Calculating the area of an equilateral triangle - GMAT Quantitative

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A given equilateral triangle has a side length and a height . What is the area of the triangle?

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For a given equilateral triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(4)(7)$

$A=\frac{1}{2}$(28)$

A given right triangle has a base of length and a height . What is the area of the triangle?

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For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(9)(8)$

$A=\frac{1}{2}$(72)$

Three straight sticks are gathered of exactly equal length. They are placed end to end on the ground to form a triangle. If the area of the triangle they form is 1.732 square feet. What is the length in feet of each stick?

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Let be the length of a side of an equilateral triangle. Then the formula for the area of an equilateral triangle with side is

So solving $1.732 = $$\frac{s^2$\sqrt{3}$}{4}$

we get .

Alternative Solution:

Without knowing this formula you can still use the Pythagorean Theorem to solve this. By drawing the height of the triangle, you split the triangle into 2 right triangles of equal size. The sides are the height, $\frac{s}{2}$ and . Letting stand for the unknown height, we solve

solving for we get

The area for any triangle is the base times the height divided by 2. So

$1.732=\frac{sh}{2}$ = $\frac{s(s\sqrt(3))}{2*2}$ = $$\frac{s^2$\sqrt(3)}{4}$$ or .

If an equilateral triangle has a side length of and a height of , what is the area of the given triangle?

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To find the area of a traingle, we need the height and base lengths. Plug the given values into the following formula:

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

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

Triangle is an equilateral triangle with side length . What is the area of the triangle?

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The area of an equilateral triangle is given by the following formula:

, where is the length of a side.

Since we know the length of the side, we can simply plug it in the formula and we have $$\frac{4\sqrt{3}$}{4}$ or $\sqrt{3}$ , which is the final answer.

is an equilateral triangle inscribed in a cirlce with radius . What is the area of the triangle ?

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Since we are given a radius for the circle, we should be able to find the length of the height of the equilateral triangle, indeed, the center of the circle is $\frac{2}{3}$ of the length of the height from any vertex.

Therefore, the height is $3=\frac{2}{3}$\cdot h$ where is the length of the height of the triangle. Therefore $h=\frac{9}{2}$$ .

We can now plug in this value in the formula of the height of an equilateral triangle $h=s$\frac{\sqrt{3}$}{2}$ , where is the length of the side of the triangle.

Therefore, $s=\frac{9}{\sqrt{3}$}$ or $3$\sqrt{3}$$ .

Now we should plug in this value into the formula for the area of an equilateral triangle where is the value of the area of the equilateral triangle. Therefore $a= 27$\frac{\sqrt{3}$}{4}$ , which is our final answer.

A given right triangle has a base length and a height . What is the area of the triangle?

Tap to see back →

For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(5)(4)$

$A=\frac{1}{2}$(20)$

Three straight sticks are gathered of exactly equal length. They are placed end to end on the ground to form a triangle. If the area of the triangle they form is 1.732 square feet. What is the length in feet of each stick?

Tap to see back →

Let be the length of a side of an equilateral triangle. Then the formula for the area of an equilateral triangle with side is

So solving $1.732 = $$\frac{s^2$\sqrt{3}$}{4}$

we get .

Alternative Solution:

Without knowing this formula you can still use the Pythagorean Theorem to solve this. By drawing the height of the triangle, you split the triangle into 2 right triangles of equal size. The sides are the height, $\frac{s}{2}$ and . Letting stand for the unknown height, we solve

solving for we get

The area for any triangle is the base times the height divided by 2. So

$1.732=\frac{sh}{2}$ = $\frac{s(s\sqrt(3))}{2*2}$ = $$\frac{s^2$\sqrt(3)}{4}$$ or .

If an equilateral triangle has a side length of and a height of , what is the area of the given triangle?

Tap to see back →

To find the area of a traingle, we need the height and base lengths. Plug the given values into the following formula:

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

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

Triangle is an equilateral triangle with side length . What is the area of the triangle?

Tap to see back →

The area of an equilateral triangle is given by the following formula:

, where is the length of a side.

Since we know the length of the side, we can simply plug it in the formula and we have $$\frac{4\sqrt{3}$}{4}$ or $\sqrt{3}$ , which is the final answer.

is an equilateral triangle inscribed in a cirlce with radius . What is the area of the triangle ?

