The Standard Equation of a Circle Formula: Everything You Need to Know!

How to Write and Graph the Standard Equation of a Circle

The following Standard Equation of a Circle Formula examples will show you everything you need to know!

Welcome to this free lesson guide that accompanies this Standard Equation of a Circle Explained! Tutorial where you will learn the answers to the following key questions and information:

• What is the standard equation of a circle definition?

• How can I find the equation of a circle given center and radius?

• How can I find the equation of a circle given center and diameter?

• How can I find the equation of a circle given a graph of the circle?

This Complete Guide to the Standard Equation of a Circle includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free equation of a circle worksheet and answer key.

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*This lesson guide accompanies our animated Standard Equation of a Circle Explained! video on YouTube.

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Standard Equation of a Circle Definition

Before you learn the standard equation of a circle formula, let’s quickly review the basic properties of circles:

The radius of a circle is any line that extends from the center of the circle to the edge.

The diameter of a circle is any straight line that extends from one edge of the circle through the center and to the opposite side. The value of the diameter is two times the radius.

• Radius = half the length of the diameter

• Diameter = twice the length of the radius

The Standard Equation of a Circle Formula:

The standard equation formula of a circle formula above is not as intimidating as it looks.

Let’s break it down:

You only need to know two pieces of information to write the standard equation of a circle:

• The Center Point Coordinates (h,k) where h is the x-value and k is the y-value

• The length of the radius r

That’s it! Now let’s look at an example:

How to Find the Standard Equation of a Circle Formula

Example 1

Write the standard form equation for circle S with a center at (4,3).

Remember that you only need two pieces of information to write the standard equation of a circle: the coordinates of the center point, and the length of the radius.

In this example, the center point is at (4,3) and the radius is 3 (you can find the length of the radius by drawing a horizontal line from the center of the circle to the edge and counting the number of spaces).

h=4, k=3, and r=3

Now you can replace h with 4 and k with 3 to complete the left side of the equals sign and then r^2 with 3^2 (which equals 9) to complete the ride side as follows:

This is the standard equation of Circle S

Example 2

Write the standard form equation for circle C with a center at (-4,-1).

Remember that you only need two pieces of information to write the standard equation of a circle: the coordinates of the center point, and the length of the radius.

In this example, the center point is at (-4,-1) and the radius is 5 (you can find the length of the radius by drawing a horizontal line from the center of the circle to the edge and counting the number of spaces).

Now you can replace h with -4 and k with -1 as follows (remember that two negatives make a positive!) and r with 5 as follows:

Remember that double negatives become positive!

This is the standard equation of Circle C.

EXAMPLE 3

Write the standard form equation for Circle J with a center at the origin and a diameter of 18.

To write the equation of Circle J, you need to know the center coordinates and the length of the radius.

You already know that the center is at the origin, which is (0,0) so h=0 and k=0

And you also know that the diameter is 18, and since a radius equals one half of the diameter, r = 18/2 = 9, so the radius is 9.

Now you can write the equation of the circle as follows:

The graph of Circle J.

This is the standard equation of Circle J.

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Example 4

*On this last example, you will work in reverse and identify the center coordinates and radius given the standard equation of a circle.

Start by finding the coordinates of the center of Circle P as follows:

Circle P has a center at (7,0)

Now you just have the find the value of the radius r. The value 196 represents r^2, so, to find the value of r, you have to figure out what number squares equals 196 as follows:

The radius of Circle P is 14 because 14^2 equals 196.

Answer: Circle P has a center at (7,0) and a radius of 14.

The trinomial factors to (x+1)(x+1) or (x+1)^2

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Standard Equation of a Circle Formula: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about the standard equation of a circle formula and to see more completing the square problems solved step-by-step:

Equation of a Circle Word Problems (Advanced Practice)

Looking for more advanced practice with the standard equation of a circle? The following video lesson covers topics including:

• equation of a circle word problems

• finding equation of a circle given two points

• using distance and midpoint formulas to find the equation of a circle

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Extra Practice: Free Equation of a Circle Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Equation of a Circle Worksheet:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

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Have thoughts? Share your thoughts in the comments section below!

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

Completing the Square Formula: Your Step-by-Step Guide

How to Solve Quadratic Equations by Completing the Square Formula

What is the Completing the Square Formula and how can you use it to solve problems?

Welcome to this free lesson guide that accompanies this Completing the Square Explained! video tutorial, where you will learn the answers to the following key questions and information:

• What is the completing the square formula?

