How To Find The Apothem

The apothem of a hexagon is the length of the line that joins the center of the hexagon with the center of ane side. The apothem is the perpendicular line that connects the heart of the hexagon with ane side. The apothem tin be very useful when we want to find the area of a hexagon since it allows us to use a simpler formula.

Nosotros can calculate the apothem by dividing the hexagon into six congruent triangles and using i of the triangles. Then, we employ the Pythagorean theorem or trigonometry to derive unlike formulas.

GEOMETRY

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Learning about the apothem of a hexagon with examples.

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GEOMETRY

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Learning well-nigh the apothem of a hexagon with examples.

Meet examples

Formula to find the apothem of a hexagon

In that location are two main methods that nosotros can use to derive a formula for the apothem of a hexagon. Both methods involve dividing the hexagon into six coinciding triangles equally in the following paradigm:

First method: We tin use the Pythagorean theorem on one of the triangles formed. We know that the triangles formed are equilateral, so their three sides are equal. Besides, cartoon the apothem, we run across that it divides the base of operations into two equal parts as in the image:

hexagon divided into six triangles with apothem

Therefore, using these lengths, nosotros have:

$latex {{s}^ii}={{a}^2}+{{(\frac{south}{2})}^2}$

$latex {{a}^two}={{s}^ii}-{{(\frac{s}{2})}^2}$

$latex {{a}^two}={{s}^2}-\frac{{{s}^2}}{4}$

$latex {{a}^2}=\frac{3{{southward}^2}}{four}$

$latex a=\frac{\sqrt{iii}s}{2}$

Second method: Similar to the previous method, we dissever the hexagon into six congruent triangles. When plotting the apothem, nosotros divide the triangle in half, which means that if we did this with all the triangles, we would have 12 small triangles in full.

To use trigonometry, we take to discover the central angle. Since the total central bending equals 360°, the central angle of each small triangle measures $latex 360 \div 12 = xxx$°:

hexagon divided into six triangles with apothem and internal angle

Now, nosotros can use the tangent. We know that the tangent of an angle is equal to the contrary side over the side by side side:

$latex \tan(thirty°)=\frac{\text{opposite}}{\text{adjacent}}$

$latex \tan(30°)=\frac{\frac{s}{2}}{a}$

$latex \tan(30°)=\frac{southward}{2a}$

$latex a=\frac{s}{2\tan(30°)}$

Apothem of a hexagon – Examples with answers

The following examples use both formulas for the apothem of a hexagon seen above. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the result.

Case 1

What is the length of the apothem of a hexagon that has sides of length 6 m?

Solution

Start formula:Using the start formula with $latex s = 6$, we have:

$latex a=\frac{\sqrt{3}s}{2}$

$latex a=\frac{\sqrt{iii}(6)}{2}$

$latex a=v.2$

The length of the apothem is 5.2 thousand.

Second formula:Using the second formula with $latex s= six$, we have:

$latex a=\frac{s}{2\tan(30°)}$

$latex a=\frac{6}{2\tan(xxx°)}$

$latex a=5.2$

We see that nosotros got the same result using both formulas.

EXAMPLE 2

A hexagon has sides of length viii m. What is the length of its apothem?

Solution

First formula:We tin apply $latex s = 8$ in the first formula:

$latex a=\frac{\sqrt{three}due south}{2}$

$latex a=\frac{\sqrt{iii}(viii)}{two}$

$latex a=6.93$

The length of the apothem is six.93 m.

Second formula: If nosotros now use the second formula with $latex s = viii$, we have:

$latex a=\frac{due south}{2\tan(30°)}$

$latex a=\frac{eight}{two\tan(30°)}$

$latex a=vi.93$

Nosotros got the same length with both formulas.

EXAMPLE 3

What is the length of the apothem of a hexagon with sides of length 10 yard?

Solution

First formula:Using the starting time formula with $latex due south = ten$, nosotros have:

$latex a=\frac{\sqrt{3}s}{2}$

$latex a=\frac{\sqrt{3}(ten)}{2}$

$latex a=8.66$

The length of the apothem is 8.66 m.

Second formula:Using the second formula with $latex s=ten$, nosotros have:

$latex a=\frac{s}{2\tan(thirty°)}$

$latex a=\frac{10}{2\tan(30°)}$

$latex a=8.66$

The aforementioned result was obtained with both formulas.

Example 4

What is the length of the sides of a hexagon that has an apothem of 7.5 one thousand?

Solution

In this case, we take the length of the apothem and we desire to find the length of the sides of the hexagon.

First formula:We utilize the starting time formula with $latex a = 7.5$ and solve fors:

$latex a=\frac{\sqrt{3}southward}{2}$

$latex seven.5=\frac{\sqrt{3}southward}{2}$

$latex 15=\sqrt{three}s$

$latex southward=8.66$

The length of the sides of the hexagon is viii.66 m.

Second formula:We use the second formula with $latex a = 7.5$ and solve fordue south:

$latex a=\frac{s}{2\tan(30°)}$

$latex 7.v=\frac{s}{2\tan(30°)}$

$latex due south=7.v(ii\tan(thirty°))$

$latex s=8.66$

Nosotros got the same length with both formulas.

EXAMPLE five

What is the length of the sides of a hexagon that has an apothem of length 12 m?

Solution

Again, we use both formulas and solve fors.

First formula:We substitute $latex a=12$ and solve fordue south:

$latex a=\frac{\sqrt{iii}southward}{2}$

$latex 12=\frac{\sqrt{3}due south}{2}$

$latex 24=\sqrt{3}s$

$latex s=13.87$

The length of the sides of the hexagon is thirteen.87 m.

Second formula:We substitute $latex a=12$ and we solve fors:

$latex a=\frac{south}{2\tan(30°)}$

$latex 12=\frac{s}{ii\tan(xxx°)}$

$latex s=12(2\tan(thirty°))$

$latex s=13.86$

In this example, nosotros got a small difference in the decimals. This is due to the rounding used when performing the operations.

Apothem of a hexagon – Practice problems

Exercise using the formulas of the hexagon apothem and solve the following issues. Select an answer and click "Check" to check your answer. If you need help with this, you tin look at the solved examples higher up.

What is the length of the apothem of a hexagon that has sides of length 11m?

Choose an reply

$latex a=8.93m$

$latex a=9.11m$

$latex a=9.53m$

$latex a=ten.12m$

A hexagon has sides of length 15m. What is the length of the apothem?

Choose an reply

$latex a=12.5m$

$latex a=13m$

$latex a=13.5m$

$latex a=fourteen.2m$

A hexagon has an apothem of length 18m. What is the length of the sides?

Choose an respond

$latex southward=sixteen.58m$

$latex s=17.22m$

$latex due south=18.35m$

$latex s=20.78m$

A hexagon has an apothem of length 20m. What is the length of the sides?

Cull an answer

$latex s=23.09m$

$latex s=24.12m$

$latex s=25.12m$

$latex s=25.87m$