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  congruent  triangles.

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

Recall that a  triangle has sides that are in ratios of . 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:

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

Now, since the perimeter of the shape consists of  of these sides, we can use the following equation to find the perimeter.

Below is regular Pentagon  with center , a segment drawn from  to each vertex - that is, each of its radii drawn.

The measure of each angle of a regular pentagon can be calculated by setting  equal to 5 in the formula

and evaluating:

By symmetry, each radius bisects one of these angles. Specifically, 

.

An equilateral triangle has three angles of measure , so  is not equilateral.

Assume . Then, since , it follows by the Isosceles Triangle Theorem that their opposite angles are also congruent. Since the measures of the angles of a triangle total , letting :

All three angles have measure , making  equiangular and, as a consequence, equilateral. Therefore, , and the perimeter, or the sum of the lengths of the sides, is

However, the perimeter is given to be 56. Therefore, .

The figure referenced is below:

 is equilateral, so .  is equilateral, so . By the Transitive Property of Congruence, . A quadrilateral with four congruent sides is a parallelogram and a rhombus. However, it is not a rectangle, and, consequently, not a square, since its angles are not right - , an angle of an equilateral triangle, measures . Also, a parallelogram is not a trapezoid. Therefore, the quadrilateral is a parallelogram and a rhombus only.

A midsegment of a triangle - a segment whose endpoints are the midpoints of two sides - is, by the Triangle Midsegment Theorem, parallel to the third side, and is half the length of that side. An equilateral triangle with perimeter 30 has three sides one third this, or 

.

Consequently, the length of each midsegment is half this, or

.

Equilateral triangles have sides of all equal length and angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = 

Given the height, we can now find the area of the triangle using the equation:

Equilateral triangles have sides of equal length, with angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = 

Given the height, we can now find the area of the triangle using the equation:

The answer is  .  

To find the area you would first need to find what the length of each side is: 54 divided by 3 is 18 for each side.  

Then you would need to draw in the altitude of the triangle in order to get its height.  Drawing this altitude will create two 30-60-90 degree triangles as shown in the picture.  The longer leg is  times the short leg.  Thus the height is . 

Next we plug in the base and the height into the formula to get 

The formula for the area of an equilateral triangle with side length  is

So, since ,

The formula of the area of an equilateral triangle is  if  is a side.

Since the sides of our triangle have doubled, they have changed from  to . We can substitute  into the equation and solve for the triangle's new area in terms of :