Question Video: Finding the Area of a Circular Sector given Its Arc Length and Central Angle Mathematics

The arc length of a circular sector is 22 cm and the central angle is 77ยฐ. Find the area of the sector giving the answer to the nearest square centimeter.

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Video Transcript

The arc length of a circular sector is 22 centimetres and the central angle is 77 degrees. Find the area of the sector giving the answer to the nearest square centimetre.

Letโs recall what we know about the arc length and area of a circular sector. For a sector with a radius ๐ and an angle ๐ radians, its arc length is given by ๐๐. And its sector area is a half ๐ squared ๐. We know that the arc length of our circular sector is 22 centimetres. And the measure of its central angle is 77 degrees.

Weโre going to use this information to form an equation in terms of ๐ using the formula for the arc length. Weโll be able to solve this to find the length of the radius of our circle, which we can then substitute into the formula for sector area.

Remember that we said that ๐ needed to be in radians. So weโre going to convert 77 degrees to radians. And to do that, we recall that two ๐ radians is equal to 360 degrees. We can then divide through by 360. And that tells us that one degree is equivalent to two ๐ over 360 radians. This simplifies to ๐ over 180.

And we can now see that, to convert from degrees into radians, we multiply by ๐ over 180. And 77 degrees is equal to 77๐ over 180 radians. This means our arc length is given by ๐ multiplied by 77๐ over 180. And since we know the arc length to be 22 centimetres, we can form an equation in terms of ๐. To solve this equation, weโll divide through by 77๐ over 180. Thatโs 16.37 and so on.

Now at this stage, weโre going to choose not to round our answer just yet. Instead, weโll use the unrounded form in the next step of our calculation. The formula for area of a sector is now one-half multiplied by 16.37 squared multiplied by 77๐ over 180. Thatโs 180.072 and so on.

We were told to round our answer correct to the nearest square centimetre. Thatโs 180 centimetres squared.