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 𝑟 squared 𝜃, where 𝑟 represents the radius of the
circle and 𝜃 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 𝜃. We can now solve this equation to
determine the value of 𝜃.
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 𝜃 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.