Illustrative Math Alignment: Grade 7 Unit 3

Measuring Circles

Lesson 3: Exploring Circumference

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Nodes
Math Clip Art--Geometry Concepts--Circle Illustrations--Unit Circle	Math Clip Art--Geometry Concepts--Circle Illustrations--Unit Circle This is part of a collection of math clip art images about selected topics in Geometry. Also look at our Geometry Basics math clip art collection.	Definition of a Circle and Applications of Coordinate Geometry
Math Example--Area and Perimeter--Circular Area and Circumference: Example 1	Math Example--Area and Perimeter--Circular Area and Circumference: Example 1 Topic Geometry Description This example presents a circle with a radius of 5 units. The task is to calculate the area of the circle using the given radius. The solution involves substituting the radius value into the area formula: A = π * r^2 = π * (5)^2 = 25π. Understanding circular area and circumference is integral to mastering geometry. Concepts such as calculating areas and circumferences of circles are fundamental, and exercises like these examples not only provide practice but also deepen the understanding of theoretical concepts in a practical way.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 2	Math Example--Area and Perimeter--Circular Area and Circumference: Example 2 Topic Geometry Description This example features a circle with a radius of 5 units. The objective is to calculate the circumference of the circle using the given radius. The solution involves applying the circumference formula: C = 2 * π * r = 2 * π * (5) = 10π. Circular area and circumference calculations are fundamental in geometry. These examples provide students with practical applications of theoretical concepts, helping them understand the relationship between a circle's radius and its circumference.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 3	Math Example--Area and Perimeter--Circular Area and Circumference: Example 3 Topic Geometry Description This example introduces a circle with an unknown radius represented by x. The task is to express the area of the circle in terms of x. The solution demonstrates how to use the area formula with a variable radius: A = π * r2 = π * (x)2 = π * x2. Working with variable expressions in geometry helps students transition from concrete to abstract thinking. This example bridges the gap between numerical calculations and algebraic representations, a crucial skill in advanced mathematics.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 4	Math Example--Area and Perimeter--Circular Area and Circumference: Example 4 Topic Geometry Description This example presents a circle with an unknown radius represented by x. The task is to express the circumference of the circle in terms of x. The solution demonstrates how to use the circumference formula with a variable radius: C = 2 * π * r = 2 * π * x = 2πx. Understanding circular area and circumference is crucial in geometry. This example helps students transition from concrete numerical values to abstract algebraic expressions, fostering a deeper comprehension of the relationship between a circle's radius and its circumference.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 5	Math Example--Area and Perimeter--Circular Area and Circumference: Example 5 Topic Geometry Description This example features two concentric circles with radii of 5 and 4 units. The task is to calculate the area of the shaded region between these circles. The solution involves subtracting the area of the smaller circle from the area of the larger circle: A = π * (5)2 - π * (4)2 = 25π - 16π = 9π. Concentric circles and shaded regions introduce students to more complex geometric concepts. This example builds upon basic circular area calculations, encouraging students to think about the relationships between different circles and how to find areas of composite shapes.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 6	Math Example--Area and Perimeter--Circular Area and Circumference: Example 6 Topic Geometry Description This example presents two concentric circles with radii of 5 and y units. The objective is to express the area of the shaded region between these circles in terms of y. The solution involves subtracting the area of the smaller circle from the area of the larger circle: A = π * (5)2 - π * y2 = 25π - πy2 = π(25 - y2).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 7	Math Example--Area and Perimeter--Circular Area and Circumference: Example 7 Topic Geometry Description This example features two concentric circles with radii x and y. The task is to express the area of the shaded region between these circles in terms of x and y. The solution involves subtracting the area of the smaller circle from the area of the larger circle: A = π * x2 - π * y2 = π(x2 - y2).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 8	Math Example--Area and Perimeter--Circular Area and Circumference: Example 8 Topic Geometry Description This example presents a circle with a radius of 5 units and a shaded sector with a central angle of 30 degrees. The task is to calculate the area of the shaded sector. The solution involves using the central angle to find the fractional amount of the total area: A = (30 / 360) * π * 52 = 25π / 12.