How to find the perimeter of an equilateral triangle

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Math › How to find the perimeter of an equilateral triangle

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An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Explanation

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 triangles.

Recall that the side lengths in a triangle are in a $1:\sqrt3:2$ ratio. Thus, the radius of the circle, which is also the base of the 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:

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

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

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

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

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

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:

$\text{side}=\text{diameter}$

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

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

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

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

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Explanation

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 triangles.

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

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

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

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

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

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

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:

$\text{side}=\text{diameter}$

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

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

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

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

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Explanation

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 triangles.

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

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

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

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

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

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

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:

$\text{side}=\text{diameter}$

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

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

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

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

The height of an equilateral triangle is $\dpi{100} \small 2\sqrt{3}$

What is the triangle's perimeter?

$\dpi{100} \small 2\sqrt{2}$

Explanation

An altitude drawn in an equilateral triangle will form two 30-60-90 triangles. The height of equilateral triangle is the length of the longer leg of the 30-60-90 triangle. The length of the equilateral triangle's side is the length of the hypotenuse of the 30-60-90.

The ratio of the length of the hypotenuse to the length of the longer leg of a 30-60-90 triangle is $\dpi{100} \small 2:\sqrt{3}$

The length of the longer leg of the 30-60-90 triangle in this problem is $\dpi{100} \small 2\sqrt{3}$

Using this ratio, we find that the length of this triangle's hypotenuse is 4. Thus the perimeter of the equilateral triangle will be 4 multiplied by 3, which is 12.

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Explanation

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 triangles.

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

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

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

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

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

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

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:

$\text{side}=\text{diameter}$

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

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

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

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

If an equilateral triangle has a height of , what would be the perimeter? Round to the nearest tenth.

Varsity record

Explanation

Since the triangle is equilateral, all sides are the same length. Therefore, we only need to find the length of one side to determine the perimeter. We can do this by means of the Pythagorean Theorem. In the attached figure, the equilateral triangle has been divided into two right triangles, for which the Pythagorean Theorem can be performed:

Varsity record

With representing the length of one side, we can solve for using the Pythagorean Theorem:

$\a^2=(13.7;cm)^2+(\frac{1}{2}a)^2\a^2=187.7;cm^2+\frac{1}{4}a^2\ \frac{3}{4}a^2=187.7;cm^2\a^2=250.3;cm^2$

$a=\sqrt{250.3;cm^2}\approx15.8;cm$

Now that we know the length of one side, we can solve for the total perimeter by summing like sides:

An equilateral triangle is placed together with a semicircle as shown in the figure below.

Find the perimeter of the figure.

Explanation

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 triangles.

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

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

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

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

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

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

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:

$\text{side}=\text{diameter}$

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

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

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

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

An equilateral triangle is placed together with a semicircle as shown by the figure below.

Find the perimeter of the figure.

Explanation

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 triangles.

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

$\frac{\text{height}}{\sqrt3}=\frac{\text{side}}{2}$

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

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

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

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

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:

$\text{side}=\text{diameter}$

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

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

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

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

Find the perimeter of the following equilateral triangle:

Triangle

Explanation

The formula for the perimeter of an equilateral triangle is:

Where is the length of the side

Plugging in our values, we get:

Determine the perimeter of the following equilateral triangle:

Screen_shot_2014-02-27_at_6.40.28_pm

$24\sqrt{3}cm$

$16\sqrt{3}cm$

Explanation

The formula for the perimeter of an equilateral triangle is:

where is the length of the side.

Plugging in our value, we get:

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