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GeoGebra

Arc Length & Sector Area Investigation

Autor:Courtney Deller, Todd Fogdall

Lesson Overview & Quick Circumference and Area Review

In our previous lesson, we defined and worked with central and inscribed angles. Today, you'll connect those ideas to what you already know about a circle's circumference and area. Before we start, let's review those concepts just in case you've forgotten. In the sketch below, you'll see a circle centered at point C, and I've given you a slider that allows you to adjust the radius of the circle. Hopefully, the names of the three buttons on the left give you an idea of what they each do, but go ahead and click the first two to see if they remind you of the difference between circumference and area. Then answer Questions #1 and #2 below the sketch.

Question #1 - Definition and Formula for Circumference

What is circumference, and what is the formula for it? (Click "Show Circumference" in the applet for help)

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Calculate circumference.

What is the circumference of a circle with a radius of 6? Use 3.14 for and round your answer to the nearest tenth.

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Question #2 - Definition and Formula for Area of a Circle

What is the area of a circle, and what is the formula for it? (Click "Show Area" in the applet for help)

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Calculate area.

What is the circumference of a circle with a radius of 5? Use 3.14 for and round your answer to the nearest tenth.

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NEW Concept - Calculating Arc Length

You just reviewed how to calculate the distance around a circle (circumference), but what if you don't want to travel all the way around the circle? This is a measurement called "arc length", and it is vital that you understand that it is different than arc measure. To understand this consider the sketch below. When the sketch loads, the radius of the circle will be 3, and you should see a central angle with a measure of 90 degrees. Notice that the length of the red arc is 4.7 units. Make the radius bigger, but leave the angle alone. Then answer the question below.

Question #3 - Arc Length & Radius

When you made the circle radius bigger, the central angle remained 90 degrees. What happened to the arc length?

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Question #4 - Arc Length & Central Angle Measure

Now make the radius three again, and then change the size of the central angle, either using the slider, or type in exactly what angle you'd like. The size of the radius of the circle isn't affected by this. What happened to the arc length?

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Question #5 - Arc Length Formula

In the sketch, two calculations are shown. The first, of course, is circumference. Describe what the circumference is multiplied by to give arc length.

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Question #7 - Calculating Arc Length

Calculate the arc length of a circle with a radius of 4, made by a central angle of . Use 3.14 for and round your answer to the nearest tenth.

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NEW Concept - Calculating Area of a Sector

In the sketch below, you will see a blue-shaded region formed by two radiuses (radii) of the circle and the arc between them. This region is called a sector. It should be obvious that we don't measure the length of a sector, but its area. When the sketch loads, the radius of the circle will be 3, and you should see a central angle with a measure of 90 degrees, just like the previous sketch. Notice that the area of the sector is 7.1 square units. Make the radius bigger, but leave the central angle alone. Then answer the question below.

Question #8 - Sector Area & Radius

When you made the circle radius bigger, the central angle remained 90 degrees. What happened to the area of the sector?

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Question #9 - Sector Area & Central Angle Measure

Now make the radius three again, and then change the size of the central angle, either using the slider, or type in exactly what angle you'd like. The size of the radius of the circle isn't affected by this. What happened to the area of the sector?

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Question #10 - Sector Area Formula

In the sketch, two calculations are shown. The first, of course, is the area of the circle. Describe what the area of the circle is multiplied by to give the area of the sector.

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Question #11 - Calculating Sector Area

Calculate the area of a sector in a circle with a radius of 5 made by a central angle of . Use 3.14 to approximate and round your answer to the nearest tenth.

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