Question Video: Circular Sectors and Area of Circles Mathematics

The radius of a circle is 5 cm and the area of a sector is 15 cmΒ². Find the central angle, in radians, rounded to one decimal place.

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Video Transcript

The radius of a circle is five centimeters and the area of a sector is 15 centimeters squared. Find the central angle, in radians, rounded to one decimal place.

In this problem, we have a sector of a circle. We know that the radius of the circle is five centimeters and we know that the area of the sector is 15 square centimetres. But we don’t know the central angle. We were asked to find the central angle in radians. So we need to recall the key formulae we need. When working in radians, the area of a sector is given by a half π‘Ÿ squared πœƒ, where π‘Ÿ represents the radius of the circle and πœƒ represents the central angle. We can therefore use the information we’re given to form an equation. The area of the sector is 15 square centimeters, and the radius of the circle is five centimeters. So we have the equation 15 equals a half multiplied by five squared multiplied by πœƒ. We can now solve this equation to determine the value of πœƒ.

Firstly, we evaluate five squared, which is 25. We can then multiply both sides of the equation by two and then divide both sides by 25, giving πœƒ equals two multiplied by 15 over 25. Two multiplied by 15 is 30, and then we can cancel a factor of five from both the numerator and denominator. So this simplifies to six over five. Six over five is equivalent to one and one-fifth or 1.2. And so we have our answer to the problem. The central angle of the sector in radians is 1.2 radians.

Notice that in this question it was important to use the area formula in radians. It would have been possible to instead use the area formula in degrees and then convert the answer from degrees to radians at the end. But that would require an extra step. And of course, we may forget to convert our answer.

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