Holt Geometry

10-Ext Spherical Geometry10-Ext Spherical Geometry

Holt Geometry

Lesson PresentationLesson Presentation

Holt Geometry

10-Ext Spherical Geometry

Understand spherical geometry as an example of non-Euclidean geometry.

Objective

Holt Geometry

10-Ext Spherical Geometry

non-Euclidean geometryspherical geometry

Vocabulary

Holt Geometry

10-Ext Spherical Geometry

Euclidean geometry is based on figures in a plane. Non-Euclidean geometry is based on figures in a curved surface. In a non-Euclidean geometry system,the Parallel Postulate is not true. One type of non-Euclidean geometry is spherical geometry, which is the study of figures on the surface of a sphere.

Holt Geometry

10-Ext Spherical Geometry

A line in Euclidean geometry is the shortest path between two points. On a sphere, the shortest path between two points is along a great circle, so “lines” in spherical geometry are defined as great circles. In spherical geometry, there are no parallel lines. Any two lines intersect at two points.

Holt Geometry

10-Ext Spherical Geometry

The two points used to name a line cannot be exactly opposite each other on the sphere. In Example 1, AB could refer to more than one line.

Caution!

Holt Geometry

10-Ext Spherical Geometry

Example 1: Classifying Figures in Spherical Geometry

Name a line, a segment, and a triangle on the sphere.

PQ is a line

PQ is a segment.

∆PQR is a triangle.

Holt Geometry

10-Ext Spherical Geometry

Check It Out! Example 1

Name a line, segment, and triangle on the sphere.

AD is a line

AD is a segment.

∆BCD is a triangle.

Holt Geometry

10-Ext Spherical Geometry

Imagine cutting an orange in half and then cutting each half in quarters using two perpendicular cuts. Each of the resulting triangles has three right angles.

Holt Geometry

10-Ext Spherical Geometry

Example 2A: Classifying Spherical Triangles

Classify the spherical triangle by its angle measures and by its side lengths.

∆XYZ

acute isosceles triangle

Holt Geometry

10-Ext Spherical Geometry

Example 2B: Classifying Spherical Triangles

Classify the spherical triangle by its angle measures and by its side lengths.

∆STU on Earth has vertex S at the South Pole and vertices T and U on the equator. TU is equal to one-fourth the circumference of Earth.

right equilateral triangle

Holt Geometry

10-Ext Spherical Geometry

Check It Out! Example 2

Classify ∆VWX by its angle measures and by its side lengths.

∆VWX is equiangular and equilateral.

Holt Geometry

10-Ext Spherical Geometry

The area of a spherical triangle is part of the surface

area of the sphere. For the piece of orange on page

726, the area is of the surface area of the orange, or

. If you know the radius of a sphere and the

measure of each angle, you can find the area of the

triangle.

Holt Geometry

10-Ext Spherical Geometry

Holt Geometry

10-Ext Spherical Geometry

Example 3A: Finding the Area of Spherical Triangles

Find the area of each spherical triangle. Round to the nearest tenth, if necessary.

A ≈ 62.8 cm2

Holt Geometry

10-Ext Spherical Geometry

Example 3B: Finding the Area of Spherical Triangles

Find the area of the spherical triangle. Round to the nearest tenth, if necessary.

∆QRS on Earth with mQ = 78°, mR = 92°, and mS = 45°.

A ≈ 9,574,506.9 mi2

Holt Geometry

10-Ext Spherical Geometry

Find the area of ∆KLM on a sphere with diameter 20 ft, where mK = 90°, mL = 90°, and mM = 30°. Round to the nearest tenth.

A ≈ 52.4 ft2

Check It Out! Example 3