Equilateral Triangles - Math
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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?
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?
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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 
The radius of the circle is 2, from the equation circumference
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
Now our formula looks like this:
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
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
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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?
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?
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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.
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.
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What is the area of an equilateral triangle with a side length of 5?
What is the area of an equilateral triangle with a side length of 5?
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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:
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:
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What is the area of an equilateral triangle with sides 12 cm?
What is the area of an equilateral triangle with sides 12 cm?
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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.
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.
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An equilateral triangle has a perimeter of 18. What is its area?
An equilateral triangle has a perimeter of 18. What is its area?
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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).
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).
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An equilateral triangle has a side length of 
. What is the triangle's area?
An equilateral triangle has a side length of
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The area of an equilateral triangle is found using the following formula.
where 
The area of an equilateral triangle is found using the following formula.
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The length of one side of an equilateral triangle is ten. What is the area of the triangle?
The length of one side of an equilateral triangle is ten. What is the area of the triangle?
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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.
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
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.
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What is the area of an equilateral triangle with side 11?
What is the area of an equilateral triangle with side 11?
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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.
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.
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What is the area of an equilateral triangle with a side length of 
?
What is the area of an equilateral triangle with a side length of
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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 
.
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
Therefore, we can find the height of the altitude of this triangle by designating a value for
Now, we can calculate the area of the triangle via the formula
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An equilateral triangle has a side length of 
find its area.
An equilateral triangle has a side length of
Tap to reveal answer
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.
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
Therefore, we can find the height of the altitude of this triangle by designating a value for
that
Now, we can calculate the area of the triangle via the formula
Now convert to meters.
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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: A right triangle with sides length
Triangle B: An equilateral triangle with side lengths
Which triangle has a greater area?
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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.
The formula for the area of a right triangle is
The formula for the area of an equilateral triangle is
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
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Find the area of the following equilateral triangle:
Find the area of the following equilateral triangle:
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The formula for the area of an equilateral triangle is:
Where 
is the length of the side
Plugging in our values, we get:
The formula for the area of an equilateral triangle is:
Where
Plugging in our values, we get:
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Determine the area of the following equilateral triangle:
Determine the area of the following equilateral triangle:
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The formula for the area of an equilateral triangle is:
,
where 
is the length of the sides.
Plugging in our value, we get:
The formula for the area of an equilateral triangle is:
where
Plugging in our value, we get:
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Find the area of an equilateral triangle whose perimeter is 
Find the area of an equilateral triangle whose perimeter is
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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
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
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What is the height of an equilateral triangle with a side length of 8 in?
What is the height of an equilateral triangle with a side length of 8 in?
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An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.
To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of 
, 
, and 
. The hypoteneuse, the side opposite the 90 degree angle, is the full length of one side of the triangle and is equal to 
. Using this information, we can find the lengths of each side fo the special triangle.
The side with length 
will be the height (opposite the 60 degree angle). The height is 
inches.
An equilateral triangle has three congruent sides, and is also an equiangular triangle with three congruent angles that each meansure 60 degrees.
To find the height we divide the triangle into two special 30 - 60 - 90 right triangles by drawing a line from one corner to the center of the opposite side. This segment will be the height, and will be opposite from one of the 60 degree angles and adjacent to a 30 degree angle. The special right triangle gives side ratios of
The side with length
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An equilateral triangle has a side length of 
. What is the triangle's height 
?
An equilateral triangle has a side length of
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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,
The altitude,
In a
In this scenario:
and
Therefore,
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What is the height of an equilateral triangle with side 6?
What is the height of an equilateral triangle with side 6?
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When you draw the height in an equilateral triangle, it makes two 30-60-90 triangles. Because of that relationship, the height (which is across from the 
) is 
.
When you draw the height in an equilateral triangle, it makes two 30-60-90 triangles. Because of that relationship, the height (which is across from the
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An equilateral triangle has a side length of 
. What is its height, 
?
An equilateral triangle has a side length of
Tap to reveal answer
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 to 
. The hypotenuse of one of the 
is also the side of the original equilateral triangle. Therefore, one can say that 
and 
.
An altitude slices an equilateral triangle into two
Therefore, we can find the height of the altitude of this triangle by designating a value to
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Find the height of the following equilateral triangle:
Find the height of the following equilateral triangle:
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Each angle in an equilateral triangle is 
.
Use the formula for 
triangles in order to find the length of the height.
The formula is:
Where 
is the length of the side opposite the 
If we were to create a 
triangle by drawing the height, the length of the side is 
, the base is 
, and the height is 
.
Each angle in an equilateral triangle is
Use the formula for
The formula is:
Where
If we were to create a
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Solve for the value of X in the following equilateral triangle:
Solve for the value of X in the following equilateral triangle:
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If we draw a line segment between X and the base of the triangle, we form a 
triangle.
We can use the relationships between the sides of a 
triangle in order to find the length of X.
We know the base opposite the 
is 
.
The value of the height opposite the 
must then be 
, or 
.
Therefore, the value of X will be twice the value of the height:
If we draw a line segment between X and the base of the triangle, we form a
We can use the relationships between the sides of a
We know the base opposite the
The value of the height opposite the
Therefore, the value of X will be twice the value of the height:
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