Geometry: An Interactive Journey to Mastery

Lesson 19: The Geometry of a Circle

([DPSOH

6\GQH\$XVWUDOLDKDVODWLWXGHƒVRXWKͼ௘6HH)LJXUH௘ͽ

The radius of the Earth is approximately 6400 km.

Find the radius rRIWKHFLUFOHRIFRQVWDQWODWLWXGHƒVRXWK

6ROXWLRQ

Note that, like a circle in two dimensions, a sphere is a set of points equidistant from

a given point in three-dimensional space.

The key to answering this question is to note that the line from the center of the Earth

WR6\GQH\LVDOVRDUDGLXVͼ௘6HH)LJXUH௘ͽ

We see cos 34 6400 r , giving r |6400 cos 34 5306 km.

([DPSOH

Find the value of the length a in )LJXUH.

6ROXWLRQ

Draw two chords, as shown in )LJXUH, to create two triangles.

The triangles have matching vertical angles and matching

LQVFULEHGDQJOHVͼ௘VXEWHQGHGIURPWKHVDPHDUF௘ͽDQGWKHUHIRUH

by AA, are similar.

Consequently, matching sides come in the same ratio:^23 a 5.

This gives a 3.^13

6WXG\7LS

x Remember that the measure of an arc is a measure of an

DPRXQWRIWXUQLQJͼ௘DQJOH௘ͽQRWDSK\VLFDOOHQJWK,WLVWKH

angle formed by the two radii to the endpoints of the arc.

Figure 19.10

Equator

Sydney

34°

6400

r

34°

Figure 19.11

5

3

a 2

Figure 19.12

5

3

a 2

Figure 19.13