SSAT Upper Level Math : How to find the area of a circle

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

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Example Questions

Example Question #11 : Geometry

Give the area of a circle that circumscribes a right triangle with legs of length \(\displaystyle 10\) and \(\displaystyle 24\).

Possible Answers:

\(\displaystyle \frac{169 \pi}{3}\)

\(\displaystyle 338 \pi\)

\(\displaystyle \frac{676\pi}{3}\)

\(\displaystyle 169 \pi\)

\(\displaystyle \frac{169 \pi}{2}\)

Correct answer:

\(\displaystyle 169 \pi\)

Explanation:

If a right triangle is inscribed inside a circle, then the arc intercepted by the right angle is a semicircle, making the hypotenuse of triangle a diameter. 

The length of the hypotenuse of this triangle can be calculated using the Pythagorean Theorem:

\(\displaystyle d = \sqrt {10^{2}+24^{2}} = \sqrt{100+576} = \sqrt{676} = 26\)

The radius is half this, or 13, so the area is

\(\displaystyle A = \pi r^{2} = \pi \cdot 13 ^{2} = 169 \pi\)

Example Question #12 : Area And Circumference Of A Circle

\(\displaystyle 120 ^{\circ }\) central angle of a circle has a chord with length \(\displaystyle 15\). Give the area of the circle.

Possible Answers:

\(\displaystyle \frac{225 \pi}{4}\)

\(\displaystyle 225 \pi\)

\(\displaystyle \frac{75 \pi}{2}\)

\(\displaystyle \frac{225 \pi}{2}\)

\(\displaystyle 75 \pi\)

Correct answer:

\(\displaystyle 75 \pi\)

Explanation:

The figure below shows \(\displaystyle \angle AOB\), which matches this description, along with its chord \(\displaystyle \overline{AB}\) and triangle bisector \(\displaystyle \overline{OM}\)

 

 Chord

We will concentrate on \(\displaystyle \Delta AOM\), which is a 30-60-90 triangle.

Chord \(\displaystyle \overline{AB}\) has length 15, so \(\displaystyle AM = \frac{1}{2} AB = \frac{1}{2} \cdot 15 = \frac{15}{2}\)

By the 30-60-90 Theorem, 

\(\displaystyle OM = AM \div \sqrt{3}= \frac{15}{2} \div \sqrt{3}= \frac{15}{2\sqrt{3}}\)

and 

\(\displaystyle AO = 2 \cdot OM = 2 \cdot \frac{15}{2\sqrt{3}} = \frac{15}{ \sqrt{3}}\)

This is the radius, so the area is

\(\displaystyle A = \pi r^{2} = \pi \cdot \left ( \frac{15}{ \sqrt{3}} \right )^{2}= \frac{225}{3} \pi = 75 \pi\)

Example Question #1 : Area Of A Circle

What is the area of a circle that has a diameter of \(\displaystyle 15\) inches?

Possible Answers:

\(\displaystyle 153.938\)

\(\displaystyle 153.938\)

\(\displaystyle 940\)

\(\displaystyle 940\)

\(\displaystyle 960\)

\(\displaystyle 176.7146\)

\(\displaystyle 960\)

\(\displaystyle 706.8583\)

Correct answer:

\(\displaystyle 176.7146\)

Explanation:

The formula for finding the area of a circle is \(\displaystyle \pi r^{2}\). In this formula, \(\displaystyle r\) 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 \(\displaystyle 2\).

\(\displaystyle \frac{15}{2}=7.5\)

Now we use \(\displaystyle 7.5\) for \(\displaystyle r\) in our equation.

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

 

Example Question #3 : Area Of A Circle

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

Possible Answers:

\(\displaystyle 3\pi\)

\(\displaystyle 18\pi\)

\(\displaystyle 36\pi\)

\(\displaystyle 9\pi\)

Correct answer:

\(\displaystyle 9\pi\)

Explanation:

First, solve for radius:

\(\displaystyle r=\frac{d}{2}=\frac{6}{2}=3\)

Then, solve for area:

\(\displaystyle A=r^2\pi=3^2\pi=9\pi\)

Example Question #1 : Radius

The diameter of a circle is \(\displaystyle 4\ cm\). Give the area of the circle.

 

 

Possible Answers:

\(\displaystyle 13\ cm^2\)

\(\displaystyle 12 \ cm^2\)

\(\displaystyle 12.56\ cm^2\)

\(\displaystyle 13.56\ cm^2\)

\(\displaystyle 11.56\ cm^2\)

Correct answer:

\(\displaystyle 12.56\ cm^2\)

Explanation:

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

\(\displaystyle Area=\frac{\pi d^2}{4}\),

where \(\displaystyle d\) is the diameter of the circle, and \(\displaystyle \pi\) is approximately \(\displaystyle 3.14\).

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

Example Question #931 : Grade 7

The diameter of a circle is \(\displaystyle 4t\). Give the area of the circle in terms of \(\displaystyle t\).

Possible Answers:

\(\displaystyle 11.56 t^2\)

\(\displaystyle 12.56 t^2\)

\(\displaystyle 11.56 t\)

\(\displaystyle 12.56 t\)

\(\displaystyle 12 t^2\)

Correct answer:

\(\displaystyle 12.56 t^2\)

Explanation:

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

\(\displaystyle Area=\frac{\pi d^2}{4}\),

where \(\displaystyle d\)  is the diameter of the circle and \(\displaystyle \pi\) is approximately \(\displaystyle 3.14\).

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

\(\displaystyle \Rightarrow Area\approx 12.56t^2\)

Example Question #4 : Area Of A Circle

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

Possible Answers:

\(\displaystyle \frac{4}{\pi }\)

\(\displaystyle 4\)

\(\displaystyle 2\pi\)

\(\displaystyle 4\pi\)

\(\displaystyle 2\)

Correct answer:

\(\displaystyle 4\)

Explanation:

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

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

Example Question #13 : How To Find The Area Of A Circle

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

Possible Answers:

\(\displaystyle 12t\)

\(\displaystyle 11.56t^2\)

\(\displaystyle 12.56t^2\)

\(\displaystyle 12.56t\)

\(\displaystyle 11.56t\)

Correct answer:

\(\displaystyle 12.56t^2\)

Explanation:

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

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

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

 

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

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

 

Example Question #5 : Area Of A Circle

The circumference of a circle is \(\displaystyle 12.56\) inches. Find the area of the circle.

Let \(\displaystyle \pi = 3.14\).

Possible Answers:

\(\displaystyle 12\ in^2\)

\(\displaystyle 11.56\ in^2\)

\(\displaystyle 11\ in^2\)

\(\displaystyle 13.56\ in^2\)

\(\displaystyle 12.56\ in^2\)

Correct answer:

\(\displaystyle 12.56\ in^2\)

Explanation:

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

\(\displaystyle 12.56=2\times 3.14\times r\Rightarrow r=2\ in\) 

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

\(\displaystyle Area=\pi r^2=3.14\times 2^2=12.56\ in^2\)

Example Question #11 : How To Find The Area Of A Circle

Find the area of a circle with a radius of 100.

Possible Answers:

\(\displaystyle 1000\pi\)

\(\displaystyle 100\textup{,}}000\pi\)

\(\displaystyle 2500\pi\)

\(\displaystyle 10\textup{,}}000\pi\)

\(\displaystyle 100\pi\)

Correct answer:

\(\displaystyle 10\textup{,}}000\pi\)

Explanation:

Write the formula for a circle.

\(\displaystyle A=\pi r^2\)

Substitute the radius.

\(\displaystyle A=\pi r^2= \pi(100)^2= 10000\pi\)

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