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Question Video: Finding the Area of a Circular Sector given Its Radius and the Measure of the Angle of the Sector Mathematics

Find the area of the colored part of the diagram, giving the answer to one decimal place.

02:48

Video Transcript

Find the area of the colored part of the diagram, giving the answer to one decimal place.

Let’s have a look at the diagram we’ve been given. It consists of two circular sectors: one with a radius of 15 centimeters and the other with a radius of seven centimeters. The central angle of these sectors — that’s the angle between the two radii that form part of the perimeter of each sector — is 60 degrees. We’re asked to find the area of the colored part of the diagram, which we can see will be the difference in the areas of these two sectors. It’s the area of the largest sector — that’s the one with the radius of 15 centimeters — minus the area of the smaller sector — that’s the one with the radius of seven centimeters.

The area of a sector with radius 𝑟 and central angle 𝜃 measured in degrees is 𝜃 over 360 multiplied by 𝜋𝑟 squared. This comes from multiplying the area of a full circle, 𝜋𝑟 squared, by the fraction 𝜃 over 360, which corresponds to the portion of the circle represented by this sector. We can find the areas then of both the larger and smaller sectors by substituting 𝜃 equals 60 for both and 𝑟 equals 15 for the largest sector and 𝑟 equals seven for the smaller sector.

We have then 60 over 360 multiplied by 𝜋 multiplied by 15 squared minus 60 over 360 multiplied by 𝜋 multiplied by seven squared. In each case, the fraction 60 over 360 can be simplified to one-sixth by canceling a common factor of 60 from the numerator and denominator. If we want, we can then factor 𝜋 over six from these two expressions. And we have 𝜋 over six multiplied by 15 squared minus seven squared. 15 squared is 225, and seven squared is 49. 225 minus 49 is 176. So, we have 176𝜋 over six.

The question specifies that we should give our answer to one decimal place. So we need to evaluate this as a decimal. It’s 92.1533 continuing. The value in the second decimal place is a five. So, we’re going to be rounding up. The units for the lengths in this question were centimeters. So, the units for the area will be square centimeters.

By recognizing then that the colored part of the diagram is the difference between the areas of two circular sectors, we found that the area of the colored part to one decimal place is 92.2 square centimeters.

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