GMAT Math : Calculating the height of an equilateral triangle

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Example Question #1 : Calculating The Height Of An Equilateral Triangle

If the area of an equilateral is , given a base of , what is the height of the triangle?

Possible Answers:

Correct answer:

Explanation:

We derive the height formula from the area of the triangle formula:

$A=\frac{(bh)}2{}$

$h=\frac{2A}{b}$

$=\frac{(2*21)}{3}$

=14

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Example Question #2 : Calculating The Height Of An Equilateral Triangle

What is the height of an equilateral triangle with sidelength 20?

Possible Answers:

$\frac{10 \sqrt{2}}{3}$

$10 \sqrt{3}$

$5 \sqrt{3}$

$10 \sqrt{2}$

Correct answer:

$10 \sqrt{3}$

Explanation:

The area of an equilateral triangle with sidelength s = 20 is

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = \frac{400\sqrt{3}}{4} = 100\sqrt{3}$

Using this area for and 20 for in the general triangle formula, we can obtain :

$\frac{1}{2} bh = A$

$\frac{1}{2}\cdot 20 \cdot h =100\sqrt{3}$

$10 \cdot h =100\sqrt{3}$

$10 \cdot h \div 10 =100\sqrt{3}\div 10$

$h=10\sqrt{3}$

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Example Question #3 : Calculating The Height Of An Equilateral Triangle

An equilateral triangle has a side length of . What is the height of the triangle?

Possible Answers:

$\frac{3\sqrt{2}}{5}$

$5\sqrt{2}$

$\frac{2\sqrt{5}}{3}$

$\frac{5\sqrt{3}}{2}$

Correct answer:

$\frac{5\sqrt{3}}{2}$

Explanation:

The height of an upright equilateral triangle is the perpendicular distance from the center of its base to its top. We can imagine that this line cuts the equilateral triangle into two congruent right triangles whose height is half the length of the original base and whose hypotenuse is the original side length. In these two congruent triangles, their base, which is the height of the equilateral triangle, is the only unknown side length, so we can use the Pythagorean theorem to solve for it:

a^2+b^2=c^2

$(\frac{5}{2})^2+b^2=5^2$

$b^2=5^2-(\frac{5}{2})^2=\frac{75}{4}$

$b=\sqrt{\frac{75}{4}}=\frac{\sqrt{3*25}}{2}=\frac{5\sqrt{3}}{2}$

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Example Question #4 : Calculating The Height Of An Equilateral Triangle

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ABC is an equilateral triangle, with a side length of . What is the height of the triangle?

Possible Answers:

$2\sqrt{3}$

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

$\frac{3}{2}$

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

$3\sqrt{3}$

Correct answer:

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

Explanation:

We know the length of the side, therefore we can use the formula for the height in an equilateral triangle:

$h=s\frac{\sqrt{3}}{2}$ , where is the length of a side and the length of the height.

Therefore, the final answer is $\frac{3\sqrt{3}}{2}$ .

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Example Question #5 : Calculating The Height Of An Equilateral Triangle

Given that an equilateral triangle has side lengths equal to , determine it's height in simplest form.

Possible Answers:

$\sqrt6$

$3\sqrt3$

$\sqrt{27}$

$\sqrt{28}$

Correct answer:

$3\sqrt3$

Explanation:

To solve, we must use pythagorean's theorem given that we know the hypotenuse is and one side length is $\left (\frac{1}{2} \textup{ of } 6 \right )$ . Therefore:

$c^2=a^2+b^2\Rightarrow 6^2=a^2+3^2\Rightarrow 27=a^2$

$a=\sqrt{27}=\sqrt{3^2*3}=3\sqrt3$

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GMAT Math : Calculating the height of an equilateral triangle

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1 : Calculating The Height Of An Equilateral Triangle

Example Question #2 : Calculating The Height Of An Equilateral Triangle

Example Question #3 : Calculating The Height Of An Equilateral Triangle

Example Question #4 : Calculating The Height Of An Equilateral Triangle

Example Question #5 : Calculating The Height Of An Equilateral Triangle

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