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if the radius of a circle is halved, how will this affect its circumference and its area? what happens to the circumference and area if the radius is doubled or tripled?

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Final answer:

Halving the radius of a circle halves its circumference and reduces its area to a quarter of the original. Doubling or tripling the radius increases the circumference proportionally, but the area increases by a factor of four or nine, respectively.

Step-by-step explanation:

Effects of Changing the Radius on Circumference and Area

When the radius of a circle is halved, the circumference of the circle is also halved since the circumference formula is C = 2πr. For example, if the original circumference was 2π(2r), halving the radius to r would change the circumference to 2π(r), which is half of the original circumference.

As for the area of the circle, when the radius is halved (r becomes r/2), the area becomes (1/2)^2 times the original area because the area of a circle is A = πr^2. Therefore, the original area π(2r)^2 is four times the size of the new area π(r/2)^2.

If the radius is doubled or tripled, the circumference would become twice or thrice as large, respectively, because the radius directly influences the circumference linearly. However, the area of a circle when the radius is doubled would become four times as large (since (2r)^2 = 4r^2), and if the radius is tripled, the area becomes nine times as large (since (3r)^2 = 9r^2), because the area is proportional to the square of the radius.

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