# Angles as Fractions of a Circle Rating

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Angles as Fractions of a Circle
CCSS.MATH.CONTENT.4.MD.C.5.A

## Exploring Angles in a Circle

Have you ever thought about the thrilling rides at amusement parks and wondered how they relate to math? Join us on an imaginary Ferris wheel adventure, where we'll delve into the fascinating world of angles in a circle.

## Introducing Angles as Fractional Parts of a Circle

Next time you find yourself on a Ferris wheel, consider how the concept of angles plays a role in this ride. Angles, circles and angles in a circle are everywhere, from the hands of a clock to the wheels of a car, and even in the thrilling rides at amusement parks.

## Angles as Fractional Parts of a Circle – Definition and Rules

Let’s revise how angles occur first:

An angle is created when two rays meet at the same point at the center of a circle.

When an angle rotates or turns, a total of three hundred sixty degrees, it forms a circle. We can divide a circle into fractional parts and measure the degrees of those angles. ## Angles as Fractional Parts of a Circle - Commonly Used Fractions

Understanding angles can be made easier by looking at fractions of a circle. A quarter turn of a circle forms a right angle that measures ninety degrees, a half-circle measures one hundred eighty degrees, and three-fourths equals two hundred seventy degrees.

Here is the table with commonly used angles that are expressed as fractions of a circle:

Fraction of a Circle Parts in a Circle Calculation Angle size
$\frac{1}{2}$ 2 360÷2 180 degrees
$\frac{1}{4}$ 4 360÷4 90 degrees
$\frac{3}{4}$ 4 360÷4 = 90, 90 x 3 = 270 270 degrees ### Understanding Reflex Angles

Did you know there's a special type of angle called a reflex angle? These angles are greater than one hundred eighty degrees but less than three hundred sixty. They are intriguing and can be found in various shapes and patterns.

## How to Find Angles in a Circle?

Calculating the degree of a fractional part of a circle is a fun puzzle to solve. Take the total degree of three hundred sixty and divide it by the number of parts, then multiply by the shaded parts. For example, four-tenths of three hundred sixty equals one hundred forty-four degrees.

Take a look at the table for more examples of how to calculate angles that are fractional parts of a circle

Fraction of a Circle Parts in a Circle Shaded Parts Calculation Angle size
$\frac{2}{6}$ 6 2 360÷6= 60, 60 x 2 = 120 120 degrees
$\frac{5}{8}$ 8 5 360÷8 = 45, 45 x 5 = 225 225 degrees
$\frac{4}{10}$ 10 4 360÷10 = 36, 36 x 4 = 144 144 degrees ## Angles in a Circle – Summary

Angles and fractions aren't just in textbooks; they're part of everyday life. From the architectural marvels to the simple door hinge, you'll find angles everywhere. Try finding angles in circles and other circular objects around you and practice your newfound skills.

Following the video, there is additional practice with exercises and angles in a circle worksheet.

Angles as fractional parts of a circle is a fascinating concept that connects math with the world around us. Whether it's a thrilling Ferris wheel ride or the hands of a clock, angles are everywhere. Dive into this adventure, explore, and have fun learning!

What is a right angle?
What is a reflex angle?
How to calculate degrees of a fraction of a circle?
What are clockwise and counterclockwise rotations?
Can angles relate to real life?
How to practice understanding angles and fractions?
What's a full turn of an angle?
How is a Ferris wheel related to angles and fractions?

