To find the perimeter of an equilateral triangle given its area, we must first find the length of the sides. This can be done by using the equation of the area of an equilateral triangle:

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

where a is the side of the triangle.

Because the sides of the equilateral triangle are equal, the perimeter is equal to 3a.

To find the perimeter of an equilateral triangle given its area, we must first find the length of the sides. This can be done by using the equation of the area of an equilateral triangle:

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

where a is the side of the triangle.

Because the sides of the equilateral triangle are equal, the perimeter is equal to 3a.

The perimeter of a triangle is the sum of all three sides.

Because an equilateral triangle has all three sides of equal length, we have

\(\displaystyle P = 10 +10+10=30cm\)

The perimeter of a triangle is the sum of all three sides.

Because an equilateral triangle has all three sides of equal length, we have

\(\displaystyle P = 5+5+5=15cm\)

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent \(\displaystyle 30-60-90\) triangles.

Recall that the side lengths in a \(\displaystyle 30-60-90\) triangle are in a \(\displaystyle 1:\sqrt3:2\) ratio. Thus, the radius of the circle, which is also the base of the \(\displaystyle 30-60-90\) triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

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

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

\(\displaystyle \text{side}=\text{diameter}\)

We have two sides of the equilateral triangle and the circumference of a semi-circle.

\(\displaystyle \text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}\)

Plug in the length of the side to find the perimeter.

\(\displaystyle \text{Perimeter}=2(2\sqrt3)+\frac{\pi(2\sqrt3)}{2}=4\sqrt3+\pi\sqrt3=12.37\)

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent \(\displaystyle 30-60-90\) triangles.

Recall that the side lengths in a \(\displaystyle 30-60-90\) triangle are in a \(\displaystyle 1:\sqrt3:2\) ratio. Thus, the radius of the circle, which is also the base of the \(\displaystyle 30-60-90\) triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

\(\displaystyle \text{side}=\frac{2\sqrt3(6)}{3}=4\sqrt3\)

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

\(\displaystyle \text{side}=\text{diameter}\)

We have two sides of the equilateral triangle and the circumference of a semi-circle.

\(\displaystyle \text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}\)

Plug in the length of the side to find the perimeter.

\(\displaystyle \text{Perimeter}=2(4\sqrt3)+\frac{\pi(4\sqrt3)}{2}=8\sqrt3+2\pi\sqrt3=24.74\)

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent \(\displaystyle 30-60-90\) triangles.

Recall that the side lengths in a \(\displaystyle 30-60-90\) triangle are in a \(\displaystyle 1:\sqrt3:2\) ratio. Thus, the radius of the circle, which is also the base of the \(\displaystyle 30-60-90\) triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

\(\displaystyle \text{side}=\frac{2\sqrt3(7)}{3}=\frac{14\sqrt3}{3}\)

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

\(\displaystyle \text{side}=\text{diameter}\)

We have two sides of the equilateral triangle and the circumference of a semi-circle.

\(\displaystyle \text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}\)

Plug in the length of the side to find the perimeter.

\(\displaystyle \text{Perimeter}=2(\frac{14\sqrt3}{3})+\frac{\pi(\frac{14\sqrt3}{3})}{2}=\frac{28\sqrt3}{3}+\frac{7\pi\sqrt3}{3}=28.86\)

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent \(\displaystyle 30-60-90\) triangles.

Recall that the side lengths in a \(\displaystyle 30-60-90\) triangle are in a \(\displaystyle 1:\sqrt3:2\) ratio. Thus, the radius of the circle, which is also the base of the \(\displaystyle 30-60-90\) triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

\(\displaystyle \text{side}=\frac{2\sqrt3(11)}{3}=\frac{22\sqrt3}{3}\)

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

\(\displaystyle \text{side}=\text{diameter}\)

We have two sides of the equilateral triangle and the circumference of a semi-circle.

\(\displaystyle \text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}\)

Plug in the length of the side to find the perimeter.

\(\displaystyle \text{Perimeter}=2(\frac{22\sqrt3}{3})+\frac{\pi(\frac{22\sqrt3}{3})}{2}=\frac{44\sqrt3}{3}+\frac{11\pi\sqrt3}{3}=45.36\)

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent \(\displaystyle 30-60-90\) triangles.

Recall that the side lengths in a \(\displaystyle 30-60-90\) triangle are in a \(\displaystyle 1:\sqrt3:2\) ratio. Thus, the radius of the circle, which is also the base of the \(\displaystyle 30-60-90\) triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

\(\displaystyle \text{side}=\frac{2\sqrt3(13)}{3}=\frac{26\sqrt3}{3}\)

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

\(\displaystyle \text{side}=\text{diameter}\)

We have two sides of the equilateral triangle and the circumference of a semi-circle.

\(\displaystyle \text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}\)

Plug in the length of the side to find the perimeter.

\(\displaystyle \text{Perimeter}=2(\frac{26\sqrt3}{3})+\frac{\pi(\frac{26\sqrt3}{3})}{2}=\frac{52\sqrt3}{3}+\frac{13\pi\sqrt3}{3}=53.60\)

In order to find the perimeter of the entire figure, we will need to find the lengths of the segments highlighted in red.

Notice that the side length of the equilateral triangle is equal to the diameter of the semicircle.

Next, you should recall that the height of an equilateral triangle splits the triangle into two congruent \(\displaystyle 30-60-90\) triangles.

Recall that the side lengths in a \(\displaystyle 30-60-90\) triangle are in a \(\displaystyle 1:\sqrt3:2\) ratio. Thus, the radius of the circle, which is also the base of the \(\displaystyle 30-60-90\) triangle, the height of the triangle, and the side length of the triangle are in the same ratio.

We can then set up the following to determine the length of the side of the equilateral triangle:

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

Rearrange the equation to solve for the length of the side.

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

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

\(\displaystyle \text{side}=\frac{2\sqrt3(17)}{3}=\frac{34\sqrt3}{3}\)

Since the diameter of the semi-circle and the length of a side of the equilateral triangle are the same, we can write the following equation:

\(\displaystyle \text{side}=\text{diameter}\)

We have two sides of the equilateral triangle and the circumference of a semi-circle.

\(\displaystyle \text{Perimeter}=2(\text{side})+\frac{\pi(\text{side})}{2}\)

Plug in the length of the side to find the perimeter.

\(\displaystyle \text{Perimeter}=2(\frac{34\sqrt3}{3})+\frac{\pi(\frac{34\sqrt3}{3})}{2}=\frac{68\sqrt3}{3}+\frac{17\pi\sqrt3}{3}=70.09\)