Let's break down this geometry problem step-by-step to understand how a percentage increase in the diameter of a circle affects its area.

Understanding Circle Diameter and Area

The area of a circle is calculated using its radius. The diameter is simply twice the radius.

• Original diameter = $D_1$
• Original radius = $r_1$
• Relationship: $D_1 = 2r_1$, so $r_1 = \frac{D_1}{2}$
• Original area = $A_1 = \pi r_1^2$

Substituting the relationship between radius and diameter into the area formula, we can also write the area in terms of diameter:

$A_1 = \pi \left(\frac{D_1}{2}\right)^2 = \pi \frac{D_1^2}{4}$

Calculating New Diameter and Radius

The question states that the diameter of the circle increases by 15%. Let the new diameter be $D_2$.

• Increase in diameter = 15% of $D_1 = 0.15 D_1$
• New diameter $D_2 = D_1 + 0.15 D_1 = (1 + 0.15) D_1 = 1.15 D_1$

Now, let's find the new radius, $r_2$. Since the new diameter is $D_2$, the new radius is half of it.

• New radius $r_2 = \frac{D_2}{2} = \frac{1.15 D_1}{2}$
• Since $D_1 = 2r_1$, we have $r_2 = \frac{1.15 \times (2r_1)}{2} = 1.15 r_1$

So, the new radius is 1.15 times the original radius.

Calculating the New Area of the Circle

Now we calculate the new area, $A_2$, using the new radius $r_2$:

$A_2 = \pi r_2^2 = \pi (1.15 r_1)^2 = \pi (1.15^2) r_1^2$

$A_2 = \pi (1.3225) r_1^2 = 1.3225 (\pi r_1^2)$

Since $A_1 = \pi r_1^2$, we can see that the new area is $A_2 = 1.3225 A_1$.

Alternatively, using the diameter form of the area formula:

$A_2 = \frac{\pi D_2^2}{4} = \frac{\pi (1.15 D_1)^2}{4} = \frac{\pi (1.15^2) D_1^2}{4} = 1.3225 \frac{\pi D_1^2}{4}$

Since $A_1 = \frac{\pi D_1^2}{4}$, we again find $A_2 = 1.3225 A_1$.

Determining Percentage Increase in Area

The percentage increase in area is calculated using the formula:

Percentage Increase = $\frac{\text{New Area} - \text{Original Area}}{\text{Original Area}} \times 100\%$

Percentage Increase = $\frac{A_2 - A_1}{A_1} \times 100\%$

Substitute $A_2 = 1.3225 A_1$ into the formula:

Percentage Increase = $\frac{1.3225 A_1 - A_1}{A_1} \times 100\%$

Percentage Increase = $\frac{(1.3225 - 1) A_1}{A_1} \times 100\%$

Percentage Increase = $\frac{0.3225 A_1}{A_1} \times 100\%$

Percentage Increase = $0.3225 \times 100\%$

Percentage Increase = $32.25\%$

Thus, if the diameter of a circle increases by 15%, its area increases by 32.25%.

Summary of Calculation

The increase in area is $A_2 - A_1 = 1.3225 A_1 - A_1 = 0.3225 A_1$.

Percentage increase is $\frac{0.3225 A_1}{A_1} \times 100\% = 32.25\%$.

Revision Table: Circle Properties

Additional Information: Percentage Change Concept

When a quantity increases by a certain percentage, the new quantity is the original quantity multiplied by $(1 + \text{percentage increase as a decimal})$.

• If diameter increases by 15%, the new diameter is $D_1 \times (1 + 0.15) = 1.15 D_1$.
• Since Area is proportional to the square of the diameter ($A \propto D^2$), the new area will be proportional to $(1.15 D_1)^2 = (1.15)^2 D_1^2 = 1.3225 D_1^2$.
• This means the new area is 1.3225 times the original area.
• The multiplier 1.3225 can be written as $(1 + 0.3225)$. This indicates a 0.3225 increase relative to the original area.
• To express this as a percentage, multiply by 100: $0.3225 \times 100\% = 32.25\%$.

This method highlights that if a linear dimension (like diameter or radius) changes by a factor 'k', the area (which depends on the square of the linear dimension) changes by a factor 'k$^2$'. Here $k=1.15$, and $k^2 = 1.15^2 = 1.3225$. The percentage change is $(k^2 - 1) \times 100\% = (1.3225 - 1) \times 100\% = 0.3225 \times 100\% = 32.25\%$.