Answer the following questions. Many of the answers have clues by the diagrams given in the problem but you should confirm the findings by drawing your own figures on the Lenart sphere (starting with question 5).
Lune
1) What is the minimum number of sides required to draw a closed figure in the plane using straight lines only? Name the figure you drew in the plane.
2) What is the minimum number of sides required to draw a closed figure on the sphere?
Lune
1) What is the minimum number of sides required to draw a closed figure in the plane using straight lines only? Name the figure you drew in the plane.
2) What is the minimum number of sides required to draw a closed figure on the sphere?
You may have decided that the term biangle would be appropriate for this shape. Another name for this figure is a lune.
3) How many lunes are formed by the intersection of two Great Circles?
4) What is the relationship between the two points of intersection of the sides of the lune?
5) How long are the sides of the lunes?
6) Measure the opposite angles of the lune. What do you notice?
Triangles
7) Construct three non-collinear points in the plane. Connect them to form a triangle. How many triangles can you form?
8) Locate three non-collinear points \(A, B,\) and \(C\), on the sphere. Draw Great Circles through \(AB\), \( AC\), and \(BC\). How many different triangles with vertices \(A, B,\) and \(C\), can be drawn?
(Use of different colors may help to identify the triangles more easily.)
3) How many lunes are formed by the intersection of two Great Circles?
4) What is the relationship between the two points of intersection of the sides of the lune?
5) How long are the sides of the lunes?
6) Measure the opposite angles of the lune. What do you notice?
Triangles
7) Construct three non-collinear points in the plane. Connect them to form a triangle. How many triangles can you form?
8) Locate three non-collinear points \(A, B,\) and \(C\), on the sphere. Draw Great Circles through \(AB\), \( AC\), and \(BC\). How many different triangles with vertices \(A, B,\) and \(C\), can be drawn?
(Use of different colors may help to identify the triangles more easily.)
We will define the triangle formed using the shorter arcs joining two points on the sphere as the
Identify and shade in the small triangle on the sphere.
9) What is the sum of the interior angles of the triangle?
10) Measure the angles of the small \(\triangle ABC\). What is the sum of the measures of the angles of the small spherical triangle?
11) Draw another larger triangle and measure its angles and find the angle sum. Is the angle sum the same for both triangles?
Triangle Sum
12) Is it possible for a triangle on the plane to have more than one right angle?
13) Is it possible for a triangle on the sphere to have more than one right angle?
Identify and shade in the small triangle on the sphere.
9) What is the sum of the interior angles of the triangle?
10) Measure the angles of the small \(\triangle ABC\). What is the sum of the measures of the angles of the small spherical triangle?
11) Draw another larger triangle and measure its angles and find the angle sum. Is the angle sum the same for both triangles?
Triangle Sum
12) Is it possible for a triangle on the plane to have more than one right angle?
13) Is it possible for a triangle on the sphere to have more than one right angle?
14) Fill in the blank: Triangle Sum Theorem: The sum of the interior angles in spherical triangle is greater than _____ and less than ______.
Exterior Angles of a Triangle
15) State the exterior angles theorem for triangles on a plane.
16) Draw \(\triangl ABC\) on the sphere. Extend \(\overline{BC}\) to \(D\) and measure \(\angle{ACD}\). Find the measure of both \(\angle{A}\) and \(\angle{B}$\). Is there a relationship between the exterior angle of a triangle on the sphere and the non-adjacent interior angles?
15) State the exterior angles theorem for triangles on a plane.
16) Draw \(\triangl ABC\) on the sphere. Extend \(\overline{BC}\) to \(D\) and measure \(\angle{ACD}\). Find the measure of both \(\angle{A}\) and \(\angle{B}$\). Is there a relationship between the exterior angle of a triangle on the sphere and the non-adjacent interior angles?
Third Angles Theorem
17) Suppose two angles of one triangle are congruent to two angles of another triangle on a plane. How does the measure of the third angles of the triangles compare?
18) Draw \(\triangle ABC\) on the sphere. Measure the size of each angle of the triangle. Construct a second \(\triangle DEF\) with \(\angle{A} \cong \angle{D}\) , \(\angle{B} \cong \angle{E}\), \(DE = 2AB\) and \(FE = 2BC\). Measure the third angle of the triangle and compare this measure with the measure of the third angle of \(\triangle ABC\). [Note, you will use these two triangle to answer question #20 as well].
In the plane, the Third Angles Theorem states that if two angles of one triangle are congruent to two angles of another, then the third angles are congruent. Does this theorem apply to triangles on the sphere?
Similar Triangles
19) Given the diagram, how do the measures of \(AC\) and \(DF\) compare?
17) Suppose two angles of one triangle are congruent to two angles of another triangle on a plane. How does the measure of the third angles of the triangles compare?