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 9	Math Example--Area and Perimeter--Circular Area and Circumference: Example 9 Topic Geometry Description This example features a circle with a radius of 5 units and a shaded arc corresponding to a central angle of 30 degrees. The task is to calculate the length of the shaded arc. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (30 / 360) * 2 * π * 5 = 5π / 6.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 10	Math Example--Area and Perimeter--Circular Area and Circumference: Example 10 Topic Geometry Description This example presents a circle with radius x and a shaded sector with central angle θ (theta). The task is to express the area of the shaded sector in terms of x and θ. The solution involves using the central angle to find the fractional amount of the total area: A = (θ / 360) * π * x2 = πx2θ / 360.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 11	Math Example--Area and Perimeter--Circular Area and Circumference: Example 11 Topic Geometry Description This example presents a circle with radius x and a shaded sector with central angle θ (theta). The task is to express the arc length of the shaded region in terms of x and θ. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (θ / 360) * (2 * π * x) = (θ * x / 180) * π.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 12	Math Example--Area and Perimeter--Circular Area and Circumference: Example 12 Topic Geometry Description This example features a circle with a radius of 5 units and a shaded sector with central angle θ (theta). The task is to express the area of the shaded sector in terms of θ. The solution involves using the central angle to find the fractional amount of the total area: Area = (θ / 360) * (π * 52) = 25π / 360 * θ = 5π / 72 * θ.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 13	Math Example--Area and Perimeter--Circular Area and Circumference: Example 13 Topic Geometry Description This example presents a circle with a radius of 5 units and a shaded sector with central angle θ (theta). The task is to express the arc length of the shaded region in terms of θ. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (θ / 360) * (2 * π * 5) = θ / 18 * π.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 14	Math Example--Area and Perimeter--Circular Area and Circumference: Example 14 Topic Geometry Description This example features a circle with radius x and a shaded sector with a central angle of 30 degrees. The task is to express the area of the shaded sector in terms of x. The solution involves using the central angle to find the fractional amount of the total area: Area = (30 / 360) * (π * x2) = π * x2 / 12.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 15	Math Example--Area and Perimeter--Circular Area and Circumference: Example 15 Topic Geometry Description This example presents a circle with radius x and a shaded sector with a central angle of 30 degrees. The task is to express the arc length of the shaded region in terms of x. The solution involves using the central angle to find the fractional amount of the circumference: Arc Length = (30 / 360) * (2 * π * x) = x / 6 * π.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 16	Math Example--Area and Perimeter--Circular Area and Circumference: Example 16 Topic Geometry Description This example features two concentric circles with radii of 5 and 4 units, and a shaded sector with a central angle of 30 degrees. The task is to calculate the area of the shaded region. The solution involves using the central angle to determine the fractional area difference between the larger and smaller circles: Area = (θ / 360) * (π * r12 - π * r22) = (30 / 360) * (π * 52 - π * 42) = π / 12 * (25 - 16) = 3π / 4.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 17	Math Example--Area and Perimeter--Circular Area and Circumference: Example 17 Topic Geometry Description This example presents two concentric circles with radii of 5 and 4 units, and a shaded sector with a central angle of 30 degrees. The task is to calculate the perimeter of the shaded region. The solution involves calculating arc lengths based on the central angle and adding straight-line segments between radii: Perimeter = (θ / 360) * (2 * π * r1 + 2 * π * r2) + 2 * (r1 - r2) = (30 / 360) * (2π * 5 + 2π * 4) + 2 * (5 - 4) = 3π / 2 + 2.	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 18	Math Example--Area and Perimeter--Circular Area and Circumference: Example 18 Topic Geometry Description This example features two concentric circles with radii x and y, and a shaded sector with a central angle θ (theta). The task is to express the area of the shaded region in terms of x, y, and θ. The solution uses the central angle to find the fractional area difference between the larger and smaller circles: Area = (θ / 360) * (π * x2 - π * y2) = (θπ / 360) * (x2 - y2).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 19	Math Example--Area and Perimeter--Circular Area and Circumference: Example 19 Topic Geometry Description This example presents two concentric circles with radii x and y, and a shaded sector with a central angle θ (theta). The task is to express the perimeter of the shaded region in terms of x, y, and θ. The solution involves calculating arc lengths based on the central angle and adding straight-line segments between radii: Perimeter = (θ / 360) * (2πx + 2πy) + 2(x - y) = π/180 * θ(x + y) + 2(x - y).