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Circle math
Here you will learn about the different parts of a circle including how to identify the key parts of a circle, properties of circles, circle formulas such as circumference and area, and how to solve various circle problems.
Students will first learn about circle math in the geometry standards in middle school and build upon that knowledge as they progress into a geometry course in high school.
What is circle math?
Circle math provides students with the definition of a circle, the parts of a circle, properties of circles and how to problem solve using the area and circumference of a circle.
By definition, a circle is a closed two dimensional figure where the set of all points that make up the edge of the circle are equidistant from a given single point, called the center of the circle.
There are essential parts of a circle that are necessary to identify in order to problem solve. Those key elements are as follows and are pictured below:
- Center of a circle
- Radius of a circle
- Chord of a circle
- Segment of a circle (minor segment and major segment)
- Secant of a circle
- Arcs
- Diameter of a circle
- Sector of a circle (minor sector and major sector)
Step by step guide: Parts of a circle
![[FREE] Equation Of A Circle Worksheet (High School)](https://thirdspacelearning.com/wp-content/uploads/2024/11/Grade-6-to-8-Equation-of-a-circle-Worksheet-listing-image.png)
[FREE] Equation Of A Circle Worksheet (High School)
Use this worksheet to check your high school students’ understanding of equation of a circle. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREE [FREE] Equation Of A Circle Worksheet (High School)
Use this worksheet to check your high school students’ understanding of equation of a circle. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREECircumference
The circumference of a circle is defined to be the distance around the circle or the perimeter of the circle.
In order to find the circumference of a circle, there is a special numerical value that is needed, called pi, and symbolized by the Greek letter \Pi.
\ Pi is an irrational number, but also the ratio between the circumference of a circle and the diameter of the circle.
For example, let’s look at the two circles below.
Let’s write the ratio comparing the two circles as \cfrac{\text { circumference }}{\text { diameter }}.
Circle A has a ratio: \cfrac{\text { circumference }}{\text { diameter }}=\cfrac{25.1}{8}
Circle B has a ratio: \cfrac{\text { circumference }}{\text { diameter }}=\cfrac{56.5}{18}
Now, let’s divide both ratios in order to compare them.
Circle A has a ratio: \cfrac{\text { circumference }}{\text { diameter }}=\cfrac{25.1}{8} \approx 3.14
Circle B has a ratio: \cfrac{\text { circumference }}{\text { diameter }}=\cfrac{56.5}{18} \approx 3.14
What do you notice? Both ratios divide out to be about 3.14. This happens with all circles because the ratio of the circumference to the diameter of all circles is about 3.14 which is the rounded approximation of \Pi.
So, in order to calculate the circumference, you can use, C=\Pi{d} or C=2\Pi{r}.
Remember that the diameter is twice the length of the radius d=2r so the two formulae for the circumference are equivalent.
Step by step guide: Circumference of a circle
Area
The area of a circle is defined to be the interior space within the circle and is calculated by using the equation, A=r^2 \Pi , where you take the radius of the circle, square it, and then multiply that value to \Pi or the approximation of \Pi which is 3.14.
Let’s take a look at circle F below.
The diameter of circle F is 16 units. Using the formula to calculate, find the area and the circumference of the circle.
Since the diameter is 16 units and the radius is half of this length 16\div{2}=8, the radius is 8 units. Substitute that value into the formula to find the area.
\begin{aligned} A&=r^2\Pi\\\\ &=(8)^2\Pi\\\\\ &=64\Pi\approx{201.1}\text{ units}^2 \end{aligned}Step by step guide: Area and circumference of a circle
Step by step guide: Area of a circle
Step by step guide: Pi \ r squared
What is circle math?
Common Core State Standards
How does this relate to middle and high school math?
- Grade 7 – Geometry: (7.G.B.4)
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- High School: Geometry (HS.G.CO.A.1)
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
- High School: Geometry (HS.G.C.A.2)
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
- High School Geometry (HSG-GMD.A.1)
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
How to answer questions involving circles
There are a lot of ways to use circles. For more specific step-by-step guides, check out the individual pages linked in the “what is circle math?” section or read through the examples below.
In order to solve problems involving circles:
- Recall the formula or definition.
