area of a regular decagon with sides four times the sides of the smaller decagon?
I'm not understanding
24
So Steve is exactly right. Since our smaller decagon is 50 cm2, and our larger decagon's sides increase by 4 times, we multiply 50 * 4^2 or 50*16 which = 800cm2. Remember that the base equation for finding the area of any polygon is 1/2ap. So since the perimeter is being multiplied by 4 and we are calculating area and not just perimeter, the entire area increases by 4^2.
To find the area of the larger decagon, we need to determine the length of its sides.
Given that the area of the smaller decagon is 50 cm², let's calculate the length of its sides.
For a regular decagon, we know that the formula for its area (A) in terms of side length (s) is:
A = (5/4) * s² * cot(π/10)
We can rearrange the formula to solve for s:
s = sqrt((4 * A) / (5 * cot(π/10)))
Let's substitute the area into the formula:
s = sqrt((4 * 50) / (5 * cot(π/10)))
Now, we can find the length of the sides of the smaller decagon.
Once we have that value, we can multiply it by 4 to get the length of the sides of the larger decagon.
To find the area of the larger decagon, we can use the formula:
A = (5/4) * (4s)² * cot(π/10)
Simplifying the formula:
A = (5/4) * 16 * s² * cot(π/10)
A = 20 * s² * cot(π/10)
Now, we can substitute the value of s we obtained earlier:
A = 20 * (sqrt((4 * 50) / (5 * cot(π/10))))² * cot(π/10)
Calculating these values will give us the area of the larger decagon.
if linear dimension grows by a factor of n
area grows by n^2
volume grows by n^3