If two circles have the same radius, is having the combined area of the two circles the same as having twice the circumference? This has been bothering me for the past hour and I cant seem to solve it. It would seem likely to be the same.

nope!

Area of circle with radius r = πr^2
so combined area of two of those circles = 2πr^2

twice the circumference = 2(2πr) = 4πr

e.g. suppose the radius = 5 cm
area of 2 such circles = 2(π(25)) = 50π

twice the circumference = 2(2π(5)) = 20π
clearly different!

To determine whether the combined area of two circles with the same radius is equal to twice the circumference, let's break down the problem step by step:

1. First, let's compute the area of a circle. The formula to calculate the area of a circle is given by: A = πr², where A represents the area and r represents the radius.

2. Now, let's calculate the combined area of two circles with the same radius. If both circles have the same radius, we can denote it as r. Therefore, the area of the first circle would be A₁ = πr², and the area of the second circle would be A₂ = πr². Thus, the combined area is A_total = A₁ + A₂ = πr² + πr² = 2πr².

3. Next, let's determine the circumference of a circle. The formula to calculate the circumference of a circle is given by: C = 2πr, where C represents the circumference and r represents the radius.

4. Now, let's determine twice the circumference by multiplying the circumference by 2. Therefore, 2C = 2(2πr) = 4πr.

By comparing the combined area of the two circles (2πr²) to twice the circumference (4πr), we can conclude that they are not the same. The combined area of the two circles is only half the value of twice the circumference. Therefore, the two quantities are not equal.

So, it's important to remember that while both calculations involve circles, the measurements being compared are different. Area is a two-dimensional measure, while circumference is a one-dimensional measure.