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Title:
Apothem and Area
Problem Statement:
Determine an equation of the area of a regular hexagon in terms of its apothem and perimeter. The apothem is
the shortest distance from the center of the polygon to one of the sides.
Will this equation be true for other regular polygons? Explain.
Problem setup:
I will find the area of the polygon using the perimeter of the polygon and the apothem. The apothem is the
perpendicular segment from the center of the polygon to the midpoint of any side segment. I will begin thinking
about forming a regular hexagon by looking at the possible internal triangles that can be formed, especially
looking at angle degrees. Internal triangles can be formed by joining any two adjacent segment endpoints and
the center point of the polygon. The apothem will be the altitude (height) of an internal triangle.
Plans to Solve/Investigate the Problem:
Using GSP, I will construct a regular hexagon. After constructing the regular hexagon, I will construct the
hexagon interior, which will enable me to measure the area and perimeter. I will closely explore the triangles
that make up the regular hexagon and use this information to help me determine an equation of the area in terms
of the regular hexagon’s apothem (the shortest distance from the center of a polygon to one of its side) and
perimeter. I will also examine other regular polygons to see if the equation relates to them as well.
Investigation/Exploration of the Problem:
First, I had to figure out how to construct a regular hexagon using GSP. The measurements could not be
determined by using trial and error methods. I know this because I tried multiple approaches before figuring
out how to go about it the correct way. First, I constructed a circle using the GSP tool bar. I highlighted the
outside point first and then the point in the center and went to the construct menu and chose circle by center +
point. These two circles are interlocked and ones center point is the others side point and vice versa. I placed
another side point directly opposite the first side point of the original circle, highlighted this point and then the
center point and constructed a third circle.
I placed points where the circles overlapped. To make sure they are in the correct position, I dragged the mouse
over the area and waited until both circles were highlighted, then I placed the point. I did this in four different
areas – two near the tops of the circles and two near the bottoms.
Next I highlighted the points in the shape of a triangle and one by one I created six triangles, which, by hiding
the circles you can see I constructed a regular hexagon.
C
A
G
D
B
E
F
To prove I had constructed a regular hexagon I measured each of the
angles coming from the center point of
each triangle. They should each be 60 degrees because they were created in a circle and a circle can only add
up to 360 and there were 6 triangles –> 360/6=60.
mABC = 60.00
mCBD = 60.00
mDBE = 60.00
mABC+mCBD+mDBE+mEBF+mFBG+mGBA = 360.00
mEBF = 60.00
mFBG = 60.00
mGBA = 60.00
Next, I highlighted all of the points along the outside so that I could construct the hexagon interior, which
enabled me to measure the perimeter and area of the hexagon. I also created the interior of triangle ABC, which
also enabled me to measure its area. I measured the midpoints of each of the sides to find the apothem. I could
have used any of the sides, I decided on the bottom side (see picture below). The apothem is equal to the height
of the individual triangles. If you break down the equation for the area of a triangle and add or combine the
measurements of the hexagon, then you would see ½ x side (base) x apothem (height).
I examined several measurements of this regular hexagon. The total area of the regular hexagon divided by a
single triangle that is within the hexagon is equal to 6. This should be the case since six congruent triangles
make up the regular hexagon – the number or triangles is equal to the number of sides.
C
A
Area
Area DCAGFE = 44.26 cm2
apothem = 3.57 cm
Perimeter DCAGFE = 24.77 cm
Area DCAGFE
Area
ABC 
ABC = 7.38 cm2
0.5m CDapothem = 7.38 cm2
= 6.00
Area DCAGFE
D
G
apothem
= 12.38 cm
Perimeter DCAGFE 
B

1
6
= 4.13 cm
m FE = 4.13 cm
apothem
Area
F
H
ABC6 = 44.26 cm2
E
As noted above, since the apothem is equal to the height the triangles and the base is equal to 1/6 of the
perimeter, the equation for the area of a regular hexagon in terms of its apothem and perimeter is the equal to
[½ x one side length (base) x apothem (height)] x 6 or area of one of the interior triangles x # of sides.
This will work for other regular polygons because within regular polygons there are congruent triangles and as
we saw above the base of the triangles are equal to the sides, the height is equal to the apothem, and then when
multiplied by the number of sides you will get the area of the polygon.
**Examples:
Pentagon
Area HIDI'J = 23.49 cm2
F
H
apothem
apothem = 2.54 cm
J
K
G
m KH = 3.70 cm
I
E
B
0.5m KHapothem5 = 23.49 cm2
I'
D
Octagon
Area P 1 = 10.25 cm2
W
T
V
P1
apothem = 1.76 cm
m XY = 1.46 cm
U
0.5m XYapothem8 = 10.25 cm2
X apothem
Y
GPS:
Geometry
Students will further develop their understanding of plane and solid geometric figures, incorporating the
use of appropriate technology and using this knowledge to solve authentic problems.
M6G1. Students will further develop their understanding of plane figures.
a. Determine and use lines of symmetry.
c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar
plane figures.
d. Interpret and sketch simple scale drawings.
e. Solve problems involving scale drawings.
Measurement
Students will understand how to determine the volume and surface area of solid figures. They will
understand and use the customary and metric systems of measurement to measure quantities efficiently
and to represent volume and surface area appropriately.
M6M2. Students will use appropriate units of measure for finding the perimeter, area, and volume and
express the answer using the appropriate unit.
b. Select and use units of appropriate size and type to measure length,
perimeter, area and volume.
Author & Contact:
Mandie Pulley
[email protected]
Important Note: You should compose your write-up targeting an audience in mind rather than just the
instructor for the course. You are creating a page to publish it on the web.