Non-Euclidean Geometry

All the lessons prior to this one have dealt with Euclidean geometry, which is a system of geometry described by the Greek mathematician Euclid. The primary difference between Euclidean geometry and other geometries is that Euclidean geometry assumes the Parallel Postulate holds.

Geometry Lesson 109

Non-Euclidean geometry is based on figures in a curved surface and is a system of geometry where the Parallel Postulate does not hold true. One type of non-Euclidean geometry is spherical geometry, a system of geometry defined on a sphere.

Geometry Lesson 109

Recall that a circle dividing a sphere into two hemispheres is called a great circle. On a sphere, the shortest distance between any two points is a minor arc of a great circle. Therefore, instead of referring to lines in spherical geometry, we refer to arcs of great circles. An angle on a sphere is the angle between the planes that two intersecting great circles lie on.

Geometry Lesson 109

Identify a great circle, a segment, an angle, and a triangle on the sphere.

SOLUTION

There are three great circles: great circle

AB, great circle CD, and great circle AD.

A segment on the sphere is 𝐶𝐷. An angle on the sphere is the one formed by the intersection of great circles AB and AD.

A triangle consists of three connected points. Therefore, ACD is a triangle.

Geometry Lesson 109

In this exploration, you will investigate spherical geometry using a globe. 1. Choose three cities on a globe. Their positions should form a triangle. Make sure they are far enough apart to easily measure. 2. With a tape measure, measure the distances between the cities as accurately as possible. 3. Use the Law of Cosines to find the angles in the triangle. 4. Use a protractor to measure the angles in the triangle on the sphere. Are they the same as the answers you found with trigonometry? 5. Now add the angle measures of the triangle. Do they add to 180°? If not, is it more or less?

Geometry Lesson 109

In spherical geometry, the angle measures of a triangle will always sum to more than 180°. Imagine three points on a sphere that are close together. The triangle they form will be relatively flat, and the sum of its angles will be close to 180°. As the points get farther away from each other, the triangle becomes more curved, and the sum of its angle measures gets further from 180°.

Geometry Lesson 109

Classify the triangle shown according to its angle measures and side lengths.

SOLUTION 𝑚∠𝐴 = 90° 𝑚∠𝐵 = 90° 𝑚∠𝐶 = 90°

This is an equiangular and equilateral

triangle, since all the side lengths and

all the angle measures are equal.

Notice that the sum of the angles of

the triangle is 270°.

Geometry Lesson 109

The area of a spherical triangle is part of the surface area of the sphere. If the radius of the sphere and the measures of each angle of the triangle are known, then the area of the triangle can be determined. Area of a Spherical Triangle - The area of a triangle on a sphere is given by the following formula, where L, M, and N are the vertices of the triangle and he sphere has a radius of r.

𝐴 =𝜋𝑟2

180°𝑚∠𝐿 + 𝑚∠𝑀 + 𝑚∠𝑁 − 180°

Geometry Lesson 109

Find the area of the spherical triangle when the radius of the sphere is 3 meters. Round your answer to the nearest hundredth. SOLUTION Using the formula for area of a spherical triangle:

𝐴 =𝜋𝑟2

180°𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 − 180°

𝐴 =𝜋32

180°85° + 85° + 80° − 180°

𝐴 =7𝜋

2

𝐴 ≈ 11.00 The area of the spherical triangle is about 11 square meters.

Geometry Lesson 109

a. Identify a great circle, a segment, and a triangle on the sphere.

b. Classify the triangle as equilateral, isosceles, or scalene.

Geometry Lesson 109

c. Find the area of the spherical triangle to the nearest tenth. The diameter of the sphere is 24 inches.

Geometry Lesson 109

Page 706

Lesson Practice (Ask Mr. Heintz)

Page 706

Practice 1-30 (Do the starred ones first)

Geometry Lesson 109