• How can I solve by completing the square?

• How can I master solving quadratic equations by completing the square?

• What are the completing the square steps?

This Complete Guide to the Completing the Square includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

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*This lesson guide accompanies our animated Completing the Square Explained! YouTube.

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When can you use the completing the square method to solve quadratic equations?

Solving by completing the square is used to solve quadratic equations in the following form:

Note that a quadratic can be rearranged by subtracting the constant, c, from both sides as follows:

Figure 1

These are two different ways of expressing a quadratic.

Keep this in mind while solving the following problems:

Completing the Square Formula

The following method is less of a formula and more like completing the square steps:

Example: Solve the following quadratic by completing the square.

Note that a=1 and b=-6

Notice that a=1, and b=-6, but what about the constant, c?

Completing the Square Step 1 of 3: Rearrange if Possible

To complete the square, you need to have all of the constants (numbers that are not attached to variables) on the right side of the equals sign.

In this example, you can achieve this by subtracting 9 from both sides and simplifying as follows:

View fullsize

Now that you have rearranged the quadratic so that all of the constants are on the right side of the equals sign, you are ready for step 2!

Completing the Square Step 2 of 3: +(b/2)^2 to both sides

The second step to solving by completing the square is to add (b/2)^2 to both sides of the equation.

Remember the alternate way to write a quadratic from Figure 1 earlier on? Let’s look at it again with our current equation directly below it for reference.

Figure 1

Step two requires that you add (b/2)^2 to both sides and it should be clear that, in this example, b equals -6.

So to find the value of (b/2)^2, just plug in -6 for b and solve as follows:

In this case, (b/2)^2 equals 9

Since (b/2)^2 equals 9, go ahead and add 9 to both sides of the equals sign as follows:

You can simplify the right side of the equal sign by adding 16 and 9.

Notice that you can simplify the right side of the equal sign by adding 16 and 9 to get 25.

Now you are ready for the final step!

Completing the Square Step 3 of 3: Factor and Solve

Notice that, on the left side of the equation, you have a trinomial that is easy to factor.

The factors of the trinomial on the left side of the equals sign are (x-3)(x-3) or (x-3)^2

Completing the square will allows leave you with two of the same factors.

Expressing the factors are (x-3)^2 instead of (x-3)(x-3) is very important because it allows you to solve the problem as follows:

Simplify by taking the square root of both sides.

This is what is left after taking the square root of both sides.

After taking the square root of both sides, you are left with x-3 = +/- 5.

Next, to get x by itself, add 3 to both sides as follows.

And to find your solutions, simply perform x = 3 + 5 AND x = 3 - 5 to get your answer as follows:

Answer: x= 8 and x = -2

This method will apply to solving any quadratic equation! Let’s quickly review the completing the square formula method steps below and then take a look at a few more examples.

Solving by Completing the Square Steps Method Review:

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Solve by Completing the Square Problems

Example 1:

Solve for x by completing the square.

STEP 1/3: REARRANGE IF NECESSARY

Start by moving all of the constants to the right side of the equals sign as follows:

Leave yourself some room to work with!

STEP 2/3: +(b/2)^2 to both sides

In this example, b=2, so (b/2)^2 = (2/2)^2 = (1)^2 = 1

So, the next step is to add 1 to both sides as follows:

STEP 3/3: Factor and Solve

For the final step, factor the trinomial on the left side of the equals sign and solve for x as follows:

The trinomial factors to (x+1)(x+1) or (x+1)^2

The square root of 8 is approximately 2.83

These are the solutions!

Answer: x=1.83 and x=-3.83

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Solve by Completing the Square Examples

Example 2:

Solve for x by completing the square.

On this final example, follow the complete the square formula 3-step method for finding the solutions* as follows:

*Note that this problem will have imaginary solutions.

Step 1/3: Move the constants to the right side.

Step 2/3: Add (b/2)^2 to both sides.

Step 3/3: Factor and Solve

You can get a more detailed step-by-step explanation of how to solve the above example by watching the video tutorial below starting from minute 7:36.

Completing the Square Explained: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about the completing the square and to see more completing the square problems solved step-by-step:

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Extra Practice: Free Completing the Square Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Completing the Square Practice Worksheet:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

Keep Learning with More Free Lesson Guides:

Have thoughts? Share your thoughts in the comments section below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

How to Use Volume of a Sphere Formula: Your Complete Guide

Here is Everything You Need to Know About Finding the Volume of a Sphere

How can you use the volume of a sphere formula to solve problems?

