Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both $\displaystyle 30-60-90$ triangles. Therefore, you can create a ratio to help you find $\displaystyle h$.

The ratio of the small base to the height is the same as $\displaystyle 1:\sqrt{3}$.  Therefore, you can write the following equation:

$\displaystyle \frac{0.5b}{h}=\frac{1}{\sqrt{3}}$

This means that $\displaystyle 0.5b\sqrt{3}=h$.

Now, the area of a triangle can be written:

$\displaystyle A=\frac{1}{2}bh$, and based on our data, we can replace $\displaystyle h$ with $\displaystyle \frac{b\sqrt{3}}{2}$.  This gives you:

$\displaystyle 56.25\sqrt{3}=\frac{1}{2}*b*\frac{b\sqrt{3}}{2}=\frac{b^2\sqrt{3}}{4}$

Now, let's write that a bit more simply:

$\displaystyle 56.25\sqrt{3}=\frac{b^2\sqrt{3}}{4}$

Solve for $\displaystyle b$. Begin by multiplying each side by $\displaystyle 4$:

$\displaystyle 225\sqrt{3}=b^2\sqrt{3}$

Divide each side by $\displaystyle \sqrt3$:

$\displaystyle b^2 = 225$

Finally, take the square root of both sides. This gives you $\displaystyle b=15$. Therefore, the perimeter is $\displaystyle 15*3 = 45$.

An equilateral triangle has 3 equal length sides.

Therefore the perimeter equation is as follows,

$\displaystyle P=3s$.

So divide the perimeter by 3 to find the length of each side.

Thus the answer is:

$\displaystyle 48=3s$

$\displaystyle \frac{48}{3}=s$

$\displaystyle 16 cm$

Since the triangle is equilateral, the base and the legs are equal, so the first step is to set the two equations equal to each other. Start with $\displaystyle 5x-1=3x+13$, add $\displaystyle 1$ to both sides giving you $\displaystyle 5x=3x+14$. Subtract $\displaystyle 3x$ from both sides, leaving $\displaystyle 2x=14$. Finally divide both sides by $\displaystyle 2$, so you're left with $\displaystyle x=7$. Plug $\displaystyle 7$ back in for $\displaystyle x$  into either of the equations so that you get a side length of $\displaystyle 34$. To find the perimeter, multiply the side length $\displaystyle \left ( 34\right )$, by $\displaystyle 3$, giving you $\displaystyle 102$.

To find perimeter of an quilateral triangle, simply multiply the side length by $\displaystyle 3$. Thus,

$\displaystyle P=s\cdot3=2.1\cdot3=6.3$

To solve, simply multiply the side length by $\displaystyle 3$. Thus,

$\displaystyle P=3s=3*4=12$

To solve, simply multiply the side length by 3 since they are all equal. Thus,

$\displaystyle P=3*s=3*2=6$

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of $\displaystyle x$, $\displaystyle x\sqrt{3}$, and $\displaystyle 2x$. The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to $\displaystyle 2x$. Using this information, we can find the lengths of each side fo the special triangle.

$\displaystyle 8=2x\rightarrow 4=x\rightarrow 4\sqrt{3}=x\sqrt{3}$

The side with length $\displaystyle x\sqrt{3}$ will be the height (opposite the 60 degree angle). The height is $\displaystyle 4\sqrt{3}$ inches.

Draw a vertical line from the vertex. This will divide the equilateral triangle into two congruent right triangles. For the hypothenuse of one right triangle, the length will be $\displaystyle 2\:cm$. The base will have a dimension of $\displaystyle 1\:cm$.  Use the Pythagorean Theorem to solve for the height, substituting in $\displaystyle 2$ for $\displaystyle c$, the length of the hypotenuse, and $\displaystyle 1$ for either $\displaystyle a$ or $\displaystyle b$, the length of the legs of the triangle:

$\displaystyle a^2+b^2=c^2$

$\displaystyle 1^2+b^2=2^2$

$\displaystyle b^2=2^2-1^2$

$\displaystyle b^2 = 3$

$\displaystyle b=\sqrt3$

Recall that from any vertex of an equilateral triangle, you can drop a height that is a bisector of that vertex as well as a bisector of the correlative side. This gives you the following figure:

Notice that the small triangles within the larger triangle are both $\displaystyle 30-60-90$ triangles. Therefore, you can create a ratio to help you find $\displaystyle h$.

The ratio of $\displaystyle 6$ to $\displaystyle h$ is the same as the ratio of $\displaystyle 1$ to $\displaystyle \sqrt{3}$.

As an equation, this is written:

$\displaystyle \frac{6}{h}=\frac{1}{\sqrt{3}}$

Solving for $\displaystyle h$, we get: $\displaystyle h=6\sqrt{3}$

To calculate the height of an equilateral triangle, first draw a perpendicular bisector for the triangle. By definition, this splits the opposite side from the vertex of the bisector in two, resulting in two line segments of length $\displaystyle 4$ inches. Since it is perpendicular, we also know the angle of intersection is $\displaystyle 90^{\circ}$. 

So, we have a new right triangle with two side lenghts $\displaystyle 8$ and $\displaystyle 4$ for the hypotenuse and short leg, respectively. The Pythagorean theorem takes over from here:

$\displaystyle 8^2 - 4^2 = x^2$ ---> $\displaystyle x^2 = 48 = 4\sqrt{3}$

So, the height of our triangle is $\displaystyle 4\sqrt{3}$.