How to find the area of a circle - ISEE Upper Level Quantitative Reasoning

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Question

What is the area of a circle with a diameter of , rounded to the nearest whole number?

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Answer

The formula for the area of a circle is

pi $r^{2}$

Find the radius by dividing 9 by 2:

$\frac{9}{2}$=4.5

So the formula for area would now be:

pi $r^{2}$=pi $(4.5)^{2}$=20.25pi approx 63.6= 64

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Question

What is the area of a circle that has a diameter of inches?

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Answer

The formula for finding the area of a circle is $\pi $r^{2}$$ . In this formula, represents the radius of the circle. Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius. In order to do this, we divide the diameter by .

$$\frac{15}{2}$=7.5$

Now we use for in our equation.

$\pi $(7.5)^{2}$=176.7146 : $in^{2}$$

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Question

What is the area of a circle with a diameter equal to 6?

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Answer

First, solve for radius:

$r=\frac{d}{2}$=\frac{6}{2}$=3$

Then, solve for area:

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Question

The diameter of a circle is $4\ cm$ . Give the area of the circle.

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Answer

The area of a circle can be calculated using the formula:

$Area=\frac{\pi $d^2$}{4}$$ ,

where is the diameter of the circle, and $\pi$ is approximately .

$Area=\frac{\pi $d^2$}{4}$=\frac{\pi\times $4^2$}{4}$=4\pi \Rightarrow Area\approx 4\times 3.14\Rightarrow Area\approx 12.56 \ $cm^2$$

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Question

The diameter of a circle is . Give the area of the circle in terms of .

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Answer

The area of a circle can be calculated using the formula:

$Area=\frac{\pi $d^2$}{4}$$ ,

where is the diameter of the circle and $\pi$ is approximately .

$Area=\frac{\pi $(4t)^2$}{4}$=\frac{16\pi $t^2$}{4}$=4\pi $t^2$ \Rightarrow Area\approx 4\times 3.14\times $t^2$$

$\Rightarrow Area\approx $12.56t^2$$

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Question

The radius of a circle is $$\frac{2}{\sqrt{\pi }$}$ . Give the area of the circle.

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Answer

The area of a circle can be calculated as $Area=\pi $r^2$$ , where is the radius of the circle, and $\pi$ is approximately .

$Area=\pi $r^2$=\pi\times $($\frac{2}{\sqrt{\pi}$})^2$=\pi\times $\frac{4}{\pi}$\Rightarrow Area=4$

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Question

The circumference of a circle is inches. Find the area of the circle.

Let $\pi = 3.14$ .

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Answer

First we need to find the radius of the circle. The circumference of a circle is $Circumference =2\pi r$ , where is the radius of the circle.

$12.56=2\times 3.14\times r\Rightarrow r=2\ in$

The area of a circle is $Area=\pi $r^2$$ where is the radius of the circle.

$Area=\pi $r^2$=3.14\times $2^2$=12.56\ $in^2$$

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Question

The perpendicular distance from the chord to the center of a circle is $$\sqrt{2}$t$ , and the chord length is $2$\sqrt{2}$t$ . Give the area of the circle in terms of .

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Answer

Chord length = , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length =

$\Rightarrow $8t^2$$=4(r^2$$-2t^2$)\Rightarrow $4r^2$$=16t^2$\Rightarrow $r^2$$=4t^2$\Rightarrow r=2t$

$Area=\pi $r^2$$ , where is the radius of the circle and $\pi$ is approximately .

$Area=\pi $r^2$=3.14\times $(2t)^2$$=12.56t^2$$

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Question

In the above figure, .

What percent of the figure is shaded gray?

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Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ :

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$

$A _{2}= \pi $r^{2}$ = \pi \cdot $2^{2}$ =4 \pi$

$A _{3}= \pi $r^{2}$ = \pi \cdot $3^{2}$ =9 \pi$

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

The outer gray ring is the region between the largest and second-largest circles, and has area

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$

The inner gray ring is the region between the second-smallest and smallest circles, and has area

$A _{2} - A _{1} = 4 \pi - \pi = 3 \pi$

The total area of the gray regions is

$7 \pi + 3 \pi = 10 \pi$

Since $10 \pi$ out of total area $16 \pi$ is gray, the percent of the figure that is gray is

$$\frac{10 \pi}{16 \pi }$ \times 100 % = $\frac{5}{8}$ \times 100 % =62 $\frac{1}{2}$ %$ .

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Question

In the above figure, .

Give the ratio of the area of the outer ring to that of the inner circle.

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Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ .

The areas of the largest circle and the second-largest circle are, respectively,

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

$A _{3}= \pi $r^{2}$ = \pi \cdot $3^{2}$ =9 \pi$

The difference of their areas, which is the area of the outer ring, is

$A _{4} - A _{3} = 16 \pi - 9 \pi = 7 \pi$ .

