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‘Ÿ and an angle ๐œƒ radians, its arc length is given by ๐‘Ÿ๐œƒ. And its sector area is a half ๐‘Ÿ squared ๐œƒ. 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‘Ÿ 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œƒ needed to be in radians. So weโ€™re going to convert 77 degrees to radians. And to do that, we recall that two ๐œ‹ 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œ‹ over 360 radians. This simplifies to ๐œ‹ over 180.

And we can now see that, to convert from degrees into radians, we multiply by ๐œ‹ over 180. And 77 degrees is equal to 77๐œ‹ over 180 radians. This means our arc length is given by ๐‘Ÿ multiplied by 77๐œ‹ over 180. And since we know the arc length to be 22 centimetres, we can form an equation in terms of ๐‘Ÿ. To solve this equation, weโ€™ll divide through by 77๐œ‹ 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œ‹ 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.

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