Welcome to this free lesson guide that accompanies this Volume of a Sphere video tutorial, where you will learn the answers to the following key questions and information:

• How do you find volume of a sphere?

• How do you use the volume of a sphere formula?

• What is the radius of a sphere?

• What is the circumference of a sphere?

This Complete Guide to Finding Volume of a Sphere includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

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*This lesson guide accompanies our animated Volume of a Sphere Tutorial on YouTube.

Want more free math lesson guides and videos? Subscribe to our channel for free!

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Volume of a Sphere: Properties

Before you learn about finding the volume of a sphere and how to use the volume of a sphere formula, you must be familiar with the following properties of a sphere:

• A sphere is a three-dimensional circle (like a ball)

• The radius of a sphere is any line that extends from the center to the edge.

• The radius, r, is equal to half of the diameter.

Volume of a Sphere Formula and Definition

Next, you need to understand what the volume of a sphere is.

The Volume of a Sphere is how much room is INSIDE of a sphere.

The Volume of a Sphere Formula:

Not that, unlike area or surface area, volume is measured in cubic units.

Now you are ready to solve some problems involving volume of a sphere!

Using The Volume of a Sphere Formula Examples

EXAMPLE: How many cubic centimeters of air would be needed to completely fill a soccer ball with a diameter of 22cm?

Let’s start by identifying the key information as follows:

Since you are concerned with filling the amount of space inside of the soccer ball, you know that this problem can be solved using the volume of a sphere formula.

And you also know that the ball has a diameter of 22cm…

Since you know the length of the diameter and that the radius is equal to half of the diameter, you can conclude that the radius of the soccer ball equals 11cm, because (22/2)=11…

Now you are ready to use the formula to solve the problem as follows:

Start by replacing r with 11 in the volume of a sphere formula.

Then evaluate using your calculator.

Finally, you can conclude that:

ANSWER: The volume of the sphere is approximately 5,575.3 cubic centimeters!

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Using The Volume of a Sphere Formula Explained: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about how to use the volume of a sphere formula:

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Free Volume of a Sphere Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Volume of a Sphere Practice Worksheet:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

Keep Learning with More Free Lesson Guides:

Have thoughts? Share your thoughts in the comments section below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

The Commutative Property: Everything You Need to Know

Here is Everything You Need to Know About the Commutative Property

What is the commutative property in math and what does it look like?

Welcome to this free lesson guide that accompanies this Commutative Property Explained! video lesson, where you will learn the answers to the following key questions and information:

• What is the commutative property of addition?

• What is the commutative property of multiplication?

• Commutative Property Examples

• Commutative Property Proof

This Complete Guide to the Commutative Property includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

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*This lesson guide accompanies our animated Commutative Property Tutorial on YouTube.

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What is the Commutative Property of Addition?

The Commutative Property of Addition states that for any real numbers a and b:

Notice that the terms are in reverse order!

If you replace the a and b terms with real numbers, like a = 2 and b =8 as shown below:

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is equal to 10.

This is an example of why the commutative property holds under addition!

Does the Commutative Property work with subtraction?

Now that you understand the commutative property of addition, what about subtraction?

Is this true???

For example, if you replace the a and b terms with real numbers, like a = 2 and b =8 as shown below:

View fullsize

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is NOT equal ( -6 does not equal 6)

This is an example of why the commutative property does NOT hold under addition!

Summary: Addition is commutative, but Subtraction is not!

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What is the Commutative Property of Addition?

The Commutative Property of Multiplication states that for any real numbers a and b:

Notice that the terms are in reverse order!

For example, if you replace the a and b terms with real numbers, like a = 4 and b =8 as shown below:

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is equal to 32.

This is an example of why the commutative property holds under multiplication!

Does the Commutative Property work with division?

Now that you understand the commutative property of multiplication, what about division?

Is this true???

For example, if you replace the a and b terms with real numbers, like a = 4 and b =8 as shown below:

You can see from this example that, even though the terms are in reverse order on each side of the equal sign, that each side is NOT equal ( one-half does not equal 2)

This is an example of why the commutative property does NOT hold under division!

Summary: Multiplication is commutative, but Division is not!

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Final Word: Commutative Property Definitions

In summary, the commutative property only works with addition and multiplication. It does not work with subtraction and division.

For all real numbers a and b:

Commutative Property of Addition Definition: a + b = b + a

Commutative Property of Multiplication Definition: (a)(b) = (b)(a)

The terms are the same, but the order is reversed!