18) Draw \(\triangle ABC\) on the sphere. Measure the size of each angle of the triangle. Construct a second \(\triangle DEF\) with \(\angle{A} \cong \angle{D}\) , \(\angle{B} \cong \angle{E}\), \(DE = 2AB\) and \(FE = 2BC\). Measure the third angle of the triangle and compare this measure with the measure of the third angle of \(\triangle ABC\). [Note, you will use these two triangle to answer question #20 as well].
In the plane, the Third Angles Theorem states that if two angles of one triangle are congruent to two angles of another, then the third angles are congruent. Does this theorem apply to triangles on the sphere?
Similar Triangles
19) Given the diagram, how do the measures of \(AC\) and \(DF\) compare?
20) Given the two triangles drawn on the sphere in problem #17, measure \(AC\) and \(DF\). How do the measures of \(AC\) and \(DF\) compare?
In the plane, two triangles are said to be similar if all of their corresponding angles are congruent and all of their corresponding sides are proportional. Does this theorem apply to triangles on the sphere?
21) Draw a triangle on the sphere. Measure the angles of the triangle. Construct a second triangle with angles equal in size to the angles of the first triangle. Now measure the sides of both triangles. What do you notice? Summarize your findings in terms related to your study of plain geometry.
Pythagorean Theorem
22) State the Pythagorean Theorem with respect a right triangle in the plane.
23) Investigate whether this theorem is relevant on the sphere. We have already discovered that it is possible to draw a triangle on the sphere with one, two, or three right triangles. Construct one of each of these triangles on the sphere and investigate whether there is any relationship between the sides of the triangle. Does the Pythagorean Theorem hold in Spherical Geometry?
SSS Triangle Congruency
24) In plane geometry, if three sides of one triangle are congruent to three sides of a second triangle, then are the two triangles congruent?
25) Construct a triangle on the sphere. Measure the length of the sides of the triangle. Construct a second triangle with sides equal in measure to the sides of the first triangle. Then measure the angles of the two triangles. Are the two triangles congruent?
SAS Triangle Congruency
26) In plane geometry, if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then are the two triangles congruent?
27) Draw \(\triangle ABC\) on the sphere. Measure the lengths of the \(\overline{AB}\) and \(\overline{BC}\) and the measure of \(\angle{DEF}\) where the measure of \(AB = DE\), \(BC = EF\) and \(\angle{ABC} \cong \angle{DEF}\) Are the two triangles congruent?
ASA Triangle Congruency
28) In plane geometry, if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then are the two triangles congruent?
29) Construct \(\triangle ABC\) on the sphere. Measure the length of \(\(\overline{BC}\) and the measure of \(\angle{B}\) and \(\angle{C}\). Construct \(\triangle DEF\) with the measure of \(BC = EF\), \(\angle{B} \cong \angle{E}\) and \(\angle{C} \cong \angle{F}\). Are the two triangles congruent?
Area of a Sphere and Lune
30) The formula for the surface area of a sphere is \(4r^{2}\pi\) where \(r\) is the radius of the sphere in the Euclidean sense (the distance from the interior center of the sphere to any point on the surface of the sphere).
a) What is the area of this lune with an interior angle of \(60^{\circ}\)?
b) What is the area of this lune with an interior angle of \(90^{\circ}\)?
Write down a generalized formula for the area of a lune.
Finding the area of a spherical triangle (Girard’s Theorem)
31) Find area of a triangle on the plane.
32) We will now derive a formula for the area of a triangle on the sphere. In order to understand how the formula is derived, you are encouraged to draw triangles on the sphere and use colors to identify the different triangles under discussion. This is very important to a clear understanding of the derivation of the formula for the area of a triangle on the sphere. This formula is commonly known as Girard’s Theorem.
Draw a triangle on the sphere and label the angles \(\alpha, \beta, \gamma\) (alpha, beta, gamma) as shown in the diagram below
In the plane, two triangles are said to be similar if all of their corresponding angles are congruent and all of their corresponding sides are proportional. Does this theorem apply to triangles on the sphere?
21) Draw a triangle on the sphere. Measure the angles of the triangle. Construct a second triangle with angles equal in size to the angles of the first triangle. Now measure the sides of both triangles. What do you notice? Summarize your findings in terms related to your study of plain geometry.
Pythagorean Theorem
22) State the Pythagorean Theorem with respect a right triangle in the plane.
23) Investigate whether this theorem is relevant on the sphere. We have already discovered that it is possible to draw a triangle on the sphere with one, two, or three right triangles. Construct one of each of these triangles on the sphere and investigate whether there is any relationship between the sides of the triangle. Does the Pythagorean Theorem hold in Spherical Geometry?
