two circles have the radii of 2cm and 6cm. how many times larger is the area of the larger circle than the smaller circle? i know the answer is 9 because i did the areas out, but im looking for a smarter and easier way to find the answer...help!!!
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Circles are proportional to the squares of their radii. What this means is that the ratio their areas is the same as the ratio of the squares of the radii. So the ratio of the 2 areas is the same as 6^2/2^2 = 36/4 = 9.
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The area of a circle is proportional to the square of the radius. Thus, the ratio of the areas between two circles is the square of the ratio of the radii: (6/2)^2 = 3^2 = 9.
Here is your problem stated differently: You have two circles, the larger of which has a radius 3 times greater than the smaller one. How many times larger is the area of the bigger circle to the smaller circle?
Answer: Let R = 3r, where R is the larger circle radius, and r the smaller circle radius. Then the ratio of their areas is (pi*R^2) / (pi*r^2) = (pi*(3r)^2) / (pi*r^2) = (pi*9*r^2) / (pi*r^2) = 9 because everything else cancels out.
Here is your problem stated differently: You have two circles, the larger of which has a radius 3 times greater than the smaller one. How many times larger is the area of the bigger circle to the smaller circle?
Answer: Let R = 3r, where R is the larger circle radius, and r the smaller circle radius. Then the ratio of their areas is (pi*R^2) / (pi*r^2) = (pi*(3r)^2) / (pi*r^2) = (pi*9*r^2) / (pi*r^2) = 9 because everything else cancels out.
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The radius is a 1-dimensional measure and area is a 2-dimensional measure. Since the radius of the bigger circle is 3 times the radius of the smaller, the area must be 3^2 or 9 times bigger.