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# polygons [Solved!]

### My question

Trying to find the area of a honey comb. My answer does not equal the online answer. The actual question is Question 7 at the url I entered below.

### Relevant page

Polygon Area Practice - MathBitsNotebook(Geo - CCSS Math)

### What I've done so far

From vertex to vertex the distance is 5. Therefore the radius is 2.5.

a = apothem

Using r = a/cos (180/n) the apothem then is 2.17.

Constructing a right triangle, the unknown side is 1.24.

Multiply this by 2 to get the length of one side: 2.48

perimeter = 2.48 x 6 = 14.88

Area = .5 x 14.88 x 2.17 = 16.1448

16.1448 x 36 1/2 cells = 589.2852 mm^2

X

Trying to find the area of a honey comb.  My answer does not equal the online answer.  The actual question is Question 7 at the url I entered below.
Relevant page

<a href="https://mathbitsnotebook.com/Geometry/Polygons/POpolygonAreaPractice.html">Polygon Area Practice - MathBitsNotebook(Geo - CCSS Math)</a>

What I've done so far

From vertex to vertex the distance is 5.  Therefore the radius is 2.5.

a = apothem

Using r = a/cos (180/n) the apothem then is 2.17.

Constructing a right triangle, the unknown side is 1.24.

Multiply this by 2 to get the length of one side: 2.48

perimeter = 2.48 x 6 = 14.88

Area = .5 x 14.88 x 2.17 = 16.1448

16.1448 x 36 1/2 cells = 589.2852 mm^2

## Re: polygons

But I think there is an easier way to find it. If the vertex-to-vertex length is 5 mm, then the side of each triangle (they are equilateral triangles) is 2.5 mm.

Using the formula

"Area hexagon" = (3sqrt(3))/2 s^2,

where s is the side length, we obtain for one cell:

"Area" = (3sqrt(3))/2 (2.5)^2 ~~ 16.23798 "mm"^2

Since there are 36.5 cells in the illustration, the total area would be:

"Area" = 36.5 xx 16.23798 ~~ 592.687 "mm"^2, which is closest to their second answer.

(Their accepted answer, 2371, would mean each cell has area 2371/36.5 = 64.96. However, if we take a rough estimate for the hexagon's area via the circle through the vertices, we get "Area" = pi r^2 = pi(2.5)^2 = 19.63. The actual area has to be less than this for each cell.)

X

@phinah: I (roughly) agree with your answer.

But I think there is an easier way to find it. If the vertex-to-vertex length is 5 mm, then the side of each triangle (they are equilateral triangles) is 2.5 mm.

Using the formula

"Area hexagon" = (3sqrt(3))/2 s^2,

where s is the side length, we obtain for one cell:

"Area" = (3sqrt(3))/2 (2.5)^2 ~~ 16.23798 "mm"^2

Since there are 36.5 cells in the illustration, the total area would be:

"Area" = 36.5 xx 16.23798 ~~ 592.687 "mm"^2, which is closest to their second answer.

(Their accepted answer, 2371, would mean each cell has area 2371/36.5 = 64.96. However, if we take a rough estimate for the hexagon's area via the circle through the vertices, we get "Area" = pi r^2 = pi(2.5)^2 = 19.63. The actual area has to be less than this for each cell.)

## Re: polygons

We took "vertex-to-vertex" to mean non-adjacent vertices as well as adjacent vertices.

So we measured from the furthest left vertex to the furthest right vertex to be 5 mm. That is how we arrived at 589. We reworked the exercise identifying the segments between adjacent vertices to be = to 5 mm.

So this is how we went forward:

apothem = s/[2 tan (180/n)] = 5/[2 tan 30] = 4.33

So Area = 1/2(apothem)(perimeter) = (1/2)(4.33)(30) = 64.95 times 36.5 = 2370.6 = 2371.

Thanks for helping.

X

We took "vertex-to-vertex" to mean non-adjacent vertices as well as adjacent vertices.

So we measured from the furthest left vertex to the furthest right vertex to be 5 mm. That is how we arrived at 589. We reworked the exercise identifying the segments between adjacent vertices to be = to 5 mm.

So this is how we went forward:

apothem = s/[2 tan (180/n)] = 5/[2 tan 30] = 4.33

So Area = 1/2(apothem)(perimeter) = (1/2)(4.33)(30) = 64.95 times 36.5 = 2370.6 = 2371.

Thanks for helping.

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