We can use sector areas to compare the value of product offers. Suppose the manager of a store wants to buy several dozen mirrors shaped like sectors of a circle to decorate the store. The manager is choosing between these two brands of mirrors at a tradeshow. Brand A’s mirror has radius 14 inches and costs $35 for each sector. Brand B’s mirror has radius 15 inches and costs $42. Which is the better deal in terms of cost per square inch of mirror?

Brand A’s mirror is made from a circle cut into 6 congruent slices. Using the expression with radius 14 inches, we find that the area of the full circle is , or about 616, square inches. Divide that by 6 to find that each sector-shaped mirror has an area of about 103 square inches. At $35 per sector, this mirror costs about $0.34 per square inch.

For Brand B, we don’t know how many slices the mirror was cut into. However, we can estimate the measure of the central angle using arc length and radius. Suppose the manager uses a flexible measuring tape to find that the length of the arc around the outside of the sector is 19 inches. The ratio of the arc length to the radius gives the measure of the central angle in radians, . This is about 1.27 radians, but to avoid rounding errors, let’s use the exact value, , in our calculations.

Next, we can find the area of the sector with the formula , where is the radius and is the radian measure of the central angle. Substitute in our values to get , or 142.5, square centimeters. At $42 per sector, this mirror costs about $0.29 per square inch. Brand B’s mirror is a better deal.