The Sum of the Angles of a Spherical Triangle

Let Ar be the surface area of a sphere of radius r. (Extra credit: Use our surface area calculation from the geometry of the earth question 7 with y=sqrt(r2- x2) in order to calculator the surface area Ar.) A great circle divides the sphere into two congruent hemispheres. Each of these will have area Ar/2.

Area of a Lune

A lune is any one of the four regions determined by two (non-coinciding) great circles. The angle of a lune is the angle swept out by the lune as in the figure on the left. We can measure the angle of a lune formed between two intersecting great circles as the angle formed by the intersection of the two defining planes with the plane tangent to the sphere at the point of intersection. Imagine holding a stick straight in front of us as we walk along the first great circle on the surface of the earth. We can measure the angle between the tangent vectors to the curves at the given point within the tangent plane by measuring the change of direction or angle swept out by our stick as we turn to walk along the second great circle. Response 1: Draw the two tangent vectors or "sticks" to the great circles that sweep out the marked angle in the figure on the right.


Use masking tape to create two non-coinciding great circles on your beach ball as in the picture below. Notice that masking tape bends easily to stay on the surface along a great circle, but it puckers if you try to turn (intrinsically) along a curve that is not "straight". Choose a lune on your sphere as in the figure below. Response 2: Find a formula for the area of a lune with lune angle v in terms of v, the angle, and Ar, the surface area of the sphere. Write down your formula here. Hints:
  • What if v is Pi/2? Notice that a Pi/2 lune will take up 25% or 1/4 = (Pi/2 radians) / (2 Pi radians) of the sphere. Hence a Pi/2 lune will have area Ar/4.
  • What if v is Pi/4?
  • In general, find out what fraction of the sphere the lune of angle v occupies (use radian measure - not degrees) and use this to find the desired area as a fraction of Ar.

    Area of a Spherical Triangle - Follow the Directions Very Carefully

    Use masking tape to create a third great circle in order to make a spherical triangle on your beach ball. Label the vertices of your triangle A, B and C. Label the angles of the spherical triangle as a, b and c at the respective vertices A, B and C (ie angle a is at vertex A ...). Fix your attention to this triangle and label its area T. Opposite angle a is side BC. Across this side is another region on the sphere - call its area X. Label X on your sphere. Notice that X and T together make up a lune of angle a. Similarly, label the area of the other region across side AC as Y, and across side AB as Z. Response 3: Use your formula for the area of a lune of a given angle to write down the areas of T+X, T+Y, T+Z in terms of Ar and a, Ar and b, and Ar and c, respectively. Response 4: What is the sum of all of these areas, T+X + T+Y + T+Z, in terms of Ar, a, b and c? Show your work here and be sure to factor out common terms.

    Notice that on the other side of the sphere there is another triangle with the same area X that is antipodal to the region you labeled as X. Label this antipodal triangle X' on your sphere. Response 5: What part of the sphere does T+X'+Y+Z take up? Explain. Response 6: Use this to find a formula for the area of T+X'+Y+Z in terms of just Ar. Recall that X'=X and so this gives a formula for T+X+Y+Z.

    The formula in Response 4 gives T+X + T+Y +T+Z = 3T + X + Y + Z in terms of a, b c and Ar. The formula in Response 6 gives T+X+Y+Z in terms of only Ar. Response 7: Subtract the formula in Response 6 from the formula in Response 4 to solve for T in terms of a, b, c and Ar. Be sure to find a common denominator and factor out common terms. Show work.

    Sum of the Angles of a Spherical Triangle

    Given any general spherical triangle T, it must have a positive area. We can extend the sides of triangle T to form 3 great circles. We will then obtain the situation above and the corresponding formula for T that you just solved for. What does the positivity of the area of the spherical triangle T tell you about the sum of the angles? Explain and show work.