Calculating Percentage Increase in Circle Area

Let's calculate the percentage increase in the area of a circle when its radius increases. We are given the original radius and the new radius, and we need to find how much the area has increased in terms of percentage.

Understanding Circle Area

The area of a circle is calculated using the formula:

\(A = \pi r^2\)

where:

• \(A\) is the area of the circle.
• \(\pi\) (pi) is a mathematical constant, approximately 3.14159.
• \(r\) is the radius of the circle.

To find the percentage increase in area, we first need to calculate the original area and the new area.

Step-by-Step Calculation

1. Calculate the Original Area

The original radius of the circle is given as 90 cm.

Original Radius (\(r_1\)) = 90 cm

Original Area (\(A_1\)) = \(\pi r_1^2 = \pi (90 \text{ cm})^2\)

\(A_1 = \pi \times 8100 \text{ cm}^2\)

\(A_1 = 8100\pi \text{ cm}^2\)

2. Calculate the New Area

The radius is increased to 99 cm.

New Radius (\(r_2\)) = 99 cm

New Area (\(A_2\)) = \(\pi r_2^2 = \pi (99 \text{ cm})^2\)

\(A_2 = \pi \times (99 \times 99) \text{ cm}^2\)

\(A_2 = \pi \times 9801 \text{ cm}^2\)

\(A_2 = 9801\pi \text{ cm}^2\)

3. Calculate the Increase in Area

The increase in area is the difference between the new area and the original area.

Increase in Area = \(A_2 - A_1\)

Increase in Area = \(9801\pi \text{ cm}^2 - 8100\pi \text{ cm}^2\)

Increase in Area = \((9801 - 8100)\pi \text{ cm}^2\)

Increase in Area = \(1701\pi \text{ cm}^2\)

4. Calculate the Percentage Increase in Area

The percentage increase is calculated by dividing the increase in area by the original area and multiplying by 100.

Percentage Increase = \(\frac{\text{Increase in Area}}{\text{Original Area}} \times 100\)

Percentage Increase = \(\frac{1701\pi \text{ cm}^2}{8100\pi \text{ cm}^2} \times 100\)

Notice that \(\pi\) cancels out from the numerator and the denominator.

Percentage Increase = \(\frac{1701}{8100} \times 100\)

Let's simplify the fraction \(\frac{1701}{8100}\). Both numbers are divisible by 9.

\(1701 \div 9 = 189\)

\(8100 \div 9 = 900\)

So, the fraction is \(\frac{189}{900}\). Both numbers are still divisible by 9.

\(189 \div 9 = 21\)

\(900 \div 9 = 100\)

The simplified fraction is \(\frac{21}{100}\).

Percentage Increase = \(\frac{21}{100} \times 100\)

Percentage Increase = \(21\)

So, the percentage increase in the area of the circle is 21%.

Summary of Calculations

Alternative Approach: Using Percentage Change in Radius

Let the original radius be \(r\). The original area is \(A = \pi r^2\).

The radius increases from 90 cm to 99 cm.

Percentage increase in radius = \(\frac{\text{Increase in Radius}}{\text{Original Radius}} \times 100\)

Increase in Radius = 99 cm - 90 cm = 9 cm

Percentage increase in radius = \(\frac{9}{90} \times 100 = \frac{1}{10} \times 100 = 10\%\)

So, the new radius is \(r_{new} = r + 10\% \text{ of } r = r + 0.10r = 1.10r\).

The new area is \(A_{new} = \pi (r_{new})^2 = \pi (1.10r)^2 = \pi (1.10)^2 r^2\).

\(A_{new} = \pi (1.21) r^2 = 1.21 (\pi r^2)\)

Since the original area is \(A = \pi r^2\), the new area is \(1.21 A\).

The increase in area is \(A_{new} - A = 1.21 A - A = 0.21 A\).

Percentage increase in area = \(\frac{0.21 A}{A} \times 100 = 0.21 \times 100 = 21\%\).

This confirms the previous calculation. A 10% increase in radius leads to a (1.10)^2 = 1.21 times increase in area, which is a 21% increase.

Revision Table: Circle Area Percentage Increase

Additional Information: Related Concepts

Understanding how changes in dimensions affect area and volume is important in geometry.

• Scaling of Area: If all linear dimensions (like radius or side length) of a shape are scaled by a factor \(k\), the area is scaled by a factor \(k^2\). In our case, the radius was scaled by a factor of \(\frac{99}{90} = \frac{11}{10} = 1.1\). The area is scaled by \((1.1)^2 = 1.21\). A scale factor of 1.21 means the new area is 1.21 times the original, which is a 21% increase.
• Scaling of Volume: Similarly, if all linear dimensions of a 3D shape are scaled by a factor \(k\), the volume is scaled by a factor \(k^3\).
• Percentage Change Formula: The general formula for percentage change is \(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%\) or \(\frac{\text{Change}}{\text{Original Value}} \times 100\%\). Both were used here.

These principles apply to many geometric shapes, not just circles.