can the circumference of a circle be rational

Area and circumference of a circle both involve the constant π and the radius of the circle, but their units vary. So, how are the circumference and area of a circle related? The circumference of a circle has linear units, while area of a circle has square units. The value of the area is equal to half the circumference multiplied by the radius of the circle. The circumference also tells us how the area of a circle changes as the radius increases or decreases. Of course, we can also find specific R values (or ranges of values) to tell us when the circumference, area, and diameter satisfy certain relationships. In this article, we’ll answer 8 common questions about circumference, area, radius, and diameter of a circle. Let’s get going. Questions About Circles (Area, Circumference, Diameter, & Radius) We can use formulas for the area and circumference of a circle to relate these two concepts and also to relate them to radius and diameter of the circle. Let’s start off with a common question regarding area and circumference of a circle. Can The Area Of A Circle Be The Same As The Circumference? The area of a circle can have the same value as the circumference in two special cases. However, the units will be different (area is in square units, while circumference is in linear units). The area of a circle can be the same as the circumference when R = 0 or R = 2. The proof is below. Remember that the area (A) of a circle with radius R is given by the equation A = πR2 The circumference (C) of a circle with radius R is given by the equation C = 2πR To find out when the area has the same value as the circumference, we must set the area equal to the circumference and solve for R: A = C [set area equal to circumference]πR2 = 2πR [use formulas for area and circumference of a circle]R2 = 2R [divide both sides by π]R2 – 2R = 0R(R – 2) = 0 So the solutions are R = 2 and R = 0. The first solution, R = 2, gives us a circle with an area of 4π square units and a circumference of 4π linear units. The solution R = 0 gives us a circle of radius zero, with zero area and zero circumference (more on this later). For any other values of R, the area and circumference will be different: For 0 2, the area will be greater than the circumference. For example, if R = 3, the area is 9π square units, while the circumference is 6π square units. Remember that it is not possible to have a negative radius for a circle. The table below summarizes the relationship between area and circumference of a circle. Value Of RArea & CircumferenceR = 0A = C(zero)0 2A > CThis table summarizes how area and circumference of a circle are related, depending on the value of the radius. Can The Area Of A Circle Be Smaller Than The Diameter? The area of a circle can have a smaller value than the diameter of the circle in some special cases. However, the units will be different (area is in square units, while diameter is in linear units). The value of the area of a circle can be smaller than the value of the diameter in certain cases. Read below to fine out when. Remember that the area (A) of a circle with radius R is given by the equation A = πR2 The diameter (D) of a circle with radius R is given by the equation D = 2R To find out when the area is smaller than the diameter, we must set up an inequality and solve for R: A < DπR2 < 2R [use formulas for area and diameter of a circle]πR2 - 2R < 0 [subtract 2R from both sides]R(πR – 2) < 0 [factor out R from the left side] The only way this can happen is when one factor is positive and the other is negative. Since R is the radius of a circle, it cannot be negative. So, the other factor is negative: πR – 2 < 0πR < 2R < 2/π So, the area of a circle has a value less than the value of the diameter whenever the radius is less than 2/π (the value is approximately 0.6366). On the other hand, the area of a circle has a value greater than the value of the diameter whenever the radius is greater than 2/π. The two values are the same when R = 2/π. The table below summarizes the relationship between area and diameter of a circle. Value Of RArea & DiameterR 2/πA>DThis table summarizes how area & diameter of a circle are related, depending on the value of the radius. Can The Area Of A Circle Be A Whole Number? The area of circle can be a whole number in certain cases. We will need to solve for the value of the radius R to find out when this will happen. Let’s say that a circle has an area that is a nonnegative whole number W. Then we have: A = WπR2 = W [use the formula for area of a circle]R2 = W / π [divide by π on both sides]R = (W / π)1/2 [take the square root on both sides – just the positive one, since radius cannot be negative] So, given a whole number W, we can find the radius of the circle that will give us an area of W. For example, let’s say we want a circle with an area