Video: Finding the Area of a Sector given Its Perimeter and the Central Angle

The perimeter of a circular sector is 36 cm and the central angle is 0.4 rad. Find the area of the sector giving the answer to the nearest square centimeter.

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Video Transcript

The perimeter of a circular sector is 36 centimeters and the central angle is 0.4 radians. Find the area of the sector giving the answer to the nearest square centimeter.

We recall the formula for area of a sector with radius 𝑟 and angle 𝜃 radians is a half 𝑟 squared 𝜃. We also know the arc length is 𝑟𝜃. This is useful because we’ve been told the perimeter of the circular sector. The perimeter is the entire way around the shape. So if we look at the diagram we’ve drawn, we can see that’s the arc length plus the two radii.

And in general, we can say that the perimeter of a circular sector can be found by adding two lots of the radii to its arc length. It’s two 𝑟 plus 𝑟𝜃. In fact, we know that our circular sector has a perimeter of 36 centimeters. We’re also told that its central angle is 0.4 radians. We can use this information to form an equation in terms of 𝑟. Solving this for 𝑟 will allow us to find the area of our sector.

Since 𝜃 is equal to 0.4, we can say that the perimeter of our circular sector is two 𝑟 plus 0.4𝑟. And in fact, this must be equal to 36. Simplifying by collecting like terms on the right-hand side of our equation, we get 2.4𝑟 is equal to 36. And we solve this for 𝑟 by dividing both sides by 2.4. 36 divided by 2.4 is 15. So we found that the radius of our circular sector is 15 centimeters.

All that’s left is to substitute what we know now about the area of our sector into the formula. It’s a half multiplied by 15 squared multiplied by 0.4, which is equal to 45.

So the area of our sector is 45 centimeters squared.

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