Geometry - How to find the length of the side of an equilateral triangle

Triangle

The height of the triangle is feet.

What is the length of the base of the triangle to the nearest tenth?

Explanation

Since it is an equilateral triangle, the line that represents the height bisects it into a 30-60-90 triangle.

Here you may use $sin(60)=\frac{4}{hypotenuse}$ and solve for hypotenuse to find one of the sides of the triangle.

Use the definition of an equilateral triangle to know that the answer of the hypotenuse also applies to the base of the triangle.

Therefore,

$hypotenuse=\frac{4}{sin(60)}=4.6$

Equilateral

Refer to the above diagram. $\bigtriangleup ABC$ has perimeter 56.

True or false: $m \angle A = 60^{\circ }$

False

True

Explanation

Assume $m \angle A = 60^{\circ }$ . 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 $180^{\circ }$ , letting $t = m \angle B = m \angle C$ :

$m \angle A + m \angle B + m \angle C = 180^{\circ }$

$60^{\circ }+ t+t = 180^{\circ }$

$60^{\circ }+2t = 180^{\circ }$

$60^{\circ }+2t - 60^{\circ } = 180^{\circ } - 60^{\circ }$

$2t = 120^{\circ }$

$\frac{2t }{2}= \frac{120^{\circ }}{2}$

$t = 60^{\circ }$

All three angles have measure $60^{\circ }$ , making $\bigtriangleup ABC$ 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, $m \angle A e 60^{\circ }$ .

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Explanation

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 $1:\sqrt3:2$ . 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:

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

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

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

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

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

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{4\sqrt3}{3})=\frac{20\sqrt3}{3}=11.55$

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Explanation

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.

Thus, we can use the ratio and the length of the height to set up the following equation:

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

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

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

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

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

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(2\sqrt3)=10\sqrt3=17.32$

An equilateral triangle is placed on top of a square, as shown by the figure below.

Find the perimeter of the shape.

Explanation

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.

Thus, we can use the ratio and the length of the height to set up the following equation:

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

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

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

$\text{side}=\frac{2\sqrt3(4)}{3}=\frac{8\sqrt3}{3}$

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

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{8\sqrt3}{3})=\frac{40\sqrt3}{3}=23.09$

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Explanation

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.

Thus, we can use the ratio and the length of the height to set up the following equation:

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

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

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

$\text{side}=\frac{2\sqrt3(5)}{3}=\frac{10\sqrt3}{3}$

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

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{10\sqrt3}{3})=\frac{50\sqrt3}{3}=28.87$

An equilateral triangle is placed on top of a square as shown by the figure below.

Find the perimeter of the shape.

Explanation

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.

Thus, we can use the ratio and the length of the height to set up the following equation:

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

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

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

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

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

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(4\sqrt3)=20\sqrt3=34.64$

An equilateral triangle is placed on a square as shown by the figure below.

Find the perimeter of the shape.

Explanation

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.

Thus, we can use the ratio and the length of the height to set up the following equation:

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

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

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

$\text{side}=\frac{2\sqrt3(8)}{3}=\frac{16\sqrt3}{3}$

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

$\text{Perimeter}=5(\text{side})$

$\text{Perimeter}=5(\frac{16\sqrt3}{3})=\frac{80\sqrt3}{3}=46.19$

The area of an equilateral triangle is $16\sqrt3$ , what is the length of each side?

Explanation

An equilateral triangle can be broken down into 2 30-60-90 right triangles (see image). The length of a side (the base) is 2x while the length of the height is $x\sqrt3$ . The area of a triangle can be calculated using the following equation: