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MTH 250 Graded Assignment 10, part two
Polygons in Euclidean and hyperbolic geometry
Material from Kay, sections 4.4, 6.3
Q1:
a) What is the sum of the angle measures of a convex decagon in Euclidean
geometry?
b) If the decagon is regular, what is the measure of each vertex angle in
Euclidean geometry?
c) And the measure of each central angle?
d) If the length of a side measures 42, what is the length of the apothem?
e) What is the perimeter?
f)
And what is the area?
Q2:
a) Suppose you have a regular nonagon in hyperbolic geometry, with defect
equal to 20. What
is the sum of the angle measures of this nonagon?
b) What is the measure of each vertex angle of this nonagon?
c) What is the area of this nonagon?
Q3:
Compute the value of the defect, perimeter, and area for the hyperbolic
heptagon shown below:
Q4: In the figure shown below, the two non-triangular polygons are
regular polygons, and the defects of the three polygons are as
indicated. Find the measures of each of the angles of the triangle.
[Kay, section 6.3, p443, #14]
Q5: We've seen that rectangles (and therefore squares) don't exist in
hyperbolic geometry. However,
we do have a process (described in the lecture) for constructing regular
polygons (polygons where all
sides have the same length, and all angles have the same measure) – you
radiate rays out from a
center point. Go over to NonEuclid and construct a regular quadilateral (all
four sides and angles
equal). So regular quadrilaterals are possible in hyperbolic geometry... they
just aren't squares! What
are the angle measures of YOUR regular quadrilateral? Please attach a
printout of the quadrilateral
image with this assignment.
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