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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

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9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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Example Question #61 : Plane Geometry

A circle is inscribed in a square that has an area of 400 . What is the area of the circle?

Possible Answers:

$100\pi$

100

$200\pi$

150

Correct answer:

$100\pi$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{400}=20$

$\text{side length}=\text{diameter}=20$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{20}{2}=10$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times 10^2=100\pi$

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Example Question #62 : Plane Geometry

A circle is inscribed in a square that has an area of 784 . What is the area of the circle?

Possible Answers:

$196\pi$

$122\pi$

$148\pi$

$216\pi$

Correct answer:

$196\pi$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{784}=28$

$\text{side length}=\text{diameter}=28$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{28}{2}=14$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times 14^2=196\pi$

Report an Error

Example Question #63 : Plane Geometry

A circle is inscribed in a square that has an area of 324 . What is the area of the circle?

Possible Answers:

$90\pi$

$128\pi$

$81\pi$

Correct answer:

$81\pi$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{324}=18$

$\text{side length}=\text{diameter}=18$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{18}{2}=9$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times 9^2=81\pi$

Report an Error

Example Question #64 : Plane Geometry

A circle is inscribed in a square that has an area of 484 . What is the area of the circle?

Possible Answers:

$144\pi$

$100\pi$

$121\pi$

Correct answer:

$121\pi$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{484}=22$

$\text{side length}=\text{diameter}=22$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{22}{2}=11$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times 11^2=121\pi$

Report an Error

Example Question #61 : Plane Geometry

A circle is inscribed in a square that has an area of $\frac{1}{4}$ . What is the area of the circle?

Possible Answers:

$\frac{\pi}{32}$

$\frac{\pi}{16}$

$\frac{\pi}{8}$

$\frac{\pi}{4}$

Correct answer:

$\frac{\pi}{16}$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

$\text{side length}=\text{diameter}=\frac{1}{2}$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{\frac{1}{2}}{2}=\frac{1}{4}$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times (\frac{1}{4})^2=\frac{\pi}{16}$

Report an Error

Example Question #66 : Plane Geometry

A circle is inscribed in a square that has an area of 1024 . What is the area of the circle?

Possible Answers:

$200\pi$

$212\pi$

$256\pi$

$164\pi$

Correct answer:

$256\pi$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{1024}=32$

$\text{side length}=\text{diameter}=32$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{32}{2}=16$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times 14^2=256\pi$

Report an Error

Example Question #62 : Plane Geometry

A circle is inscribed in a square that has an area of 900 . What is the area of the circle?

Possible Answers:

$300\pi$

$275\pi$

$30\pi$

$225\pi$

Correct answer:

$225\pi$

Explanation:

The following figure illustrates the situation put forth by the question.

Notice that the diameter of the circle has the same length as a side of the square.

First, find the side length of the square.

$\text{Area of Square}=\text{side length}^2$

$\text{side length}=\sqrt{\text{Area of square}}$

Plug in the value given for the area to find the side length.

$\text{side length}=\sqrt{900}=30$

$\text{side length}=\text{diameter}=30$

Now, recall how to find the area of a circle.

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

We also know the relationship between the radius and the diameter.

$\text{diameter}=2\times\text{radius}$

Use the value you found earlier for the diameter to get the length of the radius.

$\text{radius}=\frac{\text{diameter}}{2}$

$\text{radius}=\frac{30}{2}=15$

Plug this value for the radius back into the equation for the area of a circle.

$\text{Area}=\pi \times 15^2=225\pi$

Report an Error

Example Question #68 : Plane Geometry

Find the area of the circle with the given diameter.

Possible Answers:

$240\pi$

$24\pi$

$144\pi$

$168\pi$

Correct answer:

$144\pi$

Explanation:

Recall how to find the area of a circle.

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

Now, since we have the diameter, recall the relationship between the radius and the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the value of the diameter to find the value of the radius.

$\text{Radius}=\frac{24}{2}$

Simplify.

$\text{Radius}=12$

Substitute in the value of the radius to find the area.

$\text{Area}=\pi\times 12^2$

Solve.

$\text{Area}=144\pi$

Report an Error

Example Question #69 : Plane Geometry

Find the area of the circle with the given diameter.

Possible Answers:

$676\pi$

$104\pi$

$338\pi$

$169\pi$

Correct answer:

$169\pi$

Explanation:

Recall how to find the area of a circle.

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

Now, since we have the diameter, recall the relationship between the radius and the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the value of the diameter to find the value of the radius.

$\text{Radius}=\frac{26}{2}$

Simplify.

$\text{Radius}=13$

Substitute in the value of the radius to find the area.

$\text{Area}=\pi\times 13^2$

Solve.

$\text{Area}=169\pi$

Report an Error

Example Question #70 : Plane Geometry

Find the area of the circle with the given diameter.

Possible Answers:

$140\pi$

$170\pi$

$196\pi$

$156\pi$

Correct answer:

$196\pi$

Explanation:

Recall how to find the area of a circle.

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

Now, since we have the diameter, recall the relationship between the radius and the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the value of the diameter to find the value of the radius.

$\text{Radius}=\frac{28}{2}$

Simplify.

$\text{Radius}=14$

Substitute in the value of the radius to find the area.

$\text{Area}=\pi\times 14^2$

Solve.

$\text{Area}=196\pi$

Report an Error

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Basic Geometry : Circles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #61 : Plane Geometry

Example Question #62 : Plane Geometry

Example Question #63 : Plane Geometry

Example Question #64 : Plane Geometry

Example Question #61 : Plane Geometry

Example Question #66 : Plane Geometry

Example Question #62 : Plane Geometry

Example Question #68 : Plane Geometry

Example Question #69 : Plane Geometry

Example Question #70 : Plane Geometry

All Basic Geometry Resources

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