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Answer :
Final answer:
The reason the circumference is equal to πr is that it is a property of circles (d). The ratio of the circumference to its radius is a constant 2π in Euclidean geometry, which is why the circumference can be accurately calculated as the product of 2π and the circle's radius. So, the correct option is option d.
Explanation:
The circumference being equal to πr (pi times the radius) is a fundamental property of circles. When the student asks why the circumference of a circle is 176 cm given that its radius is 28 cm, and why this is equal to πr, the answer is: d) It is a property of circles. The ancient Greeks, over 2000 years ago, understood that the perimeter, or circumference, of a circle could be expressed as 2πr, where π denotes a constant approximately equal to 3.14159, often simply referred to as pi. In your case, given the radius r is 28 cm, multiplying it by 2π indeed yields the circumference C as 176 cm. This relationship holds because the ratio of the circumference of a circle to its radius is always constant and equal to 2π in Euclidean geometry.
Furthermore, when covering distances along a circular arc, the arc length (As) for a given angle is proportional to the radius (r) of the circle, following the relation As = rθ where θ is the angle in radians. If θ is an angle making a complete revolution (a full circle), then it measures 2π radians, and therefore the arc length As, which is also the circumference of a circle, equals 2πr.
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