High School Math : Geometry

Study concepts, example questions & explanations for High School Math

varsity tutors app store varsity tutors android store

Example Questions

Example Question #114 : Circles

What is the ratio of the diameter of a circle to the circumference of the same circle?

Possible Answers:

\(\displaystyle \pi\)

\(\displaystyle \frac{1}{\pi}\)

\(\displaystyle D\pi\)

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

Correct answer:

\(\displaystyle \frac{1}{\pi}\)

Explanation:

To find the ratio we must know the equation for the circumference of a circle. In this equation, \(\displaystyle C\) is the circumference and \(\displaystyle D\) is the diameter.

\(\displaystyle C=D\pi\)

Once we know the equation, we can solve for the ratio of the diameter to circumference by solving the equation for \(\displaystyle \frac{D}{C}\). We do this by dividing both sides by .

\(\displaystyle \frac{C}{\pi}=\frac{D\pi}{\pi}\)

\(\displaystyle \frac{C}{\pi}=D\)

Then we divide both sides by the circumference.

\(\displaystyle \frac{C}{\pi C}=\frac{1}{\pi}=\frac{D}{C}\)

We now know that the ratio of the diameter to circumference is equal to \(\displaystyle \frac{1}{\pi}\).

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

What is the ratio of the diameter and circumference of a circle?

 

Possible Answers:

\(\displaystyle 2\pi\)

\(\displaystyle \pi\)

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

Correct answer:

Explanation:

To find the ratio we must know the equation for the circumference of a circle is

\(\displaystyle Circumference=\pi d\)

Once we know the equation we can solve for the ratio of the diameter to circumference by solving the equation for \(\displaystyle \frac{diameter}{Circumference}\)

we divide both sides by the circumference giving us 

\(\displaystyle \frac{d}{\pi d}=\frac{1}{\pi }\)

We now know that the ratio of the diameter to circumference is equal to .

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

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:

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

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

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

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

\(\displaystyle C^2\)

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

What is the ratio of any circle's circumference to its radius?

Possible Answers:

\(\displaystyle \pi ^{2}:1\)

\(\displaystyle 2\pi :1\)

 

\(\displaystyle 3.14:1\)

\(\displaystyle \pi :1\)

Undefined.

Correct answer:

\(\displaystyle 2\pi :1\)

 

Explanation:

The circumference of any circle is 

\(\displaystyle 2\pi*r\)

So the ratio of its circumference to its radius r, is

\(\displaystyle \frac{2\pi*r}{r}\)

\(\displaystyle =2\pi\)

 

Example Question #1 : How To Find The Angle Of Clock Hands

Find the angle between the minute and hour hand at 8:20 PM.  

Possible Answers:

\(\displaystyle 130^{\circ}\)

\(\displaystyle 115^{\circ}\)

\(\displaystyle 125^{\circ}\)

\(\displaystyle 120^{\circ}\)

\(\displaystyle 127.5^{\circ}\)

Correct answer:

\(\displaystyle 130^{\circ}\)

Explanation:

The distance between each notch on the clock is 6 degrees because there are 360 degrees on the clock, and there are 60 notches total. The minute hand is at notch #20, and so it is 120 degrees from the top. The hour hand is a little past notch #40 because the time is a little past hour 8. Thus, the hour hand is a little past 240 degrees from the top, going clockwise (\(\displaystyle 40\times 6\) degrees).

In each hour, the hour hand goes 5 notches, or 30 degrees.  Because it is now 20 minutes past the hour, a third of an hour has passed.  One third of 30 degrees is 10 degrees.  Thus, the hour hand is 10 degrees past notch #40.  The hour hand is 250 degrees from the top, going clockwise.  The angle between the two hands is thus 130 degrees.  

Example Question #1 : How To Find The Angle Of Clock Hands

When making a pie chart, how many degrees should be allotted for \(\displaystyle 12.5\) percent?

