Question

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Answer 1

The formula for the circumference of a circle, C, is derived using the concept of angles and arc length. The formula is C = 2πr, where r is the radius of the circle. The arc length is given by A = rθ, where θ is the angle of rotation.

Step-by-step explanation:

The formula for the circumference of a circle, C, is derived using the concept of angles and arc length. When a line from the center of a circle is rotated through an angle, it covers a certain distance along the circumference, which is known as the arc length, A. The formula for arc length is A = rθ, where r is the radius of the circle and θ is the angle of rotation.

To find the circumference, we need to consider a full rotation of 360 degrees or 2π radians. Since the angle θ is equal to 2π, the formula for the circumference becomes C = 2πr.

For example, if the radius of a circle is 5 units, the circumference can be calculated as: C = 2π(5) = 10π units or approximately 31.42 units.

Answer 2

The formula for the circumference of a circle, C = 2πr, is derived by considering the limiting case of a regular polygon as the number of sides approaches infinity.

Formula for the circumference of a circle

The formula for the circumference of a circle, which is given by C = 2πr, can be derived using the following steps:

Start with a circle of radius r.

Divide the circle into a large number of equal-sized sectors (or "slices").

Arrange these sectors along the circumference of the circle, forming a shape similar to a polygon.

As the number of sectors increases, the shape becomes closer to a regular polygon.

The perimeter of a regular polygon is given by multiplying the length of one side by the number of sides.

In the case of a circle, as the number of sectors increases, the length of each side of the polygon approaches the length of the circle's circumference.

The number of sides of the polygon is equal to the number of sectors, which can be considered as approaching infinity.

Therefore, the length of the circle's circumference is given by the limit of the perimeter of the polygon as the number of sides approaches infinity.

The formula for the perimeter of a regular polygon with n sides is P = ns, where s is the length of one side.

Substituting n = ∞ and s = r (since the length of each side approaches the radius of the circle), we get C = 2πr.

Thus, the formula for the circumference of a circle, C = 2πr, is derived by considering the limiting case of a regular polygon as the number of sides approaches infinity.