Tap to see back →

Since we are given a radius for the circle, we should be able to find the length of the height of the equilateral triangle, indeed, the center of the circle is $\frac{2}{3}$ of the length of the height from any vertex.

Therefore, the height is $3=\frac{2}{3}$\cdot h$ where is the length of the height of the triangle. Therefore $h=\frac{9}{2}$$ .

We can now plug in this value in the formula of the height of an equilateral triangle $h=s$\frac{\sqrt{3}$}{2}$ , where is the length of the side of the triangle.

Therefore, $s=\frac{9}{\sqrt{3}$}$ or $3$\sqrt{3}$$ .

Now we should plug in this value into the formula for the area of an equilateral triangle where is the value of the area of the equilateral triangle. Therefore $a= 27$\frac{\sqrt{3}$}{4}$ , which is our final answer.

A given right triangle has a base length and a height . What is the area of the triangle?

Tap to see back →

For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(5)(4)$

$A=\frac{1}{2}$(20)$

A given equilateral triangle has a side length and a height . What is the area of the triangle?

Tap to see back →

For a given equilateral triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(4)(7)$

$A=\frac{1}{2}$(28)$

A given right triangle has a base of length and a height . What is the area of the triangle?

Tap to see back →

For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(9)(8)$

$A=\frac{1}{2}$(72)$

A given equilateral triangle has a side length and a height . What is the area of the triangle?

Tap to see back →

For a given equilateral triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(4)(7)$

$A=\frac{1}{2}$(28)$

A given right triangle has a base of length and a height . What is the area of the triangle?

Tap to see back →

For a given right triangle with a side length and a height , the area is

$A=\frac{1}{2}$bh$ . Plugging in the values provided:

$A=\frac{1}{2}$(9)(8)$

$A=\frac{1}{2}$(72)$

Three straight sticks are gathered of exactly equal length. They are placed end to end on the ground to form a triangle. If the area of the triangle they form is 1.732 square feet. What is the length in feet of each stick?

Tap to see back →

Let be the length of a side of an equilateral triangle. Then the formula for the area of an equilateral triangle with side is

So solving $1.732 = $$\frac{s^2$\sqrt{3}$}{4}$

we get .

Alternative Solution:

Without knowing this formula you can still use the Pythagorean Theorem to solve this. By drawing the height of the triangle, you split the triangle into 2 right triangles of equal size. The sides are the height, $\frac{s}{2}$ and . Letting stand for the unknown height, we solve

solving for we get

The area for any triangle is the base times the height divided by 2. So

$1.732=\frac{sh}{2}$ = $\frac{s(s\sqrt(3))}{2*2}$ = $$\frac{s^2$\sqrt(3)}{4}$$ or .

If an equilateral triangle has a side length of and a height of , what is the area of the given triangle?

Tap to see back →

To find the area of a traingle, we need the height and base lengths. Plug the given values into the following formula:

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

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

Triangle is an equilateral triangle with side length . What is the area of the triangle?

Tap to see back →

The area of an equilateral triangle is given by the following formula:

, where is the length of a side.

Since we know the length of the side, we can simply plug it in the formula and we have $$\frac{4\sqrt{3}$}{4}$ or $\sqrt{3}$ , which is the final answer.

is an equilateral triangle inscribed in a cirlce with radius . What is the area of the triangle ?

Tap to see back →

Since we are given a radius for the circle, we should be able to find the length of the height of the equilateral triangle, indeed, the center of the circle is $\frac{2}{3}$ of the length of the height from any vertex.

Therefore, the height is $3=\frac{2}{3}$\cdot h$ where is the length of the height of the triangle. Therefore $h=\frac{9}{2}$$ .

We can now plug in this value in the formula of the height of an equilateral triangle $h=s$\frac{\sqrt{3}$}{2}$ , where is the length of the side of the triangle.

Therefore, $s=\frac{9}{\sqrt{3}$}$ or $3$\sqrt{3}$$ .

Now we should plug in this value into the formula for the area of an equilateral triangle where is the value of the area of the equilateral triangle. Therefore $a= 27$\frac{\sqrt{3}$}{4}$ , which is our final answer.