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Basic Geometry : Basic Geometry

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #236 : High School Math

In the following diagram, the radius is given. What is area of the shaded region?

Circle_box

Possible Answers:

$32 x^{2} -8\pi x^{2}$

$32 x^{2} - 16\pi x^{2}$

$16x^{2}$

$64 x^{2} - 16\pi x^{2}$

$64 x^{2} -8\pi x^{2}$

Correct answer:

$32 x^{2} -8\pi x^{2}$

Explanation:

This question asks you to apply the concept of area in finding both the area of a circle and square. Since the cirlce is inscribed in the square, we know that its diameter (two times the radius) is the same length as one side of the square. Since we are given the radius, , we can find the area of both the circle and square.

Square:

This gives us the area for the entire square.

The bottom half of the square has area $\frac{1}{2} \cdot 64x^{}2 = 32x^{}2$ .

Now that we have this value, we must find the area that the circle occupies. The area of a circle is given by $\pi \cdot r^{}2$ .

So the area of this circle will be $\pi \cdot 16x^{}2$ .

The bottom half of the circle has half that area:

$\rightarrow \pi \cdot 8x^{}2$

Now that we have both our values, we can subtract the bottom half of the circle from the bottom half of the square to give us the shaded region:

$32 x^{}2 - \pi \cdot 8x^{}2$

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Example Question #41 : Basic Geometry

Find the area of a circle that has a radius of .

Possible Answers:

$6378\pi$

$192\pi$

$8100\pi$

$9216\pi$

Correct answer:

$9216\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

$\text{Area}=\pi \times (96)^2=9216\pi$

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Example Question #42 : Basic Geometry

Find the area of a circle that has a radius of $\frac{1}{5}$ .

Possible Answers:

$\frac{1}{10}\pi$

$\frac{1}{25}\pi$

$\frac{1}{5}$

$\frac{2}{5}\pi$

Correct answer:

$\frac{1}{25}\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

In our particular case the radius is $\frac{1}{5}$ .

$\text{Area}=\pi \times \left(\frac{1}{5}\right)^2=\frac{1}{25}\pi$

When squarring a fraction we need to square both the numerator and the denominator.

$\pi \times \left(\frac{1}{5}\right)^2=\frac{1^2}{5^2}\pi=\frac{1\cdot 1}{5\cdot 5}\pi=\frac{1}{25}\pi$

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Example Question #12 : Area Of A Circle

Find the area of a circle that has a radius of $\sqrt5$ .

Possible Answers:

$\sqrt{10}\pi$

$\sqrt{5}\pi$

$2\sqrt5 \pi$

$5\pi$

Correct answer:

$5\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

In this problem the known radius is $\sqrt5$

Now plug the radius into the area equation.

$\text{Area}=\pi \times \text{radius}^2$

Therefore we get,

$\text{Area}=\pi \times (\sqrt5)^2=5\pi$

Recall that when a square root is squared you are left with the number under the square root sign. This happens because when you square a number you are multiplying it by itself. In our case this is,

$(\sqrt{5})^2=\sqrt{5}\cdot \sqrt{5}$ .

From here we can use the property of multiplication and radicals to rewrite our expression as follows,

$\sqrt{5}\cdot \sqrt{5}=\sqrt{5\cdot 5}$

and when there are two numbers that are the same under a square root sign you bring out one and the other number and square root sign go away.

$\sqrt{5\cdot 5}=5$

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Example Question #43 : Plane Geometry

Find the area of a circle that has a radius of $\sqrt7$ .

Possible Answers:

$14\pi$

$7\pi$

$2\sqrt7\pi$

$\sqrt7\pi$

Correct answer:

$7\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

We know the radius is $\sqrt7$ .

Now, plug the radius into the area equation.

$\text{Area}=\pi \times (\sqrt7)^2=7\pi$

Recall that when a square root is squared you are left with the number under the square root sign. This happens because when you square a number you are multiplying it by itself. In our case this is,

$(\sqrt{7})^2=\sqrt{7}\cdot \sqrt{7}$ .

From here we can use the property of multiplication and radicals to rewrite our expression as follows,

$\sqrt{7}\cdot \sqrt{7}=\sqrt{7\cdot 7}$

and when there are two numbers that are the same under a square root sign you bring out one and the other number and square root sign go away.

$\sqrt{7\cdot 7}=7$

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Example Question #43 : Basic Geometry

Find the area of a circle that has a radius of $\sqrt{11}$ .

