HCPSS Worthwhile Math Task
How on Earth?
Common Core Standard
G.C.B.5 - Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius.
Common Core Geometry, Unit 3
The Task
The diameter of the Earth is approximately 7,900 miles.
Lines of latitude are circles that are parallel to each other and wrap around the Earth from east to
west. (The equator is the only latitude line that forms a circle with the full diameter of the earth.)
Lines of longitude are slightly different. They are circles that run north and south, but they all
intersect at the north and south poles. (They all have the full diameter of the earth.)
Both are marked in degrees. Using a globe or Google Earth and the degree measures of the
latitude and longitude lines:


Determine which continent (land) contains the most length at the equator? What is that
length?
Find the two points farthest apart directly north and south of each other in the United
States. Where are these points? How far apart are they?
Facilitator Notes
1. Assign students to groups of 2 or 3.
2. Each group will need access to a globe or Google Earth.
3. Give some basic instruction about latitude and longitude lines and let groups explore
briefly
4. Have students use the latitude and longitude lines to discover the number of degrees in
the distances they are to discover.
5. Have students use the degrees to compute the actual distances on the surface of the earth
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
Follow-Up Questions
1. It is possible to generalize how to find the length of an arc on a circle. Find the general
method of determining the length of such an arc.
2. Why do you think the lines of latitude and longitude are drawn differently, one group
parallel to each other, and one group intersecting at the poles?
3. Why do the measures for both, in degrees, only go to 180 and not 360?
Solution
Using the diameter of the earth of 7900 miles, the radius is 3950 miles. The circumference of the
“full” circles on the Earth is 2πr = πd ≈ 24800 miles
At the equator, the only two land masses that have significant east/west length are South
America and Africa. Africa is longer, extending from approximately 10o to 45o for 35o, while
South America extends from approximately 50o to 80o, for only 30o. The length of the equator
across Africa is 35/360 (24800) ≈ 2411 miles
There are two potential answers here.
1. The first is the distance is the continental distance from the southern tip of Texas to the
US/Canada border. It extends from about 25o to 49o, for a total of 24o. As in the first
exploration, 24/360 (24800) = 1/15(24800) ≈ 1653 miles
2. There is a greater north/south distance between points in the US between Hawaii and
Alaska from about 19o to 71o for a total of 52o. (It can be argued that this is not the
correct answer as the arc is, in a large part, out of the US). As before 52/360(24800) =
13/90(24800) ≈3582 miles.
Answers to Follow Up Questions:
1. The arc length on a circle can be found by finding arc length in degrees or central angle
that captures the arc in degrees (these are equivalent). Calling that measure x, the actual
arc length is (x/360)(2πr) or (x/360)(circumference). Some may find it more accessible
to set up a ratio: x/360=arc length/circumference to solve for arc length.
2. The Earth rotation is designated on a north/south axis. There are absolute north and south
on the Earth. So lines that run north and south must cross through the poles. There is no
absolute east and west as Earth rotation is in this direction. So lines that run east and west
are equally spaced.
3. The fact that we are actually dealing with a sphere rather than an isolated circle means we
need a uniform system for all the circles. In the north/south orientation there are an
infinite number of circles that all need the same measures. So the equator is 0o and the
degree measures extend in both directions (north and south) from all points on the
equator 180o to the north and south poles. In the east west orientation, the prime meridian
(through England) is designated as zero and the degree measures extend in both
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.
HCPSS Worthwhile Math Task
directions (east and west) from all points on the meridian 180o to the international date
line. This could have been avoided, since all these circles are parallel, with a 3600 system
going strictly east or west, but this was probably avoided to be somewhat consistent with
the north/south orientation.
Extension Activities
A web exploration of navigation techniques, GPS location, and early navigation by clocks.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.