Final answer:
To determine the measure of the area of a sector of 90 degrees in a circle with a diameter of 42 centimeters, we need to find the area of the whole circle and then multiply it by the proportion of the sector angle to the total angle of the circle.
Step-by-step explanation:
To determine the measure of the area of a sector of 90 degrees in a circle, we need to first find the area of the whole circle. The formula for the area of a circle is A = πr², where r is the radius. Given that the diameter measures 42 centimeters, the radius would be half of that or 21 centimeters. Plugging in the values, we have A = 3.14 x 21² = 1385.94 square centimeters.
Next, we need to find the proportion of the angle measure of the sector to the total angle of the circle. In this case, the sector angle is 90 degrees and the total angle of the circle is 360 degrees. Therefore, the proportion is 90/360 = 1/4.
To find the measure of the area of the sector, we multiply the area of the whole circle by the proportion. So, the area of the sector is 1385.94 x 1/4 = 346.48 square centimeters.