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.
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.
Priya, Han, and Mai each measured one of the circular objects from earlier.
Priya says that the bike wheel is 24 inches.
Han says that the yo-yo trick is 24 inches.
Mai says that the glow necklace is 24 inches.
Do you think that all these circles are the same size?
What part of the circle did each person measure? Explain your reasoning.
11.4
Activity
Drawing Circles
Standards Alignment
Building On
Addressing
7.G.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
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.
Circle C, with a radius that is equal to Circle A’s diameter.
Circle D, with a diameter that is equal to Circle B’s radius.
If you have time, use a compass to recreate one of these designs.
First of 4 images of circles in different shapes. Image 1, circles within circles.
Second of 4 images of circles in different shapes. Image 2, caterpillar made of 5 circles.
Third of 4 images of circles in different shapes. Image 3, 3 interlocking circles.
Fourth of 4 images of circles in different shapes. Image 4, 7 circles in the shape of a flower.
Student Lesson Summary
A circle consists of all of the points that are the same distance away from a particular point called the center of the circle.
A segment that connects the center with any point on the circle is called a radius. For example, segments , , , and are all radii of Circle 2. (We say one radius and two radii.) The length of any radius is always the same for a given circle. For this reason, people also refer to this distance as the radius of the circle.
Two circles labeled circle 1 and circle 2. Circle 1 has a center at P and the points A, C, F, B, D, and E lie on the circle. Diameters A B, C D, and E F are drawn. Circle 2 has a center at Q and the points G, H, I and J lie on the circle. Line segments Q G, Q H, Q I, and Q J are drawn.
A segment that connects two opposite points on a circle (passing through the circle’s center) is called a diameter. For example, segments , , and are all diameters of Circle 1. All diameters in a given circle have the same length because they are composed of two radii. For this reason, people also refer to the length of such a segment as the diameter of the circle.
The circumference of a circle is the distance around it. If a circle was made of a piece of string and we cut it and straightened it out, the circumference would be the length of that string. A circle always encloses a circular region. The region enclosed by Circle 2 is shaded, but the region enclosed by Circle 1 is not. When we refer to the area of a circle, we mean the area of the enclosed circular region.
Glossary
circle
A circle is made of all the points that are the same distance from a given point. That given point is the center of the circle.
Every point on this circle is 5 cm away from point .
circumference
The circumference of a circle is the distance around the circle. If a circle has radius units, its circumference is units.
For example, a circle has a radius of 3 inches. Its circumference is , or inches. This is about 18.85 inches.
diameter
A diameter is a line segment that goes from one point on a circle to another and passes through the center. The length of this segment is also called the diameter. Every diameter of a circle is the same length.
radius
A radius is a line segment that goes from the center of a circle to any point on the circle. The length of this segment is also called the radius. Every radius of a circle is the same length.
For example, is the radius of this circle with center .
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