of 28. Then W = 28 in our formula, and we get: R = (W / π)1/2R = (28 / π)1/2R ~ (8.9127)1/2R ~ 2.9854 Can The Circumference Of A Circle Be A Whole Number? The circumference of circle can be a whole number in certain cases. We will need to solve for the value of the radius R to find out when this will happen. Let’s say that a circle has a circumference that is a nonnegative whole number W. Then we have: C = W2πR = W [use the formula for circumference of a circle]R = W / 2π [divide by 2π on both sides] So, given a whole number W, we can find the radius of the circle that will give us a circumference of W. For example, let’s say we want a circle with a circumference of 25. Then W = 25 in our formula, and we get: R = W / 2πR = 25 / 2πR ~ 25 / 6.2832R ~ 3.9789 Can The Circumference Of A Circle Be Rational? The circumference of circle can be rational in certain cases. We will need to solve for the value of the radius R to find out when this will happen. The calculation is the same as above, and to get a rational circumference of W, the formula is R = W / 2π Can The Radius Of A Circle Be Zero? The radius of a circle can be zero. However, this will just be a point (the circle has diameter, circumference, and area of zero). In other words, a circle of radius zero centered at the point (a, b) is really just the point (a, b). How Are The Area & Circumference Of A Circle Related? We can connect the values of area and circumference of a circle with this equation: A = CR/2 Just remember that the units vary for these two: area is in square units, while circumference is in linear units. The circumference of a circle can tell us how fast the area of the circle changes as we increase or decrease the radius. It helps to know some calculus to see why. Define the area function of a circle with radius R as: A(R) = πR2 Then the first derivative of the area function (with respect to the variable R) is: A’(R) = 2πR This is the same as the circumference of the circle. So, A’(R) gives us the circumference of the circle. Since the derivative of a function tells us how the function changes, then the circumference of a circle tells us how the area changes as we increase or decrease R. The circumference tells us how the area of a circle changes as the radius increases or decreases. To put it another way: if we add up all of the circumferences of the circles with radius 0 to R, we will get the area of the circle with radius R. In fact, we see something similar for the volume and surface area of a sphere. If the volume of a sphere with radius R is defined by the volume function: V(R) = (4πR3) / 3 Then the first derivative of the volume function (with respect to the variable R) is: V’(R) = 4πR2 This is the same as the surface area of the sphere. So, V’(R) gives us the surface area of the sphere. Since the derivative of a function tells us how the function changes, then the surface area of a sphere tells us how the volume changes as we increase or decrease R. To put it another way: if we add up all of the surface areas of the spheres with radius 0 to R, we will get the volume of the sphere with radius R. How To Use The Circumference To Find The Area Of A Circle There are a few steps involved to use the circumference to find the area of a circle. Since we are given the circumference C = 2πR, we need to divide C by 2π to find R: R = C/2π Then we can calculate the area with the usual formula, A = πR2. Putting it all together, we get: A = πR2 [formula for area of a circle]A = π(C/2π)2 [formula for radius of a circle, given the circumference]A = π(C2/4π2) [distribute the power of 2 into parentheses]A = C2 / 4π [cancel out the π] For example, let’s say we have a circle with circumference 8π. Then C = 8π in our equation, and we have: A = C2 / 4πA = (8π)2 / 4πA = 64π2 / 4πA = 16π This calculation works out both ways (for area and circumference) if we use a radius of R = 4. Conclusion Now you know the answers to some common questions about the radius, diameter, circumference, and area of circles. You can learn more about what area is used for in real life with this article. To learn about applications of circles, check out my article on how circles are used. You can learn how to find the perimeter and area of circular sectors (parts of a circle) in my article here. For a reminder on the equation for a circle, check out this resource from Lamar University. To learn about where it all started, check out this resource on the definition of a circle from Euclid's elements from Clark University. You might also want to check out my article on how to find the center and radius of a circle in various situations. You can find out more about squares and circles here. I hope you found this article helpful. If so, please share it with someone who can use the information. Don’t forget to subscribe to my YouTube channel & get updates on new math videos! ~Jonathon