The diagram shows a sector of a circle of radius three centimeters. Work out the length of the arc 𝐴𝐵𝐶. Give your answer correct to three significant figures.
An arc is a portion of the circumference of a circle. To find the circumference of a circle, we either use the formula 𝜋𝑑 or two 𝜋𝑟,
where 𝑑 represents the diameter of the circle and 𝑟 the radius. We’re given that the radius of this circle is three centimeters. So we can find the circumference. It’s two 𝜋 multiplied by three, which is six 𝜋. We’ll keep this value in terms of 𝜋 for now so that we don’t introduce any rounding
Now, we don’t have the full circumference of this circle. We only have a portion of it. We need to work out what fraction of the circumference we have. To do so, we need to know the central angle of this sector, the angle that I’ve
marked in pink.
We’ve been told that the obtuse angle 𝐴𝑂𝐶 is 150 degrees. To find the reflex angle 𝐴𝑂𝐶, which is the central angle of our sector, we can
subtract this from 360 degrees as angles around a point sum to 360 degrees. This gives 210 degrees. This means that the fraction of the full circumference of the circle represented by
the arc 𝐴𝐵𝐶 is 210 out of 360.
We can simplify this fraction by first dividing the numerator and denominator by a
factor of 10 and then by a factor of three. The arc length 𝐴𝐵𝐶 is, therefore, seven twelfths of the circumference of the full
circle. The arc length is, therefore, seven twelfths of six 𝜋.
When we see the word “of” in a calculation involving fractions, it means we can
multiply. So to calculate the arc length, we need to multiply seven twelfths by six 𝜋. You can think of six 𝜋 as six 𝜋 over one. So we’re multiplying two fractions together. Before we do the multiplication, we can cross cancel as six and 12 have a common
factor of six. Six divided by six is one and 12 divided by six is two. So the calculation becomes seven over two multiplied by 𝜋 over one.
Multiplying the numerators of the fractions together gives seven 𝜋 and multiplying
the denominators gives two. So we have seven 𝜋 over two. You may recall that there’s a general formula that we can use to calculate the arc
length of a sector. It’s 𝛳 over 360 multiplied by two 𝜋𝑟 or 𝜋𝑑. Two 𝜋𝑟 gives us the circumference of the full circle and 𝛳 over 360 gives the
fraction of the circle corresponding to the arc length.
This is exactly the calculation that we’ve used, but we’ve just broken it down into
separate stages. If we didn’t have a calculator, we could leave our answer in terms of 𝜋, but we
do. And the question has asked us to give our answer correct to three significant
figures. Evaluating seven 𝜋 over two as a decimal gives 10.99557429.
To round to three significant figures, we need to consider the fourth significant
figure in the number, which is this nine here. As this digit is a nine, this tells us that we’re rounding up. But as the digit in the next column is also a nine, we need to round up once
more. This changes the zero in the units column to a one. And we included zero in the tenths column after the decimal point as we now have
effectively ten tenths which is equal to 1.0.
Notice that we do need to include the zero after the decimal point as the question
has specifically asked for three significant figures. The length of the arc 𝐴𝐵𝐶 to three significant figures is 11.0 centimeters.