Question Video: Calculating the Area of a Sector given the Measure of Its Angle in Radians Mathematics

Video Transcript

An arc has a measure of 𝜋 by three radians and a radius of five. Give the area of the sector, in terms of 𝜋, in its simplest form.

An arc is a portion of the circumference of a circle. And we’re told that this circle has a radius of five units. We’re also told that the arc has a measure of 𝜋 by three radians. Now the measure of an arc is the measure of its central angle. That is the angle formed where the two radii from each endpoint of the arc to the center of the circle intersect. So, the central angle of this circular sector, which is bounded by the arc and the two radii, is 𝜋 by three radians.

We’re asked to calculate the area of this sector. And so we should recall the formula for doing this when the central angle is measured in radians. The formula we need is this. The area of a circular sector with a radius of 𝑟 units and a central angle of 𝜃 radians is a half 𝑟 squared 𝜃. This is simplified from the formula 𝜃 over two 𝜋 multiplied by 𝜋𝑟 squared, where 𝜋𝑟 squared gives the area of the full circle and 𝜃 over two 𝜋 gives the fraction of the circle represented by this sector.

We know both the radius of the circle and the central angle in radians. So we can substitute these two values into the formula. The area is therefore equal to one-half multiplied by five squared multiplied by 𝜋 by three. That’s one-half multiplied by 25 multiplied by 𝜋 over three. And we can combine all this into a single fraction. It’s 25𝜋 over six. The question specifies that we should give our answer in terms of 𝜋. So we don’t need to evaluate this as a decimal. And this is already in its simplest form because the numbers 25 and six in the numerator and denominator have no common factors other than one.

The area of this circular sector then, which has a central angle of 𝜋 by three radians and a radius of five units, is 25𝜋 over six square units.