Question Video: Determining the Radius of a Circle and the Area of Each Sector from the Given Perimeter of the Sector and the Arc Length Mathematics

The circle of center 𝑀 and radius π‘Ÿ is divided into four sectors, each of perimeter 75 and arc length 33. Determine the radius π‘Ÿ and the area of each sector 𝐴_(sector). Use 3.14 as an approximation value for πœ‹.

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Video Transcript

The circle of center 𝑀 and radius π‘Ÿ is divided into four sectors, each of perimeter 75 and arc length 33. Determine the radius π‘Ÿ and the area of each sector 𝐴 sector. Use 3.14 as an approximation value for πœ‹.

So, we’ve been given two pieces of information about these sectors. Firstly that the perimeter is 75, and secondly that each of their arc lengths is 33. An arc is a portion of the circumference of a circle, so this means that the curved part of the perimeter of each sector is 33. The total perimeter is found by adding the arc length to two of the radii. We know that these lines are radii because the two cords or, in fact, the diameters cross in the center of the circle. We can therefore form an equation by substituting 75 for the perimeter and 33 for the arc length, and it gives 75 equals 33 plus two π‘Ÿ.

To solve this equation for π‘Ÿ, we first subtract 33 from each side, giving 42 equals two π‘Ÿ, and then divide by two giving 21 equals π‘Ÿ. So, the radius of the circle is 21. There are no units for this value, as there were no units given for the measurements in the original question. Next, we need to find the area of each sector, and first we recall that the area of a circle is given by the formula πœ‹π‘Ÿ squared. As there are four equal sectors, each a quarter of the circle, the area of each sector then is πœ‹π‘Ÿ squared over four. We’ve found that the radius of the circle is 21, so substituting this gives that the area of each sector is equal to πœ‹ multiplied by 21 squared over four. Now, 21 squared is equal to 441, so the area is equal to 441πœ‹ over four. Now, the fraction 441 over four is equal to 110 and a quarter as a mixed number, so the area simplifies to 110.25πœ‹.

Now, the question tells us to use 3.14 as an approximation for πœ‹, which suggest that we don’t have access to a calculator for this question. As otherwise, we just use the πœ‹ button on our calculator. So, we need to work out what 110.25 multiplied by 3.14 is. To do so, we can use the grid multiplication method and break 110.25 down into the sum of 100, 10, and 0.25. 100 multiplied by 3.14 is 314. 10 multiplied by 3.14 is 31.4. 0.25 multiplied by 3.14 is the same as 3.14 divided by four, as 0.25 is the same as a quarter.

We can work this out using a short division or bus stop method. First, we see how many times four goes into three, which is zero, so we carry the three. Four goes into 31 seven times, as four multiplied by seven is 28, with a remainder of three, so we put a seven and carry the three. Four goes into 34 eight times, as four times eight is 32, but there’s a remainder of two. So, we need to put an extra zero after the decimal point and carry the two. Four goes into 20 five times exactly. So, we know that 3.14 divided by four, which is the same as 3.14 multiplied by a quarter or 0.25, is 0.785.

Finally, we need to add these values using a column addition method. We can add extra zeros after the decimal points if we want, but we must make sure that we line the decimal points up. Using our column addition method, the sum of these two values is 346.185. So, our answer to the question then is that the radius is 21 and the area of each sector, using 3.14 as an approximation for πœ‹, is 346.185. Again, there are no units for either of these values, as no units were specified in the original question.

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