- Make the calculation or provide an explanation.
Circle math examples
Example 1: identify the part of the circle from the figure
Name the part of the circle highlighted in blue in the circle below.
- Recall the formula or definition.
A chord is a segment within a circle that has endpoints on the edge of the circle.
2Make the calculation or provide an explanation.
In this case, in circle D, segment AB is a chord because the endpoints, point A and point B, are on the edge of the circle.
Example 2: calculate the circumference
Find the circumference of the circle pictured below.
Recall the formula or definition.
To find the circumference of a circle, you can use the formula, C=d\Pi, where d is the diameter and \Pi=3.141592 \ldots If you do not have access to a calculator, use the approximation \Pi\approx{3.14}.
Make the calculation or provide an explanation.
The diameter of the circle is given to be 17\mathrm{~cm}. You can substitute that value into the formula to find the circumference.
\begin{aligned} C&=17\times\Pi\\\\ C&=53.38\mathrm{~cm}\text{ (2dp)} \end{aligned}
Example 3: circumference word problem
Jordan has a circular garden that has a circumference of 69.08\mathrm{~ft}. She wants to create a walkway that connects one side of the garden to the other that goes right through the center of the garden. How long is the walkway (in feet)?
Recall the formula or definition.
When working with circumference, use the formula, C=d\Pi, where d is the diameter and \Pi=3.141592\ldots If you do not have access to a calculator, use the approximation \Pi\approx{3.14}.
Make the calculation or provide an explanation.
In this case, the circumference is given to be 69.08\mathrm{~ft} and you have to solve for the length of the diameter, substituting in value for circumference, you can solve for the diameter length.
\begin{aligned} C&=d\Pi\\\\ 69.08&=d\Pi\\\\ 22&=d \end{aligned}
The walkway should be 22 feet long.
Example 4: calculating area
Find the area of the circle pictured below.
Recall the formula or definition.
To find the area of a circle use the formula, A=r^{2}\Pi where r is the radius and \Pi=3.141592\ldots If you do not have access to a calculator, use the approximation \Pi\approx{3.14}.
Make the calculation or provide an explanation.
In this case, d=15, which means the radius is half of that value.
15\div{2}=7.5
\begin{aligned} A&=\Pi{r^2}\\\\ A&=(7.5)^2\times\Pi\\\\ A&=200.96\mathrm{~in}^2 \end{aligned}
Example 5: area word problem
Margo wants to tile her circular hallway. The hallway has a radius length of 5 feet and the tile costs \$5.20 for 1 square foot. How much does it cost for her to tile the circular hallway?
Recall the formula or definition.
To find the area of a circle use the formula, A=r^2\Pi where the radius is 5\mathrm{~ft}.
Make the calculation or provide an explanation.
First find the area of the circular hallway using,
\begin{aligned} A&=r^{2}\Pi\\\\ A&=5^{2}\times\Pi\\\\ A&=78.5\mathrm{~ft}^{2} \end{aligned}
Next, to find the total cost of tiling the hallway, multiply the amount of \$5.20 per square foot to the total area.
78.5\times{5.20}=408.2
It costs Margo \$408.20 to tile the hallway.
Example 6: calculating the diameter
Find the diameter length of a circle if the area is 380\mathrm{~cm^2}.
Recall the formula or definition.
Use the formula A=r^2\Pi
Make the calculation or provide an explanation.
Substitute the values into the formula and solve for the radius.
\begin{aligned} A&=r^{2}\Pi\\\\ 380&=r^{2}\times\Pi\\\\ 120.96&=r^{2}\\\\ 11&=r\text{ (2sf)} \end{aligned}
The radius r of the circle is 11\mathrm{~cm} which means the diameter is twice that, so the diameter is 2 \times 11=22 \mathrm{~cm}.
Teaching tips for circle math
- Instead of relying on worksheets for students to identify the parts of the circle or calculating area and/or circumference, consider interactive activities for student practice. Provide students with real-world examples and have them measure the diameter and find the longest chord on that example.
- Use technology to allow students to interact with circles in a different way. There are many educational apps and interactive websites that offer simulations and exercises that focus on problem solving and lead to deeper understanding.
- Use investigative type activities so that students can explore circle math concepts.
