Video: Finding the Radius of a Circle given Its Circular Sector’s Area and Central Angle

The area of a circular sector is 561.3 cm² and the central angle is 27°. Find the radius of the circle giving the answer to the nearest centimeter.

02:33

Video Transcript

The area of a circular sector is 561.3 centimetres squared and the central angle is 27 degrees. Find the radius of the circle giving the answer to the nearest centimetre.

We’re told the area of the sector and the measure of the central angle. Let’s recall the formula that relates these two. The formula for area of a sector with radius 𝑟 and angle 𝜃 radians is a half 𝑟 squared 𝜃. So before we can use this formula to form an equation, we’ll need to convert our angle from degrees into radians. And to do this, we recall that two 𝜋 radians is equal to 360 degrees.

We can divide through by 360. And that tells us that one degree must be equivalent to two 𝜋 over 360 radians, which simplifies to 𝜋 over 180 radians. This means we can change from degrees into radians by multiplying by 𝜋 over 180. So 27 degrees is equal to 27 multiplied by 𝜋 over 180 radians, which is equal to three 𝜋 over 20 radians.

Now, we can substitute what we know into the formula for area of a sector. We don’t yet know the radius of the circle. So the area is a half multiplied by 𝑟 squared multiplied by three 𝜋 over 20. That is of course equal to 561.3. So we formed an equation for 𝑟.

We’re going to solve this equation by multiplying both sides by 40. Now we could’ve multiplied by two and then by 20 to get rid of these two denominators. But multiplying by 40 is slightly quicker. 561.3 multiplied by 40 is 22452. And then, our next step is to divide both sides by three 𝜋. And that tells us that 𝑟 squared is equal to 2382.23 and so on.

We won’t round this number just yet. Instead, we’re going to find the square root of both sides of our equation. So we’re going to square root this number in its unrounded form.

Now, when we find the square root of a number, we should remind ourselves that there are two solutions, a positive and a negative. In this case though, we can immediately disregard the negative solution as our radius is a length. And a length cannot be negative. So 𝑟 is equal to 48.808 and so on. Correct to the nearest centimetre, that’s 49.

So the radius of our circle is 49 centimetres.

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