The inner circle has area

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$ .

The ratio of these areas is therefore

$\frac{7 \pi}{\pi}$ = $\frac{7}{1}$ , or 7 to 1.

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Question

The above figure depicts a dartboard, in which .

A blindfolded man throws a dart at the target. Disregarding any skill factor and assuming he hits the target, what are the odds against his hitting the white inner circle?

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Answer

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The inner and outer circles have radii 1 and 4, respectively, and their areas can be found by substituting each radius for in the formula $A = \pi $r^{2}$$ :

$A _{1}= \pi $r^{2}$ = \pi \cdot 1 ^{2} = \pi$ - this is the white inner circle.

$A _{4}= \pi $r^{2}$ = \pi \cdot $4^{2}$ =16 \pi$

The area of the portion of the target outside the white inner circle is $16 \pi - \pi = 15 \pi$ , so the odds against hitting the inner circle are

$\frac{15 \pi}{\pi}$ = $\frac{15}{1}$ - that is, 15 to 1 odds against.

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Question

A circle has a radius of 5 miles, what is its area?

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Answer

A circle has a radius of 5 miles, what is its area?

Find the area of a circle with the following formula:

We know that r is 5, so we can find our answer with the following:

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Question

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the area of the stamping surface?

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Answer

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the area of the stamping surface?

Find the area of a circle by using the following formula:

Our radius is 3.5 cm, so plug that in:

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Question

You have a circular window in your vacation room. It has a radius of 9 inches. What is the area of the window?

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Answer

You have a circular window in your vacation room. It has a radius of 9 inches. What is the area of the window?

To find the area of a circle, use the following formula:

$A=\pi $r^2$$

Now, we know the radius, so we just need to plug it in and solve.

$A=\pi (9 $in)^2$=81 \pi $in^2$$

So, our answer:

$81 \pi $in^2$$

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Question

Find the area of a circle with a radius of 6in.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot $r^2$$

Now, we know the radius of the circle is 6in.

Knowing this, we can substitute into the formula. We get

$A = \pi \cdot $(6$\text{in}$)^2$$

$A = \pi \cdot $36$\text{in}$^2$$

$A = 36\pi $\text{ $in}$^2$$

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Question

Find the area of a circle with a diameter of 10in.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot $r^2$$

where r is the radius of the circle.

Now, we know the diameter of the circle is 10in. We also know the diameter is two times the radius. Therefore, the radius is 5in.

Knowing this, we can substitute into the formula. We get

$A = \pi \cdot $(5$\text{in}$)^2$$

$A = \pi \cdot $25$\text{in}$^2$$

$A = 25\pi $\text{ $in}$^2$$

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Question

Find the area of a circle with a diameter of 14cm.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot $r^2$$

where r is the radius of the circle.

Now, we know the diameter of the circle is 14cm. We also know the diameter is two times the radius. Therefore, the radius is 7cm.

Knowing this, we can substitute into the formula. We get

$A = \pi \cdot $(7$\text{cm}$)^2$$

$A = \pi \cdot $49$\text{cm}$^2$$

$A = 49\pi $\text{ $cm}$^2$$

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Question

You have a tomato plant growing in a pot. If the base of the pot has a radius of 6 inches, find the area of the footprint of the pot.

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Answer

You have a tomato plant growing in a pot. If the base of the pot has a radius of 6 inches, find the area of the footprint of the pot.

To solve this problem, first recall the formula for area of a circle.

$A=\pi r ^2$

Now, we know our r=6 inches, so we just need to plug in and simplify.

$A=\pi (6in) ^2=36 \pi $in^2$$

So, our answer is:

$36 \pi $in^2$$

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Question

Let $\pi = 3.14$ .

Find the area of a circle with a diameter of 14cm. If necessary, round to the nearest tenths.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi \cdot $r^2$$

where r is the radius of the circle.

Now, we know the diameter of the circle is 14cm. We also know that the diameter is two times the radius. Therefore, the radius is 7cm.

We also know $\pi = 3.14$ .

Knowing all of this, we can substitute into the formula. We get

$A = 3.14 \cdot $(7$\text{cm}$)^2$$

$A = 3.14 \cdot $49$\text{cm}$^2$$

$A = $153.86$\text{cm}$^2$$

We were asked to round to the nearest tenth. So, we get

$A = $153.9$\text{cm}$^2$$

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Question

Find the area of a circle with a radius of 16in.

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Answer

To find the area of a circle, we will use the following formula:

$A = \pi $r^2$$

where r is the radius of the circle.

Now, we know the radius is 16in. Knowing this, we can substitute into the formula. We get

$A = \pi \cdot $(16$\text{in}$)^2$$

$A = \pi \cdot $256$\text{in}$^2$$

$A = 256\pi $\text{ $in}$^2$$

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