Possible Answers:

\(\displaystyle 45^{\circ}\)

\(\displaystyle 90^{\circ}\)

\(\displaystyle 30^{\circ}\)

\(\displaystyle 12.5^{\circ}\)

\(\displaystyle 60^{\circ}\)

Correct answer:

\(\displaystyle 45^{\circ}\)

Explanation:

This is a proportion problem

 \(\displaystyle \frac{12.5}{100}=\frac{x \; degrees}{360\; degrees}\) so there are \(\displaystyle 45^{\circ}\) in \(\displaystyle 12.5\) percent of a circle

Example Question #1 : How To Find The Angle Of Clock Hands

If it is 2:00 PM, what is the measure of the angle between the minute and hour hands of the clock?

Possible Answers:

90 degrees

30 degrees

120 degrees

45 degrees

60 degrees

Correct answer:

60 degrees

Explanation:

First note that a clock is a circle made of 360 degrees, and that each number represents an angle and the separation between them is 360/12 = 30. And at 2:00, the minute hand is on the 12 and the hour hand is on the 2.  The correct answer is 2 * 30 = 60 degrees.

Example Question #1 : How To Find The Angle Of Clock Hands

A clock currently reads 2:00. What is the size of the angle formed between the hour and minute hands?

Possible Answers:

\(\displaystyle 60^{o}\)

\(\displaystyle 90^o\)

\(\displaystyle 15^o\)

\(\displaystyle 30^o\)

Correct answer:

\(\displaystyle 60^{o}\)

Explanation:

The interior angle of a sector is equal the the angle of the sector. If the entire circumference was the sector, it would equal \(\displaystyle \small 360^o\). Additionally, if it were 12:00, the angle would be equal to \(\displaystyle \small 360^o\). We can solve the problem by setting up a proportion. 2:00 will be two-twelfths of the circle past the 12:00 mark, and will be at an angle of \(\displaystyle \small x^o\).

\(\displaystyle \frac{2}{12}=\frac{x^o}{360^o}\)

Cross multiply and solve for \(\displaystyle \small x^o\).

\(\displaystyle 720^o=12x^o\)

\(\displaystyle x=60^o\)

Example Question #1 : How To Find The Angle Of Clock Hands

It is 4 o’clock.  What is the measure of the angle formed between the hour hand and the minute hand?

Possible Answers:

\(\displaystyle 30^\circ\)

\(\displaystyle 60^\circ\)

\(\displaystyle 90^\circ\)

\(\displaystyle 120^\circ\)

\(\displaystyle 180^\circ\)

Correct answer:

\(\displaystyle 120^\circ\)

Explanation:

At four o’clock the minute hand is on the 12 and the hour hand is on the 4.  The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

Example Question #141 : Geometry

If a clock reads 8:15 PM, what angle do the hands make?

Possible Answers:

\(\displaystyle 150^o\)

\(\displaystyle 240^o\)

\(\displaystyle 90^o\)

\(\displaystyle 157.5^o\)

\(\displaystyle 138.5^o\)

Correct answer:

\(\displaystyle 157.5^o\)

Explanation:

A clock is a circle, and a circle always contains 360 degrees. Since there are 60 minutes on a clock, each minute mark is 6 degrees.

\(\displaystyle \frac{360^o\ \text{total}}{60\ \text{minutes total}}=6\ \text{degrees per minute}\)

The minute hand on the clock will point at 15 minutes, allowing us to calculate it's position on the circle.

\(\displaystyle (15\ min)(6)=90^o\)

Since there are 12 hours on the clock, each hour mark is 30 degrees.

\(\displaystyle \frac{360^o\ \text{total}}{12\ \text{hours total}}=30\ \text{degrees per hour}\)

We can calculate where the hour hand will be at 8:00.

\(\displaystyle (8\ hr)(30)=240^o\)

However, the hour hand will actually be between the 8 and the 9, since we are looking at 8:15 rather than an absolute hour mark. 15 minutes is equal to one-fourth of an hour. Use the same equation to find the additional position of the hour hand.

\(\displaystyle 240^o+(\frac{1}{4}\ hr)(30)\)

\(\displaystyle 240^o+7.5^o\)

\(\displaystyle 247.5^o\)

We are looking for the angle between the two hands of the clock. The will be equal to the difference between the two angle measures.

\(\displaystyle \angle=247.5^o-90^o=157.5^o\)

Learning Tools by Varsity Tutors