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ISEE Upper Level Math : Radius

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Example Question #5 : Area Of A Circle

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

Possible Answers:

$4\pi$

$\frac{4}{\pi }$

$2\pi$

Correct answer:

Explanation:

The area of a circle can be calculated as $Area=\pi r^2$ , where is the radius of the circle, and $\pi$ is approximately 3.14 .

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

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Example Question #1 : How To Find The Area Of A Circle

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 .

Possible Answers:

12.56t

11.56t^2

12.56t^2

11.56t

12t

Correct answer:

12.56t^2

Explanation:

Chord length = $2\sqrt{r^2-d^2}$ , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length = $2\sqrt{r^2-d^2}\Rightarrow (2\sqrt{2})t=2\sqrt{r^2-(\sqrt{2}t)^2}$

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

$Area=\pi r^2$ , where is the radius of the circle and $\pi$ is approximately 3.14 .

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

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Example Question #41 : Circles

The circumference of a circle is 12.56 inches. Find the area of the circle.

Let $\pi = 3.14$ .

Possible Answers:

$11\ in^2$

$11.56\ in^2$

$12.56\ in^2$

$13.56\ in^2$

$12\ in^2$

Correct answer:

$12.56\ in^2$

Explanation:

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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Example Question #2 : How To Find The Area Of A Circle

Target

In the above figure, OA = AB = BC = CD .

What percent of the figure is shaded gray?

Possible Answers:

$62 \frac{1}{2} \%$

$66 \frac{2}{3} \%$

$50 \%$

$75 \%$

Correct answer:

$62 \frac{1}{2} \%$

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; 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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Example Question #3 : How To Find The Area Of A Circle

Target

In the above figure, OA = AB = BC = CD .

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

Possible Answers:

12 to 1

9 to 1

16 to 1

7 to 1

Correct answer:

7 to 1

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; 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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Example Question #4 : How To Find The Area Of A Circle

Target

The above figure depicts a dartboard, in which OA = AB = BC = CD .

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?

Possible Answers:

16 to 1

8 to 1

15 to 1

7 to 1

Correct answer:

15 to 1

Explanation:

For the sake of simplicity, we will assume that OA = AB = BC = CD = 1 ; 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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Example Question #11 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

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

Possible Answers:

$\dpi{100} 64$

$\dpi{100} 255$

$\dpi{100} 254$

$\dpi{100} 81$

Correct answer:

$\dpi{100} 64$

Explanation:

The formula for the area of a circle is

$\dpi{100} \pi r^{2}$

Find the radius by dividing 9 by 2:

$\dpi{100} \frac{9}{2}=4.5$

So the formula for area would now be:

$\dpi{100} \pi r^{2}=\pi (4.5)^{2}=20.25\pi \approx 63.6= 64$

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Example Question #11 : Radius

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

Possible Answers:

$25 \pi mi^2$

$250 \pi mi^2$

$10 \pi mi^2$

$125 \pi mi^2$

Correct answer:

$25 \pi mi^2$

Explanation:

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

Find the area of a circle with the following formula:

$A_{circle}=\pi r^2$

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

$A_{circle}=\pi (5mi)^2=25 \pi mi^2$

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Example Question #12 : Radius

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?

Possible Answers:

$12.25 \pi cm^2$

$36 \pi cm^2$

$9.25 \pi cm^2$

12.25 cm

Correct answer:

$12.25 \pi cm^2$

Explanation:

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:

$A_{circle}=\pi r^2$

Our radius is 3.5 cm, so plug that in:

$A_{circle}=\pi (3.5cm)^2=12.25 \pi cm^2$

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Example Question #13 : Radius

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

Possible Answers:

$81 \pi in^2$

$81 \pi in$

$36 \pi in^2$

$18 \pi in^2$

Correct answer:

$81 \pi in^2$

Explanation:

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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ISEE Upper Level Math : Radius

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #5 : Area Of A Circle

Example Question #1 : How To Find The Area Of A Circle

Example Question #41 : Circles

Example Question #2 : How To Find The Area Of A Circle

Example Question #3 : How To Find The Area Of A Circle

Example Question #4 : How To Find The Area Of A Circle

Example Question #11 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

Example Question #11 : Radius

Example Question #12 : Radius

Example Question #13 : Radius

All ISEE Upper Level Math Resources

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