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Commutative Property Explained: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about the commutative property and to see more commutative property examples:

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Extra Practice: Free Commutative Property Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Identifying The Commutative Property:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

Keep Learning with More Free Lesson Guides:

Have thoughts? Share your thoughts in the comments section below!

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By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.

How to Convert Improper Fractions to Mixed Numbers Explained!

Converting Improper Fractions to Mixed Numbers: Your Complete Guide

How can you convert an improper fraction to a mixed number?

Welcome to this free lesson guide that accompanies this Converting Improper Fractions to Mixed Numbers video lesson, where you will learn the answers to the following key questions:

• What is a mixed number?

• What is a proper fraction?

• What is an improper fraction?

• What is the difference between a proper fraction and an improper fraction?

• How do you convert improper fractions to mixed numbers?

This Converting Improper Fractions to Mixed Numbers: Complete Guide includes several examples, a step-by-step tutorial, an animated video mini-lesson, and a free worksheet and answer key.

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*This lesson guide accompanies our animated Converting Improper Fractions to Mixed Numbers Tutorial on YouTube.

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Before you learn to convert improper fractions to mixed numbers you need to understand some key vocabulary first.

What is a Mixed Number?

A mixed number is a number consisting of an integer, like 3, and a proper fraction, like (2/5), as seen in the example below:

Figure 1

Note that a mixed number is ALSO equal to the sum of the integer and the fraction. The numbers shown in Figure 1 and Figure 2 both represent the same mixed number. The plus sign is not usually included, however, understanding this relationship will help you later on!

Figure 2

What is a Proper Fraction and What is an Improper Fraction?

What is the difference between a proper fraction and an improper fraction?

A proper fraction is a fraction where the value of the numerator is less than the value of the denominator.

The value of a proper fraction is less than one. For example, (1/4) or one fourth is a proper fraction.

An improper fraction is a fraction where the value of the numerator is greater than the value of the denominator.

The value of a proper fraction is greater than one. For example, (5/4) or five fourths is an improper fraction.

Notice that the value of an improper fractions is greater than one!

Now you’re ready to start converting improper fractions to mixed numbers, starting with the above example of (5/4).

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How To Convert Improper Fractions to Mixed Numbers

Converting Improper Fractions to Mixed Numbers Example 1

Example 1: Convert five fourths into a mixed number

Let’s start by expressing (5/4) or five fourths as the sum of 5 one fourths as follows:

Since (1/4) is equal to one quarter, we can think about this example in terms of money and replace each (1/4) with one-quarter coins as follows:

And since four quarters equals one whole dollar…

In terms of money, five quarters equals one whole dollar and one quarter.

In terms of values, five quarters (or five fourths) equals one whole and one quarter. (or one fourth)

Answer:

(5/4) expressed as a mixed number…

You can think about this situation in a more visual way by observing Figure 3 below. You will use this kind of thinking to solve future problems involving converting improper fractions to mixed numbers!

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Converting Improper Fractions to Mixed Numbers Example 2

Example 2: Convert thirteen sixths into a mixed number.

*Start by noting that 13 divided by 6 is equal to 2 with a remainder of 1, or 2R1.

Keep this fact in mind for now, as we’ll come back to it later!

(13/6) equals two with a remainder of 1

Next, let’s visualize what (13/6) looks like using fraction charts:

By combining these values (finding the sum): 1 + 1 + (1/6), we are left with a mixed number.

Answer:

(13/6) expressed as a mixed number…

This should make sense considering that we knew that 13 divided by 6 equaled 2 with a remainder of 1.

In this case, two wholes (six sixths) with a remainder of one sixth as shown in Figure 4 below.

Figure 4

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Converting Improper Fractions to Mixed Numbers: Video Tutorial

Still confused? Check out the animated video lesson below:

Check out the video lesson below to learn more about converting improper fractions to mixed numbers and for more free practice problems:

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Extra Practice: Free Converting Improper Fractions to Mixed Numbers Worksheet

Free Worksheet!

Are you looking for some extra practice? Click the links below to download your free worksheets and answer key:

Practice Converting Improper Fractions to Mixed Numbers:

CLICK HERE TO DOWNLOAD YOUR FREE WORKSHEET

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Keep Learning with More Free Lesson Guides:

Have thoughts? Share your thoughts in the comments section below!

(Never miss a Mashup Math blog--click here to get our weekly newsletter!)

By Anthony Persico

Anthony is the content crafter and head educator for YouTube's MashUp Math. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.