Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #371 : Circles

Use 3.14 for Pi in the following question.

If the circumference of a circle is 25.12cm, what is the measure of its diameter?

Possible Answers:

\displaystyle 8\text{ cm}

\displaystyle 12.5\text{ cm}

\displaystyle 11.3\text{ cm}

\displaystyle 6\text{ cm}

\displaystyle 21.98\text{ cm}

Correct answer:

\displaystyle 8\text{ cm}

Explanation:

First we need to know that the formula for circumference of a circle is 

\displaystyle \text{Circumference}=\text{Diameter}\cdot \pi.

Since we know the circumference, we can plug it into our equation, along with pi, and solve for diameter. 

\displaystyle 25.12cm=D\cdot3.14 

To solve for D (diameter) we must divide both sides by pi  

\displaystyle 25.12cm\div3.14=D 

when we divide 25.12 by 3.14 we get 8, so our final answer is 8cm.

Example Question #64 : How To Find The Length Of The Diameter

 

A circular pizza has an area of \displaystyle 36\pi square inches. What is its diameter in inches?

Possible Answers:

\displaystyle 10

\displaystyle 12

\displaystyle 14

\displaystyle 8

\displaystyle 11

Correct answer:

\displaystyle 12

Explanation:

The area of a circle with a given radius \displaystyle r is \displaystyle A=\pi r^2. Since the area of the circular pizza is given as \displaystyle 36\pi square inches, we can solve for its radius as follows:

\displaystyle 36\pi=\pi r^2

\displaystyle \Rightarrow 36 = r^2

\displaystyle \Rightarrow r = \sqrt{36} = 6

Now that we know that the radius of the pizza is \displaystyle 6 inches, we can now deduce its diameter knowing that the diameter of a circle is equal to twice its radius. Hence, the diameter of this pizza is \displaystyle 2(6)=12 inches.

 

Example Question #291 : Geometry

Let \displaystyle A represent the area of a circle and \displaystyle C represent its circumference. Which of the following equations expresses \displaystyle A in terms of \displaystyle C

Possible Answers:

\displaystyle \frac{C}{4\pi}

\displaystyle \frac{C^2}{4\pi}

\displaystyle \frac{\pi C^2}{4}

\displaystyle \frac{C^2}{\pi}

\displaystyle C^2

Correct answer:

\displaystyle \frac{C^2}{4\pi}

Explanation:

The formula for the area of a circle is \displaystyle A=\pi r^2, and the formula for circumference is \displaystyle C=2\pi r. If we solve for C in terms of r, we get
\displaystyle r=C/2\pi.

We can then substitute this value of r into the formula for the area:

\displaystyle A=\pi r^2

\displaystyle =\pi (C/2\pi )^2

\displaystyle =C^2\pi /4\pi ^2

\displaystyle =C^2/4\pi

 

Example Question #1 : How To Find The Ratio Of Diameter And Circumference

A can of soup has a base area of \displaystyle 16\pi. What is the ratio of the can's diameter to its circumference?

Possible Answers:

\displaystyle \frac{2\pi}{1}

\displaystyle \frac{\pi}{8}

\displaystyle \frac{1}{2\pi}

\displaystyle \frac{\pi}{1}

\displaystyle \frac{1}{\pi}

Correct answer:

\displaystyle \frac{1}{\pi}

Explanation:

If the cirlcle has an area of \displaystyle 16\pi then we need to find a way to determine the diameter of the circle. We must do 4 things:

1. Find the radius of the circle since all we know is the area of the circle.

2. Double the radius to find the diameter.

3. Find the circumference  by \displaystyle C=\pi d

4. Write the ratio

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

\displaystyle 16\pi=\pi r^2

Solving to find the radius we get:

\displaystyle 16=r^2

\displaystyle 4=r

 

So if the radius is 4 then the diameter is:

\displaystyle d=2(4)=8

Then the circumference is:

\displaystyle C=8\pi

The ratio of the diameter to the circumference is:

\displaystyle \frac{d}{C}=\frac{8}{8\pi}=\frac{1}{\pi}

Example Question #372 : Circles

The circumference of the base of a silo is \displaystyle 144\pi ft. What is the ratio of the silo's circumference to its diameter?

