How to find the perimeter of an equilateral triangle - Math

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Question

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

Answer

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:

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

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

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Question

Given: $\bigtriangleup ABC$ ; $\overline{AB}$ has length nine inches.

True or false: The perimeter of $\bigtriangleup ABC$ is one yard.

Answer

The perimeter of an equilateral triangle - one with three sides of equal length - is equal to three times the length of one side. Therefore, $\bigtriangleup ABC$ has perimeter

$9 \textrm{ in }\times 3 = 27 \textrm{ in }$

One yard is equal to 36 inches, making the statement false.

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Question

Locate and , the midpoints of sides $\overline{AB}$ and $\overline{AC}$ of an equilateral triangle. .

Give the perimeter of the triangle.

Answer

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. Therefore,

$MN = $\frac{1}{2}$ \cdot BC$

Substitute and solve:

$12= $\frac{1}{2}$ \cdot BC$

$12 \cdot 2 = $\frac{1}{2}$ \cdot BC \cdot 2$

Since the lengths of the sides of an equilateral triangle are equal, the perimeter is three times this, or

$3 \cdot 24 = 72$

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Question

ΔABC is an equilateral triangle with area 55.

Find the perimeter (to the nearest tenth).

Answer

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:

$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.

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Question

ΔABC is an equilateral triangle with area 28.

Find the perimeter (to the nearest tenth).

Answer

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:

$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.

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Question

An equilateral triangle has a side length of cm.

What is the perimeter of the triangle?

Answer

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

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Question

An equilateral triangle has a side length of cm.

What is the perimeter of the triangle?

Answer

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

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Question

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

Find the perimeter of the figure.

Answer

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$

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Question

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

Find the perimeter of the figure.

Answer

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

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Question

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

Find the perimeter of the figure.

Answer

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(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$

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Question

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

Find the perimeter of the figure.

Answer

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(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$

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Question

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

Find the perimeter of the figure.

Answer

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(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$

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Question

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

Find the perimeter of the figure.

Answer

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(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$

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Question

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

Find the perimeter of the figure.

Answer

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(19)}{3}$=\frac{38\sqrt3}{3}$

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{38\sqrt3}{3}$)+\frac{\pi(\frac{38\sqrt3}{3}$)}{2}=\frac{76\sqrt3}{3}$+\frac{19\pi\sqrt3}{3}$=78.34$

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Question

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

Find the perimeter of the figure.

Answer

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(23)}{3}$=\frac{46\sqrt3}{3}$

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{46\sqrt3}{3}$)+\frac{\pi(\frac{46\sqrt3}{3}$)}{2}=\frac{92\sqrt3}{3}$+\frac{23\pi\sqrt3}{3}$=94.83$

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Question

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

Find the perimeter of the figure.

Answer

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(32)}{3}$=\frac{42\sqrt3}{3}$

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{42\sqrt3}{3}$)+\frac{\pi(\frac{42\sqrt3}{3}$)}{2}=\frac{84\sqrt3}{3}$+\frac{21\pi\sqrt3}{3}$=86.59$

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Question

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

Find the perimeter of the figure.

Answer

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}$=\frac{2\sqrt3(24)}{3}$=16\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}$=2(16\sqrt3)+\frac{\pi(16\sqrt3)}{2}$=32\sqrt3+8\pi\sqrt3=98.96$

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Question

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

Find the perimeter of the figure.

Answer

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(26)}{3}$=\frac{52\sqrt3}{3}$

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{52\sqrt3}{3}$)+\frac{\pi(\frac{52\sqrt3}{3}$)}{2}=\frac{104\sqrt3}{3}$+\frac{26\pi\sqrt3}{3}$=107.20$

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Question

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

Find the perimeter of the figure.

Answer

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}$=\frac{2\sqrt3(27)}{3}$=18\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}$=2(18\sqrt3)+\frac{\pi(18\sqrt3)}{2}$=36\sqrt3+9\pi\sqrt3=111.33$

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Question

A square rug border consists of a continuous pattern of equilateral triangles, with isosceles triangles as corners, one of which is shown above. If the length of each equilateral triangle side is 5 inches, and there are 40 triangles in total, what is the total perimeter of the rug?

The inner angles of the corner triangles is 30°.

Answer

There are 2 components to this problem. The first, and easier one, is recognizing how much of the perimeter the equilateral triangles take up—since there are 40 triangles in total, there must be 40 – 4 = 36 of these triangles. By observation, each contributes only 1 side to the overall perimeter, thus we can simply multiply 36(5) = 180" contribution.

The second component is the corner triangles—recognizing that the congruent sides are adjacent to the 5-inch equilateral triangles, and the congruent angles can be found by

180 = 30+2x → x = 75°

We can use ratios to find the unknown side:

75/5 = 30/y → 75y = 150 → y = 2''.

Since there are 4 corners to the square rug, 2(4) = 8'' contribution to the total perimeter. Adding the 2 components, we get 180+8 = 188 inch perimeter.

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