GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #1 : Calculating The Length Of The Side Of An Equilateral Triangle

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Equilateral triangle ABC is inscribed in a circle with radius , what is the length of a side of the triangle?

Possible Answers:

$\frac{15}{2}$

$\sqrt{5}$

$5\sqrt{3}$

$\frac{15}{\sqrt{3}}$

$3\sqrt{5}$

Correct answer:

$\frac{15}{\sqrt{3}}$

Explanation:

Since we are given the radius, we should be able to find the height of the equilateral triangle. Indeed, the center of the circle is at the intersections of the heights of the triangle, and is located $\frac{2}{3}$ away from the edge of a given height.

Therefore 5, the radius of the circle is $\frac{2}{3}$ of the height.

Therefore, the height must be $\frac{15}{2}$ .

From here, we can use the formula for the height of the equilateral triangle $h=s\frac{\sqrt{3}}{2}$ , where is the length of the height and is the length of a side of the equilateral triangle.

Therefore, $s=2\frac{h}{\sqrt{3}}$ , then $\frac{15}{\sqrt{3}}$ is the final answer.

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

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?

Possible Answers:

$3\ feet$

$2.5\ feet$

$1.5\ feet$

$1\ foot$

$2\ feet$

Correct answer:

$2\ feet$

Explanation:

Equilateral_triangle

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

$\frac{s^2\sqrt{3}}{4}$

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

we get s=2 .

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

$(\frac{s}{2})^2 + h^2= s^2$ solving for we get

$h=\sqrt(s^2-(\frac{s}{2})^2) = \sqrt(s^2-\frac{s^2}{4}) = \sqrt(\frac{3s^2}{4}) = \frac{s\sqrt(3)}{2}$

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 s=2 .

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

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

Possible Answers:

Correct answer:

Explanation:

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

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

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

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Example Question #2 : Equilateral Triangles

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

Possible Answers:

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

$\sqrt{3}$

$\frac{\sqrt{3}}{4}$

$2\sqrt{3}$

Correct answer:

$\sqrt{3}$

Explanation:

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

$\frac{s^{2}\sqrt{3}}{4}$ , 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.

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Example Question #3 : Equilateral Triangles

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

Possible Answers:

$3\frac{\sqrt{3}}{4}$

$27\frac{\sqrt{5}}{4}$

$27\frac{\sqrt{3}}{4}$

$4\sqrt{3}$

Correct answer:

$27\frac{\sqrt{3}}{4}$

Explanation:

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 $a=s^{2}\frac{\sqrt{3}}{4}$ where is the value of the area of the equilateral triangle. Therefore $a= 27\frac{\sqrt{3}}{4}$ , which is our final answer.

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Example Question #5 : Equilateral Triangles

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

Possible Answers:

Not enough information provided

Correct answer:

Explanation:

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=14

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Example Question #6 : Equilateral Triangles

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

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

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=10

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

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

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

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=36

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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:

$10 \sqrt{2}$

$5 \sqrt{3}$

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

$10 \sqrt{3}$

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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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating The Length Of The Side Of An Equilateral Triangle

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

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

Example Question #2 : Equilateral Triangles

Example Question #3 : Equilateral Triangles

Example Question #5 : Equilateral Triangles

Example Question #6 : Equilateral Triangles

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

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

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

All GMAT Math Resources

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