Answer:
a.
The area of the large circle is 225π cm² [exact] or 706.9 cm² [approximate].
b.
area = 50π/6 cm² [exact]
area = 26.2 cm² [approximate]
Step-by-step explanation:
This problem is easier to solve than it looks.
a.
Area of the large circle.
Look at the top left small circle, the middle circle, and the bottom right circle. You can draw a single line segment that is a diameter of the large circle and also contains the diameters of the three small circles. That means that the diameter of the large circle is 3 times the diameter of each small circle.
R and D are the radius and diameter, respectively, of the large circle.
r and d are the radius and diameter, respectively, of the small circles.
D = 3d
D = 6r = 6(5 cm) = 30 cm
R = 15 cm
A = πr²
A = π(15 cm)²
A = 225π cm²
The area of the large circle is 225π cm² [exact] or 706.9 cm² [approximate].
b.
There are 6 congruent spaces between the small circles and the large circle, and there are another, different 6 congruent spaces between the center circle and the 6 circles around it. The region we need is made up of one space of each type.
The combined area of all 12 spaces is the area of the large circle minus the total area of all the small circles. The area we need to calculate is 1/6 of that combined area.
area = (1/6)(area of large circle - total are of all small circles)
area = (1/6)[225π cm² - 7π(5 cm)²]
area = (1/6)[225π - 175π]cm²
area = 50π/6 cm² [exact]
area = 26.2 cm² [approximate]
