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

The area of a circular sector is 1,888 cmยฒ and its central angle is 1.7 rad. Find the arc length of the sector giving the answer to the nearest centimeter.

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Video Transcript

The area of a circular sector is 1888 square centimetres and its central angle is 1.7 radians. Find the arc length of the sector giving the answer to the nearest centimeter.

In this problem then, we have a circular sector. We know its area is 1888 square centimeters and we know its central angle, which we can call ๐œƒ, is 1.7 radians. We want to calculate the arc length of the sector, which we often denote using the letter ๐‘ . The key thing to note is that the central angle of the sector is measured in radians. So the formulae that weโ€™re going to use for both the area of the sector and the arc length must be the formulae in radians. Those are a half ๐‘Ÿ squared ๐œƒ for the area of the sector and ๐‘Ÿ๐œƒ for the arc length.

In order to calculate the arc length then, we need to know both the radius and the central angle. Letโ€™s use the information we were given about the area to form an equation. The area is 1888 square centimeters. We donโ€™t know the radius, so weโ€™ll keep the letter ๐‘Ÿ. And central angle ๐œƒ in radians is 1.7. We can now solve this equation to work out the radius of the sector, which weโ€™ll then be able to substitute into our formula for the arc length. We can multiply both sides of the equation by two and then divide by 1.7 giving ๐‘Ÿ squared is equal to 1888 multiplied by two over 1.7, which as a decimal is 2221.176.

To solve for ๐‘Ÿ, we take the square root of each side of this equation, taking only the positive value as ๐‘Ÿ represents a length. That gives 47.129, and weโ€™ll keep this exact value on our calculator display for now. So weโ€™ve now calculated the radius of the sector, and all that remains is to calculate the arc length. Using the formula arc length equals ๐‘Ÿ๐œƒ, we multiply the value weโ€™ve just calculated by the central angle of 1.7, which gives 80.119. We were asked to give an answer to the nearest centimeter, so we round down. Using the known area and the known central angle of this circular sector then, weโ€™ve calculated that the arc length to the nearest centimeter is 80 centimeters.

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