Spherical Geometry Geometry Final Part B Problem 4

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1 Spherical Gemetry Gemetry Final Part B Prblem 4 By: Duglas A. Ruby Date: 11/10/00 Class: Gemetry Grades: 11/1 Prblem 4: Arc length and the area f a triangle in spherical gemetry are quite different frm that dne in Euclidean gemetry. Assume the radius f the Earth t be 3960 miles. Three psitins n the surface f the Earth are defined by: i) Latitude = 0, lngitude = 60 ii) Latitude = 90, lngitude = 60 iii) Latitude = 0, lngitude = 90 a) Find the perimeter and area f the spherical triangle. b) Find the area and perimeter f the triangle whse vertices are at the three pints assuming Euclidean gemetry. c) Hw d the areas in Part a) and Part b) differ? Nte: Use sftware supprt as necessary 1. Diagramming the Prblem Using Gemeter s sketchpad, we will try t apprximate a sphere by drawing a circle with a vertical diameter representing the Greenwich meridian. We will indicate the equatr with an arc created using three pints, tw n the circle and ne n the Greenwich Meridian. Final Part B Prblem 4 Dug Ruby - Page 1

2 Observatins: The three given pints, A (n the equatr at 60 East), B (at the Nrth Ple), and C (n the Equatr at 90 East), are indicated. The viewpint is frm apprximately 30 Nrth, directly ver the Greenwich Meridian. Spherical ABC is indicated in bld lines, while Euclidean ABC is shwn in dtted lines. We als see the prjectin f segments AB and CB nt the Equatrial plane as OA and OC in dtted lines. Nte that OA, OB, and OC are radii f the Earth with lengthy = 3960 miles. Finally, given the psitin f pints A and C n the equatr, AOC has a measure f 30. AOC is als the prjectin f spherical BOC nt the equatrial plane. Angles BAC = BCA = 90. a) Find the perimeter and area f the spherical triangle. The perimeter f ur spherical triangle can be calculated by using the basic frmula fr the circumference f a circle: 1. C D r (where D = diameter and r = radius) In ur case, r = 3960 miles. Thus, a great circle circumference is: miles 4, miles Since bth spherical segments BA and BC are frm the Nrth Ple t the Equatr, BOC are bth = 90. Thus: BOA and BA = BC = , miles Spherical segment AC, is a 30 arc n the equatr. Thus, its length is: AC = , miles. Adding the results f equatins 3 and 4, we get the perimeter f spherical ABC which is: 5. BA + BC + AC = perimeter f spherical ABC 14, miles T find the area (K) f spherical ABC, we use the frmula: r 6. K A B C (where A, B, and C are the measures f the vertex angles f 180 the spherical triangle; which will always be greater than 180 ) Page

3 Using A = C = 90, B = 30, and r = 3960 miles, fr the Area f ur spherical triangle, we get: K ,10, sq. miles b) Find the area and perimeter f the Euclidean triangle. T calculate the dimensins f the Euclidean triangle, we can either use Euclidean Gemetry with the Pythagrean Therem r use Trignmetry. Since ur cmparisn is Euclidean versus Spherical, we will use traditinal triangulatin withut having t knw the values f the sine, csine, r tangents f the angles frmed by the varius triangles. First, let s lk at the prjectins f AOB and COB acrss their respective lngitudinal planes (60 East and 90 East respectively): Triangle COB is a 45/45/90 right triangle. Since the legs (OC and OB) are radii f the Earth, the frmula fr the length f tw sides f Euclidean triangle ABC is: 8. AB BC , miles Finding the length f segment AC will be a bit mre difficult. T d this, we lk at a prjectin f ABC nt the Equatrial plane: Page 3

4 Angle COA is a 30 angle, therefre, DOA is a 30/60/90 triangle. Segment OA is 3960 miles. Frm this, we can calculate that: AD 1, 980 miles. OD , miles 10. CD = (OC OD) = miles 11. AC AD CD , miles Thus, we nw knw that the perimeter f Euclidean ABC is: 1. AB + BC +AC = ( 5, miles) +, miles = 13, miles Page 4

5 T find the area f Euclidean ABC, we need t lk at a prjectin f ABC thrugh lngitude 75, ½ f the way between 60 and 90. We can find the height f ABC (segment EB) by using the Pythagrean Therem. Once we d that, we will find the area f ABC using the standard frmula EB , miles 14. Area 1 1 bh AC EB ABC 5,835, sq. miles Frm this diagram, we can als calculate the measure f A, B, and C using trignmetric functins. Please see the summary fr the results f this. Page 5

6 iii) Summary Let s start with a quick cmparisn f ur Spherical and Euclidean triangles: Triangle BA=BC AC Perimeter Area Sum f s Spherical 6,0.35 miles, miles 14, miles 8,10,867 miles 10 Euclidean 5,600.9 miles, miles 13,50.4 miles 5,835,197 miles 180 We have shwn, using Spherical gemetry as well as traditinal Euclidean (nn-trig) gemetry that even given the same vertices in 3-space, the sum f the interir angles f the spherical triangle is greater than the Euclidean triangle. The Spherical triangle has a larger area and a larger perimeter. We have als shwn hw t slve the 3-space Euclidean triangle using -space prjectins. I might als nte, that in the case f ur particular spherical triangle, the chsen pints made slving the area prblem much easier, since we did nt have t calculate the,, and with respect t the rigin f the sphere. On the ther hand, calculating the vertex angles fr the Euclidean triangle requires the use f the trignmetric arctangent functin. Frm ur last diagram, used t calculate the area f Euclidean ABC, we can see that: A = C = tan , , B = * tan A summary f the differences in the vertex angles between ur spherical and Euclidean triangle fllws: Triangle A B C Sum f s Spherical Euclidean Page 6

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Geometry Strand Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.

Geometry Strand Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. 2005 Gemetry Standards Algebra Strand Nte: The algebraic skills and cncepts within the Algebra prcess and cntent perfrmance indicatrs must me maintained and applied as students are asked t investigate,