Work out the area of the quarter circle, giving your answer in terms of 𝜋.
Remember, the formula for area of a circular sector is given by a half multiplied by 𝑟 squared multiplied by 𝜃, where 𝑟 is the radius of the sector and 𝜃 is the angle in radians. We can see that our quarter circle has a radius of three units and a right angle. Remember, that’s just an angle of 90 degrees. We need this angle to be in radians. So how do we convert from degrees to radians?
Well, we begin by recalling the fact that a full term, 360 degrees, is equal to two 𝜋 radians. And then at this point, we have two options. We could find the value of one degree by dividing both sides of this equation by 360. So one degree is equal to two 𝜋 over 360 radians. This simplifies to 𝜋 over 180 radians. And we can therefore change from degrees to radians by multiplying by 𝜋 over 180.
Alternatively, and this method works nicely when the angle is a factor of 360, we spot that 90 degrees is a quarter of 360 as specified in the question. And we can divide both sides of the equation by four. Doing so, we can see that 90 degrees is equal to two 𝜋 over four radians. That simplifies to 𝜋 over two.
Now that we know the size of the angle in radians, we can substitute everything into the formula of area of a sector. Doing so, we get a half multiplied by three squared multiplied by 𝜋 over two. Three squared is nine. And we can write this as nine over one. So we can multiply these three fractions. We begin by multiplying the numerators. One multiplied by nine multiplied by 𝜋 is nine 𝜋. And we then multiply the denominators. Two multiplied by one multiplied by two is four.
And we see that the area of our sector is nine 𝜋 over four units squared.