Question Video: Finding the Length of an Arc given Its Central Angle’s Measure in Degrees and Its Circle’s Radius Mathematics

Video Transcript

Rectangle 𝐵𝐶𝑀𝐷 is drawn inside a quarter-circle, where 𝐵𝐷 equals nine centimeters and 𝐵𝐶 equals 12 centimeters. Find the length of arc 𝐴𝐵𝐸, giving the answer in terms of 𝜋.

So, we’ve been asked to find the length of the arc 𝐴𝐵𝐸. We know that we have quarter of a circle, which means that this arc length will be quarter of the circle’s full circumference. So, it’s equal to one-quarter multiplied by 𝜋𝑑. We can also think of that value of one-quarter as 90 over 360 if we wish because this quarter-circle is also a sector of a circle with a central angle of 90 degrees.

In order to answer this problem then, we need to know the diameter or perhaps the radius of this circle. Instead, we’ve been given some other measurements in the question. 𝐵𝐷 is nine centimeters and 𝐵𝐶 is 12 centimeters, but neither of these are the radius or diameter. We can identify three radii of this circle in the diagram: the lines 𝑀𝐴, 𝑀𝐵, and 𝑀𝐸. Now, as 𝐵𝐶𝑀𝐷 is a rectangle, we know that the length of the line segment 𝐶𝑀 will be the same as the length of the line segment 𝐵𝐷. So, it’s also nine centimeters.

If we look carefully at the figure, we now see that we have a right triangle, triangle 𝐵𝐶𝑀, in which we know the lengths of two of the sides; they’re 12 centimeters and nine centimeters. And the third side is the radius of the circle, which we wish to calculate. If we know two sides in a right triangle and we want to calculate the third, we can do this using the Pythagorean theorem, which tells us that the sum of the squares of the two shorter sides, 𝑎 and 𝑏, is equal to the square of the hypotenuse. So, this is often written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared.

In our triangle, the hypotenuse is the radius of the circle because it’s the side directly opposite the right angle. So, we have 𝑎 and 𝑏 as nine and 12 centimetres and 𝑐 as 𝑟 centimeters. We can, therefore, form an equation, 𝑟 squared equals 12 squared plus nine squared. And we can solve this equation to find the radius of the circle. 12 squared plus nine squared, that’s 144 plus 81, is 225. 𝑟 is, therefore, equal to the square root of 225, which is 15.

Now, you may actually have already realized this because nine, 12, 15 is an example of a Pythagorean triple. In fact, it’s an enlargement of the three, four, five Pythagorean triple with a scale factor of three. And the three, four, five triple is probably the most easily recognizable. In any case, we now know the radius of the circle; it’s 15 centimeters. We can, therefore, calculate the diameter of the circle by doubling this value. The diameter is 30 centimeters.

So, all that remains then is to substitute this diameter into our formula for the arc length, which remember was one-quarter multiplied by 𝜋𝑑. We have one-quarter multiplied by 30𝜋. And we can then simplify by canceling a factor of two from the numerator and denominator. We’re left with 15 over two 𝜋, which we can write using decimals, if we wish, as 7.5𝜋.

The question asks us to leave our answer in terms of 𝜋. So, we can include the units of centimeters and we have our answer to the problem. The length of arc 𝐴𝐵𝐸 is 7.5𝜋 centimeters. Notice that at no point in this question did we need a calculator. We left our answer in terms of 𝜋. And in our work with the Pythagorean theorem, we were working with a Pythagorean triple.