All High School Math Resources
Example Questions
Example Question #4 : Equilateral Triangles
An equilateral triangle has a side length of find its area.
Not enough information to solve
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 : Equilateral Triangles
Triangle A: A right triangle with sides length , , and .
Triangle B: An equilateral triangle with side lengths .
Which triangle has a greater area?
Triangle A
There is not enough information given to determine which triangle has a greater area.
The areas of the two triangles are the same.
Triangle B
Triangle B
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 #1 : Equilateral Triangles
What is the area of an equilateral triangle with side 11?
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 #1 : How To Find The Area Of An Equilateral Triangle
Find the area of the following equilateral triangle:
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 #2 : How To Find The Area Of An Equilateral Triangle
Determine the area of the following equilateral triangle:
The formula for the area of an equilateral triangle is:
,
where is the length of the sides.
Plugging in our value, we get:
Example Question #11 : Equilateral Triangles
Find the area of an equilateral triangle whose perimeter is
The formula for the perimeter of an equilateral triangle is:
Plugging in our values, we get
The formula for the area of an equilateral triangle is:
Plugging in our values, we get
Example Question #1 : How To Find The Area Of An Equilateral Triangle
What is the area of an equilateral triangle with a side length of 5?
Note that an equilateral triangle has equal sides and equal angles. The question gives us the length of the base, 5, but doesn't tell us the height.
If we split the triangle into two equal triangles, each has a base of 5/2 and a hypotenuse of 5.
Therefore we can use the Pythagorean Theorem to solve for the height:
Now we can find the area of the triangle:
Example Question #2 : Equilateral Triangles
An equilateral triangle has a perimeter of 18. What is its area?
Recall that an equilateral triangle also obeys the rules of isosceles triangles. That means that our triangle can be represented as having a height that bisects both the opposite side and the angle from which the height is "dropped." For our triangle, this can be represented as:
Now, although we do not yet know the height, we do know from our 30-60-90 regular triangle that the side opposite the 60° angle is √3 times the length of the side across from the 30° angle. Therefore, we know that the height is 3√3.
Now, the area of a triangle is (1/2)bh. If the height is 3√3 and the base is 6, then the area is (1/2) * 6 * 3√3 = 3 * 3√3 = 9√(3).
Example Question #2 : How To Find The Area Of An Equilateral Triangle
A circle contains 6 copies of a triangle; each joined to the others at the center of the circle, as well as joined to another triangle on the circle’s circumference.
The circumference of the circle is
What is the area of one of the triangles?
The radius of the circle is 2, from the equation circumference . Each triangle is the same, and is equilateral, with side length of 2. The area of a triangle
To find the height of this triangle, we must divide it down the centerline, which will make two identical 30-60-90 triangles, each with a base of 1 and a hypotenuse of 2. Since these triangles are both right traingles (they have a 90 degree angle in them), we can use the Pythagorean Theorem to solve their height, which will be identical to the height of the equilateral triangle.
We know that the hypotenuse is 2 so . That's our solution. We know that the base is 1, and if you square 1, you get 1.
Now our formula looks like this: , so we're getting close to finding .
Let's subtract 1 from each side of that equation, in order to make things a bit simpler:
Now let's apply the square root to each side of the equation, in order to change into :
Therefore, the height of our equilateral triangle is
To find the area of our equilateral triangle, we simply have to multiply half the base by the height:
The area of our triangle is
Example Question #471 : Plane Geometry
An equilateral triangle has a side length of . What is the triangle's height ?
Not enough information to solve
The altitude, , divides the equilateral triangle into two right triangles and divides the bottom side in half.
In a right triangle, the sides of the triangle equal , , and . In these equations equals the length of the smallest side, which in our triangle is or .
In this scenario:
and
Therefore,