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High School Math : How to find the height of an equilateral triangle

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Example Question #471 : Plane Geometry

Equilateral_triangle

An equilateral triangle has a side length of $\sqrt{3}$ . What is the triangle's height $\dpi{100} h$ ?

Possible Answers:

$\frac{\sqrt{3}}2{}$

$\frac{3}{2}$

Not enough information to solve

$\sqrt{3}$

Correct answer:

$\frac{3}{2}$

Explanation:

The altitude, $\dpi{100} h$ , divides the equilateral triangle into two $30^{\circ}-60^{\circ}-90^{\circ}$ right triangles and divides the bottom side in half.

In a $30^{\circ}-60^{\circ}-90^{\circ}$ right triangle, the sides of the triangle equal , $x\sqrt{3}$ , and . In these equations equals the length of the smallest side, which in our triangle is $\frac{s}{2}$ or $\frac{\sqrt{3}}{2}$ .

In this scenario:

$h=x\sqrt{3}$

and

$x=\frac{\sqrt{3}}2{}$

Therefore,

$h=\frac{\sqrt{3}}{2}*\frac{\sqrt{3}}{1}$

$\rightarrow\frac{3}{2}$

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Example Question #481 : Geometry

Equilateral_triangle

An equilateral triangle has a side length of s=4 cm . What is its height, ?

Possible Answers:

3.46cm

5.19cm

1.73cm

Not enough information to solve

4.36cm

Correct answer:

3.46cm

Explanation:

An altitude slices an equilateral triangle into two $30^{\circ}-60^{\circ}-90^{\circ}$ triangles. These triangles follow a side-length pattern. The smallest of the two legs equals and the hypotenuse equals . By way of the Pythagorean Theorem, the longest leg or $h=x\sqrt{3}$ .

Therefore, we can find the height of the altitude of this triangle by designating a value to . The hypotenuse of one of the $30^{\circ}-60^{\circ}-90^{\circ}$ is also the side of the original equilateral triangle. Therefore, one can say that 2x=s=4cm and x=2cm .

$h=x\sqrt{3}$

$h=2\sqrt{3}cm$

$\rightarrow3.46cm$

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Example Question #1 : How To Find The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with side 6?

Possible Answers:

$3\sqrt{3}$

$6\sqrt{2}$

Correct answer:

$3\sqrt{3}$

Explanation:

When you draw the height in an equilateral triangle, it makes two 30-60-90 triangles. Because of that relationship, the height (which is across from the $60^{\circ}$ ) is $3\sqrt{3}$ .

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Example Question #483 : Plane Geometry

Find the height of the following equilateral triangle:

Triangle

Possible Answers:

20m

$20\sqrt{3}m$

10m

$10\sqrt{3}m$

$15\sqrt{3}m$

Correct answer:

$10\sqrt{3}m$

Explanation:

Each angle in an equilateral triangle is $60^{\circ}$ .

Use the formula for 30-60-90 triangles in order to find the length of the height.

The formula is:

$a-a\sqrt{3}-2a$

Where is the length of the side opposite the $\measuredangle 30$

If we were to create a 30-60-90 triangle by drawing the height, the length of the side is 20m , the base is 10m , and the height is $10\sqrt{3}m$ .

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Example Question #4 : How To Find The Height Of An Equilateral Triangle

Solve for the value of X in the following equilateral triangle:

Screen_shot_2014-02-27_at_6.35.43_pm

Possible Answers:

$6\sqrt{3}m$

$3\sqrt{3}m$

$9\sqrt{3}m$

Correct answer:

$6\sqrt{3}m$

Explanation:

If we draw a line segment between X and the base of the triangle, we form a 30-60-90 triangle.

We can use the relationships between the sides of a 30-60-90 triangle in order to find the length of X.

We know the base opposite the $\measuredangle60$ is .

The value of the height opposite the $\measuredangle30$ must then be $\frac{9m\sqrt{3}}{3}$ , or $3\sqrt{3}m$ .

Therefore, the value of X will be twice the value of the height:

$X = 2(3\sqrt{3}m)=6\sqrt{3}m$

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Example Question #1 : How To Find The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with a side length of 8 in?

Possible Answers:

$4\sqrt{3}$

$6\sqrt{2}$

$4\sqrt{2}$

$6\sqrt{3}$

Correct answer:

$4\sqrt{3}$

Explanation:

An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.

To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of , $x\sqrt{3}$ , and . The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to . Using this information, we can find the lengths of each side fo the special triangle.

$8=2x\rightarrow 4=x\rightarrow 4\sqrt{3}=x\sqrt{3}$

The side with length $x\sqrt{3}$ will be the height (opposite the 60 degree angle). The height is $4\sqrt{3}$ inches.

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High School Math : How to find the height of an equilateral triangle

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #471 : Plane Geometry

Example Question #481 : Geometry

Example Question #1 : How To Find The Height Of An Equilateral Triangle

Example Question #483 : Plane Geometry

Example Question #4 : How To Find The Height Of An Equilateral Triangle

Example Question #1 : How To Find The Height Of An Equilateral Triangle

All High School Math Resources

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