Easy mistakes to make
- Confusing the radius and the diameter
The radius is from the center of the circle to the circumference while the diameter goes across the whole circle. Both will pass through the origin.
- Using the diameter instead of radius to calculate the area
When finding the area of a circle, be sure to use the length of the radius and not the diameter because the formula is A=r^2\Pi.
- Thinking the circumference is a squared dimension
The circumference of a circle is like the perimeter because it is the distance around a circle, so it is a one dimensional measurement, not a square dimension like area.
Circle math practice problems
1. Name the tangent line of circle D.
line BG
line segment AG
line FK
line segment CD
A tangent line is a line that touches the edge of a circle at one point. So, looking at the figure, line FK touches the circle at exactly one point which is point F.
2. A semicircle has a diameter of 15\mathrm{~cm}. Find the distance around the figure below. Round your answer to the nearest tenth.
To find the distance around the figure, first calculate the circumference of a circle, you can use the formula, C=d \Pi. To find the distance around a semicircle, find the circumference and divide it by 2. In this case the diameter is equal to 15\mathrm{~cm} so substitute that in for diameter.
C=15\Pi\approx{47.1}
47.1\div{2}=23.55
You then have to add the diameter length to that value.
23.55+15=38.55 \approx 38.6\mathrm{~cm}
3. Find the length of the diameter of the circle that has a circumference of 40.82\mathrm{~cm}.
To find the diameter length when given the circumference of a circle, you can use the formula C=d \Pi.
40.82\div\Pi=13
The diameter is 13\mathrm{~cm}.
4. Find the area of the circle that has a diameter of 10\mathrm{~cm}, round the answer to the nearest tenth.
In order to find the area of a circle, you can use the formula, A=\Pi r^2 where r represents the radius of the circle. In this case, since the diameter is 10\mathrm{~cm} the radius is half of that so, 10 \div 2=5.
Substituting r=5 in for the radius, you can calculate,
\begin{aligned} A&=r^2\Pi\\\\ &=(5)^2\times\Pi\\\\ &=25\times\Pi\\\\ &=78.5\mathrm{~cm}^2 \end{aligned}
5. Which of the following statements is true?
The radius is twice the length of the diameter.
The ratio of the circumference to the diameter of all circles is equal to \Pi.
The ratio of the circumference to the area of all circles is always \Pi
The diameter is half the length of the circumference.
The circumference of a circle is C=d \Pi which means that \cfrac{C}{d}=\Pi. This holds true for all circles. So the ratio of the circumference to the diameter of all circles is equal to \Pi.
6. Travis has an old table that has a circular table top. He wants to paint the top circular surface. The table has a diameter of 8 feet and the paint he wants to use costs \$1.20 per square foot.
Approximately, how much will it cost him to paint the top circular surface of the table?
To find the total cost, first find the area of the circular table top using the formula, A=r^2\Pi and substitute the radius length into the formula. The radius, in this case, is 4\mathrm{~ft} because the diameter of a circle is double the length of a radius.
A=(4)^2\Pi\approx 50.24 \mathrm{ft}^2
Since the cost of the paint is \$1.20 per square foot, multiply that amount to the total area, 1.20 \times 50.24=60.288 which is rounded to \$60.29.
Circle math FAQs
Congruent circles are circles that have congruent or equal radii.
Concentric circles are circles that share the same center but have different radii lengths.
The arc length part of the circumference of a circle. It’s the distance between two fixed points on the edge of the circle.
A semicircle is half of a circle.
To find the area of a sector, find the area of the circle and then multiply the area to the fractional part of the circle represented by the sector.
In order to figure out what the fractional part of the circle the sector represents, you will need to know the measurement of the central angle formed by the sector.
The equation of a circle is (x-h)^{2}+(y-k)^{2}=r^{2} where (h, k) represents the center of the circle and r is the radius.
An ellipse is a closed shape obtained by slicing a cone at an angle. An ellipse has two focal points where the sum of the two distances from a point on the edge of the ellipse to the focal points is the same.
The next lessons are
- Sectors Arcs and Segments
- Circle theorems
- Types of data
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[FREE] Common Core Practice Tests (3rd to 8th Grade)
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