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

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

Equilateral_triangle

An equilateral triangle has a side length of . What is the triangle's area?

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

The area of an equilateral triangle is found using the following formula.

 where 

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

The area of square ABCD is 50% greater than the perimeter of the equilateral triangle EFG. If the area of square ABCD is equal to 45, then what is the area of EFG?

Possible Answers:

25√3

50√3

50

25

30

Correct answer:

25√3

Explanation:

If the area of ABCD is equal to 45, then the perimeter of EFG is equal to x * 1.5 = 45. 45 / 1.5 = 30, so the perimeter of EFG is equal to 30. This means that each side is equal to 10.

The height of the equilateral triangle EFG creates two 30-60-90 triangles, each with a hypotenuse of 10 and a short side equal to 5. We know that the long side of 30-60-90 triangle (here the height of EFG) is equal to √3 times the short side, or 5√3.

We then apply the formula for the area of a triangle, which is 1/2 * b * h. We get 1/2 * 10 * 5√3 = 5 * 5√3 = 25√3.

In general, the height of an equilateral triangle is equal to √3 / 2 times a side of the equilateral triangle. The area of an equilateral triangle is equal to 1/2 * √3s/ 2 * s = √3s2/4.

Example Question #2 : How To Find The Area Of An Equilateral Triangle

What is the area of an equilateral triangle with sides 12 cm?

Possible Answers:

12√2

18√3

72√3

54√2

36√3

Correct answer:

36√3

Explanation:

An equilateral triangle has three congruent sides and results in three congruent angles. This figure results in two special right triangles back to back: 30° – 60° – 90° giving sides of x - x √3 – 2x in general. The height of the triangle is the x √3 side.  So Atriangle = 1/2 bh = 1/2 * 12 * 6√3 = 36√3 cm2.

Example Question #3 : Equilateral Triangles

The length of one side of an equilateral triangle is ten. What is the area of the triangle?

Possible Answers:

Correct answer:

Explanation:

To calculate the height, the length of a perpendicular bisector must be determined. If a perpendicular bisector is drawn in an equilateral triangle, the triangle is divided in half, and each half is a congruent 30-60-90 right triangle. This type of triangle follows the equation below.

The length of the hypotenuse will be one side of the equilateral triangle.

.

The side of the equilateral triangle that represents the height of the triangle will have a length of because it will be opposite the 60o angle.

To calculate the area of the triangle, multiply the base (one side of the equilateral triangle) and the height (the perpendicular bisector) and divide by two.

Example Question #4 : Equilateral Triangles

Equilateral_triangle

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

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

In order to find the area of the triangle, we must first calculate the height of its altitude.  An altitude slices an equilateral triangle into two  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 .

Therefore, we can find the height of the altitude of this triangle by designating a value for . The hypotenuse of one of the  is also the side of the original equilateral triangle.  Therefore, one can say that  and .

 

Now, we can calculate the area of the triangle via the formula .

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

Equilateral_triangle

An equilateral triangle has a side length of  find its area.

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

In order to find the area of the triangle, we must first calculate the height of its altitude.  An altitude slices an equilateral triangle into two  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 .

Therefore, we can find the height of the altitude of this triangle by designating a value for . The hypotenuse of one of the  is also the side of the original equilateral triangle.  Therefore, one can say

that  and .

 

Now, we can calculate the area of the triangle via the formula

 

Now convert to meters.

Example Question #5 : How To Find The Area Of An Equilateral Triangle

Triangle A: A right triangle with sides length , , and .

Triangle B: An equilateral triangle with side lengths .

Which triangle has a greater area?

Possible Answers:

The areas of the two triangles are the same.

There is not enough information given to determine which triangle has a greater area.

Triangle A

Triangle B

Correct answer:

Triangle B

Explanation:

The formula for the area of a right triangle is , where  is the length of the triangle's base and  is its height. Since the longest side is the hypotenuse, use the two smaller numbers given as sides for the base and height in the equation to calculate the area of Triangle A:

 

The formula for the area of an equilateral triangle is , where  is the length of each side. (Alternatively, you can divide the equilateral triangle into two right triangles and find the area of each). Triangle B's area is thus calculated as:

 

To determine which of the two areas is greater without using a calculator, rewrite the areas of the two triangles with comparable factors. Triangle A's area can be expressed as , and Triangle B's area can be expressed as . Since  is greater than , the product of the factors of Triangle B's area will be greater than the product of the factors of Triangle A's, so Triangle B has the greater area.

Example Question #6 : How To Find The Area Of An Equilateral Triangle

What is the area of an equilateral triangle with side 11?

Possible Answers:

Correct answer:

Explanation:

Since the area of a triangle is 

you need to find the height of the triangle first. Because of the 30-60-90 relationship, you can determine that the height is .

Then, multiply that by the base (11).

Finally, divide it by two to get 52.4.

Example Question #3 : How To Find The Area Of An Equilateral Triangle

Find the area of the following equilateral triangle:

Triangle

Possible Answers:

Correct answer:

Explanation:

The formula for the area of an equilateral triangle is:

Where  is the length of the side

Plugging in our values, we get:

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

Determine the area of the following equilateral triangle:

Screen_shot_2014-02-27_at_6.40.28_pm

Possible Answers:

Correct answer:

Explanation:

The formula for the area of an equilateral triangle is:

,

where  is the length of the sides.

Plugging in our value, we get:

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