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 20	Math Example--Area and Perimeter--Circular Area and Circumference: Example 20 Topic Geometry Description This example features two concentric circles with radii 5 and y, and a shaded sector with a central angle of 30 degrees. The task is to express the area of the shaded region in terms of y. The solution involves calculating the difference between the areas of the larger and smaller circles, accounting for the central angle: Area = π/12 * (25 - y2).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 21	Math Example--Area and Perimeter--Circular Area and Circumference: Example 21 Topic Geometry Description This example features two concentric circles with radii 5 and y, and a shaded sector with a central angle of 30 degrees. The task is to calculate the perimeter of the shaded region. Given a central angle of 30 degrees, radius 5, and unknown radius y, the perimeter is calculated as: Perimeter = π/6 * (5 + y) + 2(5 - y).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 22	Math Example--Area and Perimeter--Circular Area and Circumference: Example 22 Topic Geometry Description This example presents two concentric circles with radii 4 and x, and a shaded sector with a central angle θ (theta). The task is to express the area of the shaded region in terms of x and θ. The solution uses the central angle to find the fractional area difference between the larger and smaller circles: Area = (θ / 360) * (π * x2 - π * 42) = (θ * π / 360) * (x2 - 16).	Area and Circumference
Math Example--Area and Perimeter--Circular Area and Circumference: Example 23	Math Example--Area and Perimeter--Circular Area and Circumference: Example 23 Topic Geometry Description This example features two concentric circles with radii x and 4, and a shaded sector with a central angle θ (theta). The task is to express the perimeter of the shaded region in terms of x and θ. The solution involves calculating arc lengths based on the central angle and adding straight-line segments between radii: Perimeter = (θ / 360) * (2 * π * x + 2 * π * 4) + 2 * (x - 4) = (π / 180) * θ(x + 4) + 2(x - 4).	Area and Circumference
MATH EXAMPLES--Teacher's Guide: Circular Area and Circumference	MATH EXAMPLES--Teacher's Guide: Circular Area and Circumference This set of tutorials provides 23 examples of solving for the area and circumferences of circles and sections of circles. This is part of a collection of teacher's guides. To see the complete collection of teacher's guides, click on this link. Note: The download is a PDF file. Related Resources To see resources related to this topic click on the Related Resources tab above.	Area and Circumference
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Crossword Puzzle--Circles	Worksheet: Crossword Puzzle--Circles This is part of a collection of math worksheets that are crossword puzzles of math vocabulary. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: 2D Shapes, Worksheet 1	Worksheet: 2D Shapes, Worksheet 1 This is part of a collection of math worksheets on the topic of 2D and 3D geometry. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle and Definition of a Quadrilateral
Worksheet: Analyzing Shapes, Worksheet 1	Worksheet: Analyzing Shapes, Worksheet 1 This is part of a collection of math worksheets on the topic of analyzing shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle, Definition of a Quadrilateral and Definition of a Triangle
Worksheet: Analyzing Shapes, Worksheet 3	Worksheet: Analyzing Shapes, Worksheet 3 This is part of a collection of math worksheets on the topic of analyzing shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle
Worksheet: Identifying Shapes, Worksheet 1	Worksheet: Identifying Shapes, Worksheet 1 This is part of a collection of math worksheets on the topic of geometric shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle, Definition of a Polygon and Definition of a Triangle
Worksheet: Identifying Shapes, Worksheet 1	Worksheet: Identifying Shapes, Worksheet 1 This is part of a collection of math worksheets on the topic of geometric shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle, Definition of a Polygon and Definition of a Triangle
Worksheet: Identifying Shapes, Worksheet 1	Worksheet: Identifying Shapes, Worksheet 1 This is part of a collection of math worksheets on the topic of geometric shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle, Definition of a Polygon and Definition of a Triangle
Worksheet: Identifying Shapes, Worksheet 1	Worksheet: Identifying Shapes, Worksheet 1 This is part of a collection of math worksheets on the topic of geometric shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle, Definition of a Polygon and Definition of a Triangle
Worksheet: Identifying Shapes, Worksheet 2	Worksheet: Identifying Shapes, Worksheet 2 This is part of a collection of math worksheets on the topic of geometric shapes. To see the complete worksheet collection on this topic, click on this link. Note: The download is a PDF file. Related Resources To see additional resources on this topic, click on the Related Resources tab. Worksheet Library To see the complete collection of Worksheets, click on this link.	Definition of a Circle, Definition of a Polygon and Definition of a Triangle

Illustrative Math Alignment: Grade 7 Unit 3

Measuring Circles

Lesson 3: Exploring Circumference

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