SSS Triangle Congruency
24) In plane geometry, if three sides of one triangle are congruent to three sides of a second triangle, then are the two triangles congruent?
25) Construct a triangle on the sphere. Measure the length of the sides of the triangle. Construct a second triangle with sides equal in measure to the sides of the first triangle. Then measure the angles of the two triangles. Are the two triangles congruent?
SAS Triangle Congruency
26) In plane geometry, if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then are the two triangles congruent?
27) Draw \(\triangle ABC\) on the sphere. Measure the lengths of the \(\overline{AB}\) and \(\overline{BC}\) and the measure of \(\angle{DEF}\) where the measure of \(AB = DE\), \(BC = EF\) and \(\angle{ABC} \cong \angle{DEF}\) Are the two triangles congruent?
ASA Triangle Congruency
28) In plane geometry, if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then are the two triangles congruent?
29) Construct \(\triangle ABC\) on the sphere. Measure the length of \(\(\overline{BC}\) and the measure of \(\angle{B}\) and \(\angle{C}\). Construct \(\triangle DEF\) with the measure of \(BC = EF\), \(\angle{B} \cong \angle{E}\) and \(\angle{C} \cong \angle{F}\). Are the two triangles congruent?
Area of a Sphere and Lune
30) The formula for the surface area of a sphere is \(4r^{2}\pi\) where \(r\) is the radius of the sphere in the Euclidean sense (the distance from the interior center of the sphere to any point on the surface of the sphere).
a) What is the area of this lune with an interior angle of \(60^{\circ}\)?
b) What is the area of this lune with an interior angle of \(90^{\circ}\)?
Write down a generalized formula for the area of a lune.
Finding the area of a spherical triangle (Girard’s Theorem)
31) Find area of a triangle on the plane.
32) We will now derive a formula for the area of a triangle on the sphere. In order to understand how the formula is derived, you are encouraged to draw triangles on the sphere and use colors to identify the different triangles under discussion. This is very important to a clear understanding of the derivation of the formula for the area of a triangle on the sphere. This formula is commonly known as Girard’s Theorem.
Draw a triangle on the sphere and label the angles \(\alpha, \beta, \gamma\) (alpha, beta, gamma) as shown in the diagram below
Using colors, draw and shade the α-lunes. Notice that there is a congruent \(\alpha\)-lune on the back of the sphere. Repeat this for the two \(\beta\)-lunes and the two \(\gamma\)-lunes using different colors. Notice that the \(triangle ABC\) appears in each of the lunes.
Notice also, that there is a copy of \(triangle ABC\) in each of the lunes on the back of the sphere. Also notice that you shaded in both \(triangle ABC\) and the copy of \(\triangle ABC\) with all three colors.
If we now wished to get an expression for the area of the sphere in terms of the area of the lunes, we would get the following (lune \(\alpha\) = area of lune \(\alpha\))
Notice also, that there is a copy of \(triangle ABC\) in each of the lunes on the back of the sphere. Also notice that you shaded in both \(triangle ABC\) and the copy of \(\triangle ABC\) with all three colors.
If we now wished to get an expression for the area of the sphere in terms of the area of the lunes, we would get the following (lune \(\alpha\) = area of lune \(\alpha\))
This formula has a very interesting consequence for the area of a triangle on the sphere. It states that the area of a triangle on the sphere is directly related to the angles of the triangle.
Once again the formula for the area of \(\triangle{ABC}=\pi r^{2}(\Large\frac{\alpha + \beta + \gamma }{180^{\circ}})\) is called Girard's Theorem, and the quantity \(\alpha + \beta + \gamma - 180^{\circ}\) is called the spherical excess of the triangle
a) Calculate the area of a spherical triangle with angles \(90^{\circ}, 90^{\circ}\), and \(90^{\circ}\).
Draw these triangles on the sphere and confirm that the answer you got for the area is consistent with what you would have expected starting with the formula for the sphere. \(A=4r^{2}\pi\)
b) Calculate the area of a spherical triangle with angles \(45^{\circ}, 45\^{\circ}\), and \(145^{\circ}\).
Answer Bank
Once again the formula for the area of \(\triangle{ABC}=\pi r^{2}(\Large\frac{\alpha + \beta + \gamma }{180^{\circ}})\) is called Girard's Theorem, and the quantity \(\alpha + \beta + \gamma - 180^{\circ}\) is called the spherical excess of the triangle
a) Calculate the area of a spherical triangle with angles \(90^{\circ}, 90^{\circ}\), and \(90^{\circ}\).
Draw these triangles on the sphere and confirm that the answer you got for the area is consistent with what you would have expected starting with the formula for the sphere. \(A=4r^{2}\pi\)
b) Calculate the area of a spherical triangle with angles \(45^{\circ}, 45\^{\circ}\), and \(145^{\circ}\).
Answer Bank