Possible Answers:

$\sqrt{22}\pi$

$11\pi$

$2\sqrt{11}\pi$

$\sqrt{11}\pi$

Correct answer:

$11\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

We know the radius is $\sqrt{11}$ therefore, the area equation becomes,

$\text{Area}=\pi \times (\sqrt{11})^2=11\pi$ .

Recall that when a square root is squared you are left with the number under the square root sign. This happens because when you square a number you are multiplying it by itself. In our case this is,

$(\sqrt{11})^2=\sqrt{11}\cdot \sqrt{11}$ .

From here we can use the property of multiplication and radicals to rewrite our expression as follows,

$\sqrt{11}\cdot \sqrt{11}=\sqrt{11\cdot 11}$

and when there are two numbers that are the same under a square root sign you bring out one and the other number and square root sign go away.

$\sqrt{11\cdot 11}=11$

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Example Question #44 : Basic Geometry

Find the area of the circle that has a radius of 0.8 .

Possible Answers:

$0.8\pi$

$1.6\pi$

$0.64\pi$

$0.04\pi$

Correct answer:

$0.64\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

Our radius is 0.8 therefore,

$\text{Area}=\pi \times (0.8)^2=0.64\pi$ .

Remember when multiplying decimals together first move the decimal over so that the number is a whole integer.

$0.8\rightarrow 8$

Now we multiple the integers together.

$0.8\cdot 0.8\rightarrow 8\cdot 8=64$

From here, we need to move the decimal place back. In this particular problem we moved the decimal over one time for each number for a total of two decimal places.

Therefore our r^2 becomes,

$64\rightarrow 0.64$ .

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Example Question #45 : Radius

Find the area of the circle that has a radius of 0.1 .

Possible Answers:

$0.01\pi$

$0.2\pi$

$10\pi$

$\pi$

Correct answer:

$0.01\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

Remember when multiplying decimals together first move the decimal over so that the number is a whole integer.

$0.1\rightarrow 1$

Now we multiple the integers together.

$0.1\cdot 0.1\rightarrow 1\cdot 1=1$

From here, we need to move the decimal place back. In this particular problem we moved the decimal over one time for each number for a total of two decimal places.

Therefore our r^2 becomes,

$1\rightarrow 0.01$ .

$\text{Area}=\pi \times (0.1)^2=0.01\pi$

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Example Question #46 : Radius

Find the area of the circle with a radius of 0.3 .

Possible Answers:

$0.9\pi$

$0.09\pi$

$1.2\pi$

$0.6\pi$

Correct answer:

$0.09\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

Remember when multiplying decimals together first move the decimal over so that the number is a whole integer.

$0.3\rightarrow 3$

Now we multiple the integers together.

$0.3\cdot 0.3\rightarrow 3\cdot 3=9$

From here, we need to move the decimal place back. In this particular problem we moved the decimal over one time for each number for a total of two decimal places.

Therefore our r^2 becomes,

$9\rightarrow 0.09$ .

$\text{Area}=\pi \times (0.3)^2=0.09\pi$

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Example Question #45 : Basic Geometry

Find the area of a circle that has a radius of 0.9 .

Possible Answers:

$1.8\pi$

$0.9\pi$

$8.1\pi$

$0.81\pi$

Correct answer:

$0.81\pi$

Explanation:

Use the following formula to find the area of a circle:

$\text{Area}=\pi \times \text{radius}^2$

For the circle in question, plug in the given radius to find the area.

Remember when multiplying decimals together first move the decimal over so that the number is a whole integer.

$0.9\rightarrow 9$

Now we multiple the integers together.

$0.9\cdot 0.9\rightarrow 9\cdot 9=81$

From here, we need to move the decimal place back. In this particular problem we moved the decimal over one time for each number for a total of two decimal places.

Therefore our r^2 becomes,

$81\rightarrow 0.81$ .

$\text{Area}=\pi \times (0.9)^2=0.81\pi$

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Basic Geometry : Basic Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #236 : High School Math

Example Question #41 : Basic Geometry

Example Question #42 : Basic Geometry

Example Question #12 : Area Of A Circle

Example Question #43 : Plane Geometry

Example Question #43 : Basic Geometry

Example Question #44 : Basic Geometry

Example Question #45 : Radius

Example Question #46 : Radius

Example Question #45 : Basic Geometry

All Basic Geometry Resources

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