Calculating the height of an equilateral triangle - GMAT Quantitative

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Question

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

Answer

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

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

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

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

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Question

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

Answer

The area of an equilateral triangle with sidelength 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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Question

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

Answer

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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Question

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

Answer

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:

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

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

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

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Question

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

Answer

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:

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

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

Compare your answer with the correct one above

Question

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

Answer

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

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

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

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

Compare your answer with the correct one above

Question

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

Answer

The area of an equilateral triangle with sidelength 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 :

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

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

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

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

$h=10\sqrt{3}$

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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:

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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:

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

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

Compare your answer with the correct one above

Question

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

Answer

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

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

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

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

Compare your answer with the correct one above

Question

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

Answer

The area of an equilateral triangle with sidelength 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 :

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

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

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

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

$h=10\sqrt{3}$

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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:

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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:

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

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

Compare your answer with the correct one above

Question

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

Answer

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

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

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

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

Compare your answer with the correct one above

Question

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

Answer

The area of an equilateral triangle with sidelength 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 :

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

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

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

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

$h=10\sqrt{3}$

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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:

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

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

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

Compare your answer with the correct one above

Question

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

Answer

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:

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

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

Compare your answer with the correct one above

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