Possible Answers:

\displaystyle 1:144

\displaystyle 144:1

\displaystyle 144:\pi

\displaystyle \pi:1

\displaystyle 1:\pi

Correct answer:

\displaystyle \pi:1

Explanation:

Since the silo has base circumference of \displaystyle 144\pi, then according to the formula for the circumference of a circle:

\displaystyle C=\pi d

\displaystyle 144\pi=\pi d

Solving for diameter:

\displaystyle 144=d

So the diameter is 144 ft. Therefore the ratio of the circumference to the diameter would be:

\displaystyle 144\pi:144\rightarrow \pi:1

Example Question #2 : How To Find The Ratio Of Diameter And Circumference

The diameter of a certain circle is tripled. Compared to the circumference of the original circle, how many times as large is the circumference of the new circle?

Possible Answers:

\displaystyle 9

\displaystyle 9\pi

\displaystyle 3

\displaystyle 3\pi

\displaystyle 6

Correct answer:

\displaystyle 3

Explanation:

The easiest way to find our answer is to try actual values. Imagine we have a circle with a diameter of \displaystyle 2. Given the formula for circumference

\displaystyle C=d\pi

our circumference is simply \displaystyle 2\pi. Tripling the diameter gives a new diameter of \displaystyle 6 and therefore a new circumference of \displaystyle 6\pi. We can then determine the ratio between the two circumferences.

\displaystyle \frac{6\pi}{2\pi}=3

Therefore, the new circumference is 3 times as large as the old.

Example Question #3 : How To Find The Ratio Of Diameter And Circumference

Find the ratio of the diameter to the circumference in a circle with radius r.

Possible Answers:

\displaystyle 2\pi{r}

\displaystyle 2r

\displaystyle \frac{1}{\pi}

\displaystyle \pi

Correct answer:

\displaystyle \frac{1}{\pi}

Explanation:

To find a ratio, simply divide your two quantities. Remember,

\displaystyle D=2r

\displaystyle C=2\pi{r}

Thus,

\displaystyle ratio=\frac{D}{C}=\frac{2r}{2\pi{r}}=\frac{1}{\pi}

Example Question #6 : How To Find The Ratio Of Diameter And Circumference

What is the circumference to diameter ratio of a circle with a diameter of 15?

Possible Answers:

\displaystyle 3.75

Cannot be determined.

\displaystyle 4.14

\displaystyle 4.75

\displaystyle 3.14

Correct answer:

\displaystyle 3.14

Explanation:

First you must find the circumference of a circle with a diameter of 15 with the formula \displaystyle C=\pi*D.

\displaystyle C=\pi*15=47.1

Since the ratio of C to D is represented by \displaystyle \pi=\frac{C}{D}, all of the ratios of Circumference to Diameter should approximately equal \displaystyle \pi.

\displaystyle \frac{47.1}{15}=3.14

Example Question #7 : How To Find The Ratio Of Diameter And Circumference

 

In the game of Sumo, both wrestlers are placed on the outer edge of the ring, on opposite sides of each other. If the wrestling ring is a perfect circle and has an area of \displaystyle 804.25\;ft^2, what would be the distance between both wrestlers?

Possible Answers:

\displaystyle 32\;ft

\displaystyle 64\;ft

\displaystyle 16\;ft

\displaystyle 56\;ft

\displaystyle 256\;ft

Correct answer:

\displaystyle 32\;ft

Explanation:

With the two wrestlers standing opposite each other on the edge of the ring, the distance between them would constitute the diameter of the circle. We can use the formula for circular area to find the diameter:

\displaystyle \\area=\pi\cdot r^2\\ \\804.25\;ft^2=3.14159\cdot r^2\\ \\r^2=\frac{804.25\;ft^2}{3.14159}=256\;ft^2\\ \\radius=\sqrt{256\;ft^2}=16\;ft\\ \\diameter=2\cdot16\;ft=32\;ft

Example Question #8 : How To Find The Ratio Of Diameter And Circumference

True or false: A circle with radius 1 has circumference less than 4.

Possible Answers:

True

False

Correct answer:

False

Explanation:

The circumference of a circle is equal to its radius multiplied by \displaystyle 2 \pi, so, if the radius of a circle is 1, its circumference is

\displaystyle 1 \times 2 \pi = 2\pi \approx 2 \cdot 3.14 \approx 6.28

This is greater than 4.

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