The ratio of the radii of two circles is $4: 5$. What is the ratio of the circumferences of the two circles?
Answer (1)
Define the radii of the two circles as r 1 and r 2 , with a ratio of r 2 r 1 = 5 4 .
Express the circumferences of the circles as C 1 = 2 π r 1 and C 2 = 2 π r 2 .
Determine the ratio of the circumferences: C 2 C 1 = 2 π r 2 2 π r 1 = r 2 r 1 .
Conclude that the ratio of the circumferences is the same as the ratio of the radii: 5 4 .
Explanation
Define the radii and their ratio. Let r 1 and r 2 be the radii of the two circles. We are given that the ratio of the radii is 4 : 5 , which means:
Express the given ratio mathematically. r 2 r 1 = 5 4
State the circumference formula. The circumference of a circle is given by the formula C = 2 π r , where r is the radius of the circle. Let C 1 and C 2 be the circumferences of the two circles. Then:
Circumference of the first circle. C 1 = 2 π r 1
Circumference of the second circle. C 2 = 2 π r 2
Find the ratio of the circumferences. We want to find the ratio of the circumferences, which is C 2 C 1 . We can express this as:
Express the ratio of circumferences mathematically. C 2 C 1 = 2 π r 2 2 π r 1
Simplify the ratio. We can simplify this expression by canceling out the common factor 2 π :
Simplified ratio of circumferences. C 2 C 1 = r 2 r 1
Substitute the given ratio. Since we know that r 2 r 1 = 5 4 , we can substitute this into the equation:
Final ratio of circumferences. C 2 C 1 = 5 4
Conclusion. Therefore, the ratio of the circumferences of the two circles is 4 : 5 .
Examples
Imagine you're designing two gears for a machine. The ratio of their radii is crucial for determining how they mesh and turn each other. If one gear has a radius 4 times a unit and the other 5 times the same unit, their circumferences will also be in the same ratio, 4 : 5 . This ensures that for every rotation of the first gear, the second gear rotates proportionally, maintaining the machine's timing and efficiency.
