A target board consists of three concentric circles of radii six centimetres, 12 centimetres, and 24 centimetres, which define three regions, gold, red, and blue as shown in the diagram. If a dart is shown randomly such that it hits the target, what is the probability it will hit the red region?
In order to answer this question, we need to calculate the area of the three regions. We recall that the area of a circle is equal to 𝜋 multiplied by the radius squared. The gold circle has a radius of six centimetres. This means that the area of the target board that is gold is equal to 𝜋 multiplied by six squared. Six squared is equal to 36. Therefore, the gold area is equal to 36𝜋.
We’re told that the red circle has a radius of 12 centimetres. The area of the target board that is red will be equal to the red circle minus the gold circle. This is equal to 𝜋 multiplied by 12 squared minus 𝜋 multiplied by six squared. 12 squared is equal to 144. Therefore, the area of the red circle is 144𝜋. We already know that the area of the gold circle is 36𝜋. 144 minus 36 is equal to 108. Therefore, the red area is 108𝜋.
We can repeat this process for the blue area. The area of the blue circle is equal to 𝜋 multiplied by 24 squared. And we need to subtract 𝜋 multiplied by 12 squared. 24 squared is equal to 576. This means we need to subtract 144𝜋 from 576𝜋. This is equal to 432𝜋. Dividing all three of these values by 𝜋 tells us that the ratio of gold to red to blue is 36 to 108 to 432. All three of these numbers are divisible by 36. 36 divided by 36 is equal to one. 108 divided by 36 is equal to three. And 432 divided by 36 is equal to 12. The ratio of the gold area to the red area to the blue area simplifies to one to three to 12.
We were asked to calculate the probability that a dart hits the red region. The red region is equal to three parts of our ratio. We have a total of 16 parts, as one plus three plus 12 is equal to 16. This means that the probability that the dart hits the red region is three out of 16 or three sixteenths.