# How do you find the area of an equilateral triangle with a 3 inch apothem?

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#### Explanation

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2
Dec 31, 2015

$A = \frac{54 \sqrt{3}}{2} {\text{in"^2=46.77 "in}}^{2}$

#### Explanation:

Apothem is the distance from the center of a regular polygon perpendicular to one of its side.

Refer to this figure:

In order to find the area of the triangle, we use the formula for regular polygon $-$ means all sides and angles are equal:

$A = \frac{1}{2} P a$

Where:
$A \implies$ area
$P \implies$perimeter (sum of all side)
$a \implies$apothem

This problem requires knowledge in trigonometry since the only given is the apothem.
We now know that in order to solve for the area, we need to have both the apothem and the measurement of sides.

Since, this is an equilateral triangle, we just need to find one side and to find its perimeter, we just multiply it by 3.

Using this figure we solve for the side:

We know that each angle of the triangle is ${60}^{\circ}$ because it is an equilateral triangle. If we pass a line from the center to one of the interior angles of triangle, it will bisect the angle thus creating a ${30}^{\circ}$ $-$NOTE that this is only true for regular figures

Now, take a closer look. Since we have formed a right triangle inside, we can use trigonometric functions to solve for half of the side.

Using tangent function:

$\tan \theta = \left(\text{opposite")/("adjacent}\right)$

tan theta=("apothem")/(color(red)("half-of-side")) where half-of -side $= \textcolor{red}{\frac{s}{2}}$

$\tan {30}^{\circ} = \frac{3 \text{in}}{\frac{s}{2}}$

$\frac{s}{2} = \frac{3 \text{in}}{\tan} {30}^{\circ}$

$s = \frac{2 \cdot 3 \text{in")/(tan30^@) = (6"in")/(tan30^@) = (6"in}}{\frac{1}{\sqrt{3}}}$

$s = 6 \sqrt{3} \text{ in}$

Now we have the side, we can now compute for the perimeter:

$P = 3 s$

$P = 3 \left(6 \sqrt{3} \text{ in}\right)$

$P = 18 \sqrt{3} \text{ in}$

Then solving for area:

$A = \frac{1}{2} P a$

$A = \frac{1}{2} \left(18 \sqrt{3} \text{in")(3"in}\right)$

$A = \frac{54 \sqrt{3}}{2} {\text{in"^2 = 46.77 "in}}^{2}$

Then teach the underlying concepts
Don't copy without citing sources
preview
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#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Dec 31, 2015

$A = 3 \sqrt{3}$ inches

#### Explanation:

the apothem is the same as the height of the triangle.
there is a formula on equilateral triangles (it can on only be used if the triangle is equilateral). the formula is: $h = x \frac{\sqrt{3}}{2}$

like this:
$h = x \frac{\sqrt{3}}{2}$

$3 = x \frac{\sqrt{3}}{2}$

$3 \times 2 = x \sqrt{3}$

$\frac{6}{\sqrt{3}} = x$

$\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = x$

$\frac{6 \sqrt{3}}{3} = x$

$x = 2 \sqrt{3}$

now that you have the measure of the triangles' sides you can find the area of the triangle:

$A = \frac{b h}{2}$

$A = \frac{\left(2 \sqrt{3}\right) \left(3\right)}{2}$

$A = \frac{6 \sqrt{3}}{2}$

$A = 3 \sqrt{3}$

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