Geometry In Space

Student Worksheet

Did you know that there is more than one Geometry in the Universe? No, not easy or hard! Euclidean (or flat space) or Non-Euclidean (or curved space.) Your teacher has probably explained both of these types to you, but in this lesson you will investigate them. Let us begin in a simple fashion.

Remember this theorem (called the Triangle Sum Theorem)? The sum of the angles of a triangle equals 180 degrees.

1. Test out this theorem by using a piece of string to form the perimeter of a triangle on a sheet of paper. Because we are using a flat sheet of paper, this investigation is a test of Euclidean Geometry.

   Stretch the string tightly and tape down the corners with clear tape. Measure the angles of the triangle and record them. Measure the perimeter of this triangle and record it. Calculate the sum of the angles of this triangle and record that also. Repeat this process at least 5 times.

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   What is your hypothesis about the sum of the angles of a triangle on a flat piece of paper?

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   Does this agree with the Triangle Sum Theorem?

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2. Let's try this investigation in Elliptic Geometry. Specifically, you will measure the angles in a triangle formed on the surface of a sphere, rather than a flat piece of paper. You may be able to use tacks to hold the string in a triangular shape so that you can measure the angles. Make sure you record at least 5 sets of measurements.

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   What is your hypothesis about the sum of angles of a triangle on a spherical surface?

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   Does this agree with the Triangle Sum Theorem?

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3. With your hyperbolic model, try the same procedure. Remember that your lines are special arcs in Hyperbolic Geometry. To measure the angle formed by two lines, measure the angle formed by the tangents to the arcs at the intersection points. Make sure you record at least 5 sets of measurements.

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   What is your hypothesis about the sum of angles of a triangle on a hyperbolic surface?

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   Does this agree with the Triangle Sum Theorem?

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4. Record your measurements in the front of the room to share with your classmates. When all of the data have been recorded, look to see if you can find a relationship between the value of the perimeter of the triangle and the sum of the angles in flat, elliptic, and hyperbolic geometries.

   What is your hypothesis about the relationship between the sum of the angles of triangles and their perimeters in Euclidean Geometry?

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   What is your hypothesis about the relationship between the sum of the angles of triangles and their perimeters in Elliptic Geometry?

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   What is your hypothesis about the relationship between the sum of the angles of triangles and their perimeters in Hyperbolic Geometry?

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5. Research and discuss what each of these geometries means to our view of the Universe. Attach your work to this worksheet.