The radius of a circle is 40 centimeters and the perimeter of a sector is 106 centimeters. Find the central angle in degrees giving the answer to the nearest second and in radians giving the answer to one decimal place.
Let’s begin by recalling what a sector is. It’s part of the circle bounded by an arc and two radii, which connect the endpoints of that arc to the center of the circle. So it looks something like this. We have the arc and then the two radii. We’re told that the radius of this circle is 40 centimeters and also that the perimeter of this sector, that’s the distance all the way around its edge, is 106 centimeters. We don’t know the central angle of this sector. That’s the angle between the two radii. And that’s what we need to calculate in both degrees and in radians.
Let’s consider this problem in degrees first. Now the perimeter of this sector, remember, is the distance all the way around its edge. So if we start at the center of the circle, that is the length of the radius of the circle, the arc length which we often denote as 𝑠, and then the length of the radius of the circle again. So a formula we can use for calculating the perimeter of this sector is twice the radius plus 𝑠, the arc length. The arc length can be found using the formula two 𝜋𝑟𝜃 over 360. This comes from taking the full circumference of the circle, two 𝜋𝑟, and multiplying it by the fraction 𝜃 over 360, as the arc is only part of the full circumference.
This formula can be factored by two 𝑟 to give two 𝑟 multiplied by one plus 𝜋𝜃 over 360. We can use this formula to form an equation because we know the perimeter of this sector and we also know its radius. So substituting 106 for the perimeter and 40 for the radius 𝑟, we have 106 equals two multiplied by 40 all multiplied by one plus 𝜋𝜃 over 360. And we can now solve this equation to find the value of 𝜃. Two multiplied by 40 is of course 80. And then we can divide both sides of the equation by this value to give 106 over 80 is equal to one plus 𝜋𝜃 over 360.
We can then subtract one or 80 over 80 from each side to give 106 over 80 minus 80 over 80 is equal to 𝜋𝜃 over 360. 106 over 80 minus 80 over 80 is 26 over 80. And we can also simplify on the left-hand side by canceling a factor of two in the numerator and denominator. So we have 13 over 40 is equal to 𝜋𝜃 over 360. To find the value of 𝜃, we need to divide both sides of the equation by 𝜋 over 360, which is equivalent to multiplying both sides by its reciprocal of 360 over 𝜋. So we have 𝜃 equals 13 over 40 multiplied by 360 over 𝜋. We could evaluate this at this point or we can first cross cancel a factor of 40 to give 13 over one multiplied by nine over 𝜋. That’s 117 over 𝜋. And then evaluating on a calculator, we have 37.2422 continuing.
Now, this is a value in degrees. And in the question, we are asked to give our answer in degrees to the nearest second. So we have this value of 37 degrees and then a decimal of 0.2422 continuing that we need to convert to minutes and seconds. We can recall that one degree is equivalent to 60 minutes. So multiplying this decimal by 60 gives 14.535 continuing. We therefore have 14 entire minutes and a decimal of 0.535 continuing which we need to convert to seconds. Well, one minute is equivalent to 60 seconds. So multiplying this decimal by 60 gives 32.124 continuing, which to the nearest integer is 32. So we have 37 degrees, 14 minutes, and 32 seconds. And so this is our answer for the central angle 𝜃 in degrees given to the nearest second.
The question also asks us to give the answer in radians to one decimal place. Now we could convert this value from degrees, minutes, and seconds to radians by recalling the relationship between the two measures. But let’s also demonstrate how we would approach this problem if we’d used radians right at the start. Well, the perimeter of the sector can still be found as twice the radius plus the arc length 𝑠. But we have a different formula for arc length when working in radians. For a sector with a radius of 𝑟 units and a central angle of 𝜃 measured in radians, the arc length is simply equal to 𝑟𝜃. The formula for the perimeter then is two 𝑟 plus 𝑟𝜃, which can be factored as 𝑟 multiplied by two plus 𝜃.
We can therefore form a much simpler equation working in radians. Substituting 𝑟 for the radius and 106 for the perimeter, we have 40 multiplied by two plus 𝜃 equals 106. Dividing each side by 40, we have two plus 𝜃 is equal to 106 over 40. So 𝜃 is equal to 106 over 40 minus two. That’s 106 over 40 minus 80 over 40, which simplifies to 26 over 40 or 13 over 20. This is exactly equal to 0.65. But as the question asks us to give our answer to one decimal place, we’ll round this up to 0.7. So we have our answers in both degrees and radians. But let’s just check that these two values are indeed equivalent.
We know that 𝜋 radians is equivalent to 180 degrees. And so one degree is equivalent to 𝜋 over 180 radians. If we take our answer in degrees at this stage here, so 117 over 𝜋, and multiply it by 𝜋 over 180, this will convert it from degrees to radians. Of course, the factors of 𝜋 will cancel one another out, giving 117 over 180. This simplifies to 13 over 20, which is equal to our exact value of 𝜃 in radians.
Our answer to the problem then is that the central angle in degrees to the nearest second is 37 degrees, 14 minutes, and 32 seconds and in radians to one decimal place is 0.7 radians.