### TranscriptAngles as Fractions of a Circle

"Nico, what's going on?" "The ride has stopped! "What are we going to do?" "Don't worry,(...) I have a plan to get us out of here!" As Nico and Nia begin their daring escape down from the Ferris Wheel, we can take a look at… “Angles as Fractions of a Circle.” An angle is created when two rays meet at the same point at the center of a circle. An angle can rotate clockwise OR counterclockwise. When an angle rotates or turns, a total of three hundred sixty degrees, it forms a circle. We can divide a circle into fractional parts and measure the degrees of those angles. First, let’s look at some benchmark fractions made by turns of a circle. This is one-fourth of a circle. Look at the angle created by this quarter turn. One-fourth of a circle creates a right angle that measures ninety degrees. This is a half-circle,(...) and the angle measures one hundred eighty degrees. Three-fourths of a circle is equal to two hundred-seventy degrees. This is also known as a REFLEX ANGLE. A reflex angle is any angle that is greater than one hundred eighty degrees, but less than three hundred sixty degrees. A full turn of the angle is a circle and measures three hundred sixty degrees. We can figure out the degrees of ANY fraction of a circle by solving for how many parts out of three hundred sixty degrees it covers. Let’s use Nico and Nia’s daring escape from the Ferris wheel to measure fractions of a circle. Their climb down has covered fourth-tenths of the Ferris wheel. To solve for the degrees of this area, we are going to find four-tenths of three hundred sixty degrees. To start, we are going to take the three hundred-sixty degrees angle turn... and divide it by the total number of parts. The circle is in ten parts. Three hundred-sixty divided by ten equals thirty-six. This means each one-tenth fraction has the value of thirty-six degrees. Now, we multiply the product by the numerator or shaded parts. There are four shaded parts. Thirty-six times four equals (...) one hundred forty-four. Four-tenths of three hundred-sixty is equal to one hundred forty-four degrees. That means Nico and Nia turned one hundred forty-four degrees on the Ferris Wheel. “Woah,(...) something is happening! HANG ON!” The Ferris wheel started moving counterclockwise and traveled backwards five-twelfths of the circle. To solve for the number of degrees the wheel moved, we are going to find five-twelfths of three hundred-sixty. First, what is three hundred-sixty divided by twelve? (...) It is thirty. Each fraction has the value of thirty degrees. Now, multiply thirty by the numerator, five. What is thirty times five? (...) One hundred fifty That means they went counterclockwise(...)One hundred-fifty degrees. "This machine has gone haywire!" What fraction of the circle are we solving for? (...) We are finding three-fifths of three hundred-sixty. What do we do first? (...) We divide three hundred-sixty by five. What is three hundred sixty-five divided by five? (...) Seventy-two. What is the next step? (...) Multiply seventy-two by the numerator. What is seventy-two times three? (...) Two hundred sixteen degrees. Remember... an angle is created when two rays meet at the same point at the center of a circle. When an angle rotates or turns three hundred sixty degrees, it forms a circle. We can divide a circle into fractional parts and measure the degrees of those angles. To solve for the degree measurement of a fraction... First, divide three hundred sixty by the denominator, or number of parts the circle is divided into. Then, take the quotient and multiply it to the numerator, or number of shaded parts. [Nico and Nia near the bottom cart of the Ferris wheel, ready to jump and relieved they are safe. Just as they are about to leap off, the ride starts back up. Nico and Nia's eyes go wide! ” Here we go again!”

## Angles as Fractions of a Circle exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Angles as Fractions of a Circle .

Hints

A half circle measures 180°.

A circle can be divided into several parts.

Solution
• A circle measures a total of 360°.
• An angle that measures greater than 180° but less than 360° is called a reflex angle.
• When an angle rotates 360° it forms a circle.
• A circle can be divided into fractional parts.
• Two rays meeting at the same point at the center of a circle form an angle.
• ### Match the angles with the fractions.

Hints

A circle can be divided into four equal parts.

The total turn of an angle to form a circle is 360°.

$\frac{1}{4}$ of a circle creates a right angle that measures 90°.

Solution
• 90° = $\frac{1}{4}$
• 180° = $\frac{1}{2}$
• 270° = $\frac{3}{4}$
• 360°= $\frac{4}{4}$ or 1
• ### Which statements are true?

Hints

$\frac{1}{4}$ of a circle measures 90°.

A circle can be divided into several fractional parts.

A full circle measures 360°.

An angle is formed from the center of a circle.

There are four correct options.

Solution
• False: A circle is formed when an angle rotates a total of 270°.
• True: An angle can rotate clockwise or counter-clockwise.
• False: A circle can only be divided into four fractional parts.
• True: A half circle measures 180°.
• True: A reflex angle is greater than 180° but less than 360°.
• True: An angle is formed by two rays meeting at a center point.
• ### Can you figure out the angle of the Ferris wheel turning so far?

Hints

The circle is divided into ten parts and the fifth part is the best.

They would exit from the fifth part of the circle.

Multiply 36 with the numerator or the total number of shaded parts.

Solution
• They have to find $\frac{5}{10}$ of 360°.
• To start they divide 360 into 10 parts. Each part is 36°.
• Next they multiply 36 by 5, which equals 180°.
• So the angle at which they think they can safely exit is 180°.
• ### How many degrees must an angle rotate to form a circle?

Hints

An angle can rotate clockwise or counterclockwise.

A half circle measures 180°

Solution

A circle measures 360°.

• ### What is the angle measure of Nico's pizza?

Hints

Look for the angle size of Nico's pizza which is $\frac{4}{12}$.

First divide 360 by 12. This will give you the degrees in one fractional part.

Multiply the measure of each angle with the numerator from Nico's fraction.

Solution
• Nico's pizza measures 120°
• To find $\frac{4}{12}$ of 360, first divide 360 by 12 to get 30
• Next multiply 30 by 4 to get 120