Finding the Area of a Circle with Half the Radius of Another Circle

To find the area of a circle with a radius that is half the length of the original circle's radius, we first need to understand the relationship between the area of a circle and its radius. The area (A) of a circle is given by the formula:

Step-by-Step Calculation

Determine the Radius of the Original Circle

The formula for the area (A) of a circle is:

A πr2

Given that the area of the original circle is 28πcm2, we can set up the equation:

πr2 28π

Dividing both sides by π:

r2 28

Take the square root of both sides:

r 28 27cm

Calculate the Radius of the New Circle

The radius of the new circle is half of the original radius:

rnew r2 272 7cm

Calculate the Area of the New Circle

Using the same area formula, we calculate the area of the new circle:

Anew π rnew2 π72 π? 7 7πcm2

Thus, the area of the circle with a radius of half the length of the original circle is:

boxed{7π cm2}

Understanding the Proportionality

The area of a circle is directly proportional to the square of its radius. If you reduce the radius of a circle to half its original value, the circle’s area will be decreased to one quarter of the original value. This is derived from the formula:

A varies as X2

If the original radius (r) produces an area of (28π), the new radius (r/2) will produce an area of:

π(r2)#160;cm2πrcm2 14

Therefore, the new area is:

A 7πcm2

Conclusion

Understanding the relationship between the area of a circle and its radius can help solve a variety of geometric problems. In this case, the area of a circle with a radius that is half the length of the original circle's radius is 7π cm2.

Keywords: circle area, radius relation, proportional area

References:

1. Wikipedia - Area of a Circle

2. Math is Fun - Area of a Circle