# Question Video: Finding the Perimeters and Areas of Shapes Involving Sectors Mathematics

In the given figure, πβ π΅π΄πΆ = 30Β°, π΄π΅ = 7, and π΄π· = 4. Work out the perimeter of π΅π·πΈπΆ. Give your answer to 2 decimal places. Work out the area of π΅π·πΈπΆ. Give your answer to 2 decimal places.

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### Video Transcript

In the given figure, the measure of angle π΅π΄πΆ is 30 degrees, π΄π΅ equals seven, and π΄π· equals four. Work out the perimeter of π΅π·πΈπΆ. Give your answer to two decimal places. Secondly, work out the area of π΅π·πΈπΆ. Give your answer to two decimal places.

In each part of this question, weβre asked to consider π΅π·πΈπΆ. Thatβs this part of the figure highlighted in orange. Letβs first add the information given in the question onto our diagram. The measure of angle π΅π΄πΆ is 30 degrees, the length of π΄π΅ is seven units, and the length of π΄π· is four units. Letβs now consider how to find the perimeter of π΅π·πΈπΆ.

And if we start in one corner and trace a way round, we see that it is composed of two straight sections, π΅π· and πΈπΆ, and two arcs, π΅πΆ and π·πΈ. The lengths of the two straight sections can be worked out relatively easily. Theyβre the difference between the longer length of seven and the shorter length of four. So, π΅π· and πΈπΆ are, each, three units.

To calculate the lengths of π·πΈ and πΆπ΅, we need to recall the general formula for calculating arc length. Itβs π over 360 multiplied by ππ, where π is the central angle of the sector and π is the circleβs diameter. What weβre doing when we apply this formula is calculating the full circumference of the circle, ππ, and then multiplying it by the fraction of the circle that this arc represents. Thatβs π out of 360. For our two arcs, the central angle π is 30 degrees. But what about the diameters?

Well, the arc π·πΈ is part of a smaller circle, which has a radius of four units. Thatβs the length π΄π·. And so, the diameter will be twice this; it will be eight units. The arc πΆπ΅ is part of a larger circle with a radius of seven units, so it will have a diameter of 14 units. The length of the arc π·πΈ then will be 30 over 360 multiplied by eight π. 30 over 360 is in fact equivalent to one twelfth. And then, we can simplify again by dividing both the numerator and denominator by four to give two π over three.

For the arc πΆπ΅, we have 30 over 360 multiplied by 14π. 30 over 360, once again, simplifies to one twelfth. And then canceling a factor of two in the numerator and denominator, we have that the length of the arc πΆπ΅ is seven π over six. So, our calculation for the perimeter of π΅π·πΈπΆ is three plus two π over three plus three plus seven π over six.

Weβre asked to give our answer to two decimal places. So, we can assume that we have a calculator to help with this question. Evaluating on a calculator then and it gives 11.7595. Rounding to two decimal places and we have our answer to the first part of the problem. The perimeter is 11.76 units.

Iβm now going to delete some of this working out. So, you may need to pause the video if you want to jot anything down. In the second part of the question, weβre asked to work out the area of π΅π·πΈπΆ. We can do this by calculating the difference in the area of two circular sectors: the sector π΄π΅πΆ and the sector π΄π·πΈ. These each of the same central angle of 30 degrees. And we know the radius of the largest sector is seven units and the radius of the smaller sector is four units.

We then recall the formula for finding the area of a sector. Itβs π over 360 multiplied by ππ squared. What weβre doing here is finding the area of a full circle and then multiplying it by the fraction of the circle that our sector represents. So substituting 30 for the central angle of each sector and seven for the larger radius and four for the smaller radius. We have 30 over 360 multiplied by π multiplied by seven squared minus 30 over 360 multiplied by π multiplied by four squared.

In each case, 30 over 360 can be simplified to one twelfth. And we then have 49π over 12 minus 16π over 12. We could give an exact answer of 33π over 12. But the question asked us to give our answer to two decimal places. So evaluated on a calculator, we have 8.6393. Rounding appropriately then and we have our answer of 8.64.

Weβve completed the problem then. There are no units for our answers, as there were no units given for the measurements in the original question. But the perimeter would be in length units and the area would be in square units. Weβve found that the perimeter of π΅π·πΈπΆ is 11.76 and the area of π΅π·πΈπΆ is 8.64, each correct to two decimal places.