If the circumference of a circle is increased by 20% then the area of the circle will be increased by:

Question

Prepp Team · Accepted Answer

Understanding Circle Circumference and Area Increase

This problem involves understanding the relationship between the circumference and area of a circle and how a percentage change in circumference affects the area.

Formulas for Circle

Let \(r\) be the radius of the circle.

Circumference (\(C\)): \(C = 2\pi r\)
Area (\(A\)): \(A = \pi r^2\)

Step-by-Step Calculation

Let the original circle have radius \(r_1\), circumference \(C_1\), and area \(A_1\).

Original Circumference: \(C_1 = 2\pi r_1\)
Original Area: \(A_1 = \pi r_1^2\)

The circumference is increased by 20%. Let the new circle have radius \(r_2\), circumference \(C_2\), and area \(A_2\).

New Circumference: \(C_2 = C_1 + 20\% \text{ of } C_1\)
\(C_2 = C_1 \times (1 + 0.20) = 1.20 C_1\)

Now, we relate the new circumference to the new radius:

\(C_2 = 2\pi r_2\)
Substitute the expression for \(C_2\): \(2\pi r_2 = 1.20 C_1\)
Substitute the expression for \(C_1\): \(2\pi r_2 = 1.20 (2\pi r_1)\)
Dividing both sides by \(2\pi\): \(r_2 = 1.20 r_1\)

This shows that if the circumference increases by 20%, the radius also increases by 20%.

Now, let's find the new area \(A_2\) using the new radius \(r_2\).

\(A_2 = \pi r_2^2\)
Substitute \(r_2 = 1.20 r_1\): \(A_2 = \pi (1.20 r_1)^2\)
\(A_2 = \pi (1.20)^2 r_1^2\)
\(A_2 = \pi (1.44) r_1^2\)

We know that the original area \(A_1 = \pi r_1^2\). So,

\(A_2 = 1.44 A_1\)

The new area is 1.44 times the original area.

To find the percentage increase in area, we use the formula:

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

Change in Area \( = A_2 - A_1\)
Percentage Increase \( = \frac{A_2 - A_1}{A_1} \times 100\%\)
Substitute \(A_2 = 1.44 A_1\): Percentage Increase \( = \frac{1.44 A_1 - A_1}{A_1} \times 100\%\)
Percentage Increase \( = \frac{(1.44 - 1) A_1}{A_1} \times 100\%\)
Percentage Increase \( = \frac{0.44 A_1}{A_1} \times 100\%\)
Percentage Increase \( = 0.44 \times 100\%\)
Percentage Increase \( = 44\%\)

Therefore, the area of the circle will be increased by 44%.

Summary Table of Changes

Measure	Original (relative)	Change	New (relative)	Formula
Circumference	\(C_1\)	+20%	\(C_2 = 1.20 C_1\)	\(C = 2\pi r\)
Radius	\(r_1\)	+20%	\(r_2 = 1.20 r_1\)	Derived from \(C = 2\pi r\)
Area	\(A_1\)	+44%	\(A_2 = 1.44 A_1\)	\(A = \pi r^2\)

Revision Table: Circle Properties

Property	Formula	Units
Circumference	\(2\pi r\) or \(\pi d\) (where \(d\) is diameter)	Units of length (e.g., cm, meters)
Area	\(\pi r^2\) or \(\frac{\pi d^2}{4}\)	Units of length squared (e.g., cm<sup>2</sup>, meters<sup>2</sup>)
Radius	Distance from center to edge	Units of length
Diameter	Distance across circle through center (\(d = 2r\))	Units of length

Additional Information: Percentage Change Concepts

When a quantity changes from an original value \(V_{original}\) to a new value \(V_{new}\), the percentage change is calculated as:

\(\text{Percentage Change} = \frac{V_{new} - V_{original}}{V_{original}} \times 100\%\)

If the percentage change is positive, it's an increase. If it's negative, it's a decrease.

In our problem, the original area is \(A_1\) and the new area is \(A_2 = 1.44 A_1\). The change is \(A_2 - A_1 = 1.44 A_1 - A_1 = 0.44 A_1\).

The percentage increase is \(\frac{0.44 A_1}{A_1} \times 100\% = 44\%\).

A shortcut to calculate the new value after a percentage increase:

\(V_{new} = V_{original} \times (1 + \frac{\text{Percentage Increase}}{100})\)

And for a percentage decrease:

\(V_{new} = V_{original} \times (1 - \frac{\text{Percentage Decrease}}{100})\)

In our case, the radius \(r_2 = r_1 \times (1 + \frac{20}{100}) = r_1 \times 1.20\). The area \(A_2 = A_1 \times (1 + \frac{44}{100}) = A_1 \times 1.44\), which matches our calculation \(A_2 = \pi (1.20 r_1)^2 = 1.44 \pi r_1^2 = 1.44 A_1\).