Question Video: Circular Sectors and Areas of Circles Mathematics

Three congruent circles with a radius of 43 cm are placed touching each other. Find the area of the part between the circles giving the answer to the nearest square centimeter.

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Video Transcript

Three congruent circles with a radius of 43 centimeters are placed touching each other. Find the area of the part between the circles giving the answer to the nearest square centimeter.

So we want to find the area of the part between these circles. That’s this part here. We know that the circles are congruent, that’s identical, each with the radius of 43 centimeters. If we join the centers of the three circles, we have a triangle. Each side of this triangle consists of two radii. And so, in fact, this is an equilateral triangle with a side length of two times 43; that’s 86 centimeters. The area we’re looking to find is inside this triangle. The other areas inside this triangle, that’s these three areas here, are three congruent circular sectors. So our approach to finding the blue area is going to be to find the area of the triangle and then subtract the area of these three sectors.

The triangle, first of all. This is a nonright triangle, so we need to use the formula a half 𝑎𝑏 sin 𝑐, where 𝑎 and 𝑏 represent two sides of the triangle and 𝑐 represents the angle between them. Each of the side lengths of our triangle are 86 centimeters. And as it’s an equilateral triangle, all of the angles are 60 degrees. So we have a half multiplied by 86 multiplied by 86 multiplied by sin of 60 degrees. sin of 60 degrees is root three over two, so the area of the triangle simplifies to 1849 root three, which we’ll keep in its exact form for now.

Now, for the area of the sectors, the area of each sector working in degrees is 𝜃 over 360 multiplied by 𝜋𝑟 squared, where 𝜃 is the central angle. But we have three of these identical sectors, so we’ll multiply by three. That would be equivalent then to the formula 𝜃 over 120 multiplied by 𝜋𝑟 squared, as three over 360 is one over 120. So substituting the angle of 60 degrees and the radius of the sector which, remember, is 43 centimeters, we have 60 over 120 multiplied by 𝜋 multiplied by 43 squared. 60 over 120 simplifies to a half and 43 squared is 1849. So we have 1849𝜋 over two for the area of the sectors.

We can now go ahead and evaluate this using our calculators. Doing so gives 298.159. We’re required to give an answer to the nearest square centimeter, so we’ll be rounding down. We found then that the area of the part between the circles, which is the difference between the area of a nonright triangle and three circular sectors, is 298 square centimeters.

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