To solve this solution, work backwards using the formula: 

\(\displaystyle Area=\frac{b\cdot h}{2}\)

Plugging in the given values we are able to solve for the height.
\(\displaystyle 9=\frac{3h}{2}\)

\(\displaystyle 3h=18\)

\(\displaystyle h=18\div3=6\)

To solve this solution, work backwards using the formula: 

\(\displaystyle Area=\frac{b\cdot h}{2}\)

Plugging in the given values we are able to solve for the height.

\(\displaystyle 225=\frac{30h}{2}\)

\(\displaystyle 30h=450\)

\(\displaystyle h=\frac{450}{30}=15\)

To solve this solution, first work backwards using the formula: 

\(\displaystyle Area=\frac{b\cdot h}{2}\)

Plugging in the given values we are able to solve for the height.
\(\displaystyle 450=\frac{45h}{2}\)

\(\displaystyle 900=45h\)

\(\displaystyle h=900\div45=20\)

An equilateral triangle can be broken down into 2 30-60-90 right triangles (see image). The length of a side (the base) is 2x while the length of the height is \(\displaystyle x\sqrt3\). The area of a triangle can be calculated using the following equation:

\(\displaystyle A=\frac{1}{2}bh\)

Therefore, if \(\displaystyle 2x\) equals the length of a side:

\(\displaystyle A=16\sqrt3=\frac{1}{2}bh=\frac{1}{2}(2x)(x\sqrt3)=x^2\sqrt3\)

\(\displaystyle 16\sqrt3=x^2\sqrt3\)

\(\displaystyle 16=x^2\)

\(\displaystyle x=4\)

A length of the side equals 2x:

\(\displaystyle 2(4)=8\)

We know the formula for the area of an equilateral triangle is:

\(\displaystyle Area=\frac{\sqrt{3}}{4}a^2\)

if \(\displaystyle a\) is the side of the triangle.

So, since we are told that \(\displaystyle a=1\:cm\), we can substitute in \(\displaystyle 1\) for \(\displaystyle a\) and solve for the area of the triangle:

\(\displaystyle Area=\frac{\sqrt{3}}{4}1^2=\frac{\sqrt{3}}{4}1=\frac{\sqrt{3}}{4}\:cm^2\)

This triangle is equilateral; we can tell because each of its sides are the same length, \(\displaystyle a\). To find the length of one side, we need to divide the perimeter by \(\displaystyle 3\):

\(\displaystyle a=\frac{18}{3}=6\)

Since each of this triangle's sides is equal in length, it is equilateral. To find the length of one side of an equilateral triangle, we need to divide the perimeter by \(\displaystyle 3\).

\(\displaystyle a=\frac{\frac{3}{7}}{3}=\frac{1}{7}\:cm\)

Since it is an equilateral triangle, the line that represents the height bisects it into a 30-60-90 triangle.

Here you may use \(\displaystyle sin(60)=\frac{4}{hypotenuse}\) and solve for hypotenuse to find one of the sides of the triangle.

Use the definition of an equilateral triangle to know that the answer of the hypotenuse also applies to the base of the triangle.

Therefore,

\(\displaystyle hypotenuse=\frac{4}{sin(60)}=4.6\)

The height of an equilateral triangle, shown by the dotted line, is also one of the legs of a right triangle:

The hypotenuse is x, the length of each side in this equilateral triangle, and then the other leg is half of that, 0.5x. 

To solve for x, use Pythagorean Theorem:

\(\displaystyle (0.5x)^2 + 5^2 = x^2\) square the terms on the left

\(\displaystyle 0.25x^2 + 25 = x^2\) combine like terms by subtracting 0.25 x squared from both sides

\(\displaystyle 25 = 0.75x^2\) divide both sides by 0.75

\(\displaystyle 33. \overline 3 = x^2\) take the square root of both sides

\(\displaystyle \sqrt{ 33. \overline 3 } = x\)

Recall that the perimeter is the sum of all the exterior sides of a shape. The sides that add up to the perimeter are highlighted in red.

Since the equilateral triangle shares a side with the square, each of the five sides that are outlined have the same length.

Recall that the height of an equilateral triangle splits the triangle into \(\displaystyle 2\) congruent \(\displaystyle 30-60-90\) triangles.

We can then use the height to find the length of the side of the triangle.

Recall that a \(\displaystyle 30-60-90\) triangle has sides that are in ratios of \(\displaystyle 1:\sqrt3:2\). The smallest side in the given figure is the base, the second longest side is the height, and the longest side is the side of the triangle itself.

Thus, we can use the ratio and the length of the height to set up the following equation:

\(\displaystyle \frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}\)

\(\displaystyle \text{side}=\frac{2(\text{height})}{\sqrt3}=\frac{2\sqrt3(\text{height})}{3}\)

Plug in the given height to find the length of the side.

\(\displaystyle \text{side}=\frac{2\sqrt3(\text{2})}{3}=\frac{4\sqrt3}{3}\)

Now, since the perimeter of the shape consists of \(\displaystyle 5\) of these sides, we can use the following equation to find the perimeter.

\(\displaystyle \text{Perimeter}=5(\text{side})\)

\(\displaystyle \text{Perimeter}=5(\frac{4\sqrt3}{3})=\frac{20\sqrt3}{3}=11.55\)