In the given figure, the measure of angle 𝐵𝐴𝐶 is 30 degrees, 𝐴𝐵 equals seven, and 𝐴𝐷 equals four. Work out the perimeter of 𝐵𝐷𝐸𝐶. Give your answer to two decimal places. Secondly, work out the area of 𝐵𝐷𝐸𝐶. Give your answer to two decimal places.
In each part of this question, we’re asked to consider 𝐵𝐷𝐸𝐶. That’s this part of the figure highlighted in orange. Let’s first add the information given in the question onto our diagram. The measure of angle 𝐵𝐴𝐶 is 30 degrees, the length of 𝐴𝐵 is seven units, and the length of 𝐴𝐷 is four units. Let’s now consider how to find the perimeter of 𝐵𝐷𝐸𝐶.
And if we start in one corner and trace a way round, we see that it is composed of two straight sections, 𝐵𝐷 and 𝐸𝐶, and two arcs, 𝐵𝐶 and 𝐷𝐸. The lengths of the two straight sections can be worked out relatively easily. They’re the difference between the longer length of seven and the shorter length of four. So, 𝐵𝐷 and 𝐸𝐶 are, each, three units.
To calculate the lengths of 𝐷𝐸 and 𝐶𝐵, we need to recall the general formula for calculating arc length. It’s 𝜃 over 360 multiplied by 𝜋𝑑, where 𝜃 is the central angle of the sector and 𝑑 is the circle’s diameter. What we’re doing when we apply this formula is calculating the full circumference of the circle, 𝜋𝑑, and then multiplying it by the fraction of the circle that this arc represents. That’s 𝜃 out of 360. For our two arcs, the central angle 𝜃 is 30 degrees. But what about the diameters?
Well, the arc 𝐷𝐸 is part of a smaller circle, which has a radius of four units. That’s the length 𝐴𝐷. And so, the diameter will be twice this; it will be eight units. The arc 𝐶𝐵 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 𝐷𝐸 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 𝐶𝐵, 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 𝐶𝐵 is seven 𝜋 over six. So, our calculation for the perimeter of 𝐵𝐷𝐸𝐶 is three plus two 𝜋 over three plus three plus seven 𝜋 over six.
We’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.
I’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, we’re asked to work out the area of 𝐵𝐷𝐸𝐶. We can do this by calculating the difference in the area of two circular sectors: the sector 𝐴𝐵𝐶 and the sector 𝐴𝐷𝐸. 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. It’s 𝜃 over 360 multiplied by 𝜋𝑟 squared. What we’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.
We’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. We’ve found that the perimeter of 𝐵𝐷𝐸𝐶 is 11.76 and the area of 𝐵𝐷𝐸𝐶 is 8.64, each correct to two decimal places.