30-60-90 Triangles - Trigonometry
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What is the ratio of the side opposite the 
angle to the hypotenuse?
What is the ratio of the side opposite the
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Step 1: Locate the side that is opposite the 
side..
The shortest side is opposite the 
angle. Let's say that this side has length 
.
Step 2: Recall the ratio of the sides of a 
triangle:
From the shortest side, the ratio is 
.
is the hypotenuse, which is twice as big as the shortest side..
The ratio of the short side to the hypotenuse is 
Step 1: Locate the side that is opposite the
The shortest side is opposite the
Step 2: Recall the ratio of the sides of a
From the shortest side, the ratio is
The ratio of the short side to the hypotenuse is
What is the height of an equilateral triangle with side length 8?
What is the height of an equilateral triangle with side length 8?
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The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is 
.
To double check the answer use the Pythagorean Thereom:
The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is
To double check the answer use the Pythagorean Thereom:
In a 
triangle, the side opposite the 
degree angle is 
. How long is the side opposite the 
degree angle?
In a
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Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
In a 30-60-90 triangle, the length of the side opposite the 
angle is 
. What is the length of the hypotenuse?
In a 30-60-90 triangle, the length of the side opposite the
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By definition, the length of the hypotenuse is twice the length of the side opposite the 
angle.
Recall that the hypotenuse is the side opposite the 
angle.
Thus, using the equation below, where ss represents the short side (that opposite the 
angle) we get:
Plugging in our values for the short side we find the hypotenuse as follows:
By definition, the length of the hypotenuse is twice the length of the side opposite the
Recall that the hypotenuse is the side opposite the
Thus, using the equation below, where ss represents the short side (that opposite the
Plugging in our values for the short side we find the hypotenuse as follows:
A triangle has three angles 
, 
and 
such that 
and 
. The side opposite to 
measures 
units in length. How long is the side opposite of 
?
A triangle has three angles
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A triangle with a 
angle relation is a 
, 
, 
degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length 
. The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, 
units.
Therefore, to make the above statement true 
.
A triangle with a
Therefore, to make the above statement true
Triangle 
is equilateral with a side length of 
.
What is the height of the triangle?
Triangle
What is the height of the triangle?
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An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..
An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..
It is known that the smallest side of a 30-60-90 triangle is 5.
Find 
.
It is known that the smallest side of a 30-60-90 triangle is 5.
Find
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We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
, so 
or 
.
We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
It is known that for a 30-60-90 triangle,
.
Find the area of the triangle.
Note:
It is known that for a 30-60-90 triangle,
Find the area of the triangle.
Note:
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First, we know that in a 30-60-90 triangle,
.
Also, the base is the smallest side times 
, so in our case it is 
.
The height is just the smallest side, 
.
Substituting these values into the formula given for area of a triangle, we obtain the answer 
.
First, we know that in a 30-60-90 triangle,
Also, the base is the smallest side times
The height is just the smallest side,
Substituting these values into the formula given for area of a triangle, we obtain the answer
What is the ratio of the side opposite the angle to the hypotenuse?
What is the ratio of the side opposite the
Tap to see back →
Step 1: Locate the side that is opposite the side..
The shortest side is opposite the angle. Let's say that this side has length .
Step 2: Recall the ratio of the sides of a triangle:
From the shortest side, the ratio is .
is the hypotenuse, which is twice as big as the shortest side..
The ratio of the short side to the hypotenuse is
Step 1: Locate the side that is opposite the
The shortest side is opposite the
Step 2: Recall the ratio of the sides of a
From the shortest side, the ratio is
The ratio of the short side to the hypotenuse is
What is the height of an equilateral triangle with side length 8?
What is the height of an equilateral triangle with side length 8?
Tap to see back →
The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is .
To double check the answer use the Pythagorean Thereom:
The altitude of an equilateral triangle splits it into two 30-60-90 triangles. The height of the triangle is the longer leg of the 30-60-90 triangle. If the hypotenuse is 8, the longer leg is
To double check the answer use the Pythagorean Thereom:
In a triangle, the side opposite the degree angle is . How long is the side opposite the degree angle?
In a
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Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
In a 30-60-90 triangle, the length of the side opposite the angle is . What is the length of the hypotenuse?
In a 30-60-90 triangle, the length of the side opposite the
Tap to see back →
By definition, the length of the hypotenuse is twice the length of the side opposite the angle.
Recall that the hypotenuse is the side opposite the angle.
Thus, using the equation below, where ss represents the short side (that opposite the angle) we get:
Plugging in our values for the short side we find the hypotenuse as follows:
By definition, the length of the hypotenuse is twice the length of the side opposite the
Recall that the hypotenuse is the side opposite the
Thus, using the equation below, where ss represents the short side (that opposite the
Plugging in our values for the short side we find the hypotenuse as follows:
A triangle has three angles , and such that and . The side opposite to measures units in length. How long is the side opposite of ?
A triangle has three angles
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A triangle with a angle relation is a , , degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, units.
Therefore, to make the above statement true .
A triangle with a
Therefore, to make the above statement true
Triangle is equilateral with a side length of .
What is the height of the triangle?
Triangle
What is the height of the triangle?
Tap to see back →
An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..
An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..
It is known that the smallest side of a 30-60-90 triangle is 5.
Find .
It is known that the smallest side of a 30-60-90 triangle is 5.
Find
Tap to see back →
We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
, so or .
We know that in a 30-60=90 triangle, the smallest side corresponds to the side opposite the 30 degree angle.
Additionally, we know that the hypotenuse is 2 times the value of the smallest side, so in this case, that is 10.
The formula for
It is known that for a 30-60-90 triangle,
.
Find the area of the triangle.
Note:
It is known that for a 30-60-90 triangle,
Find the area of the triangle.
Note:
Tap to see back →
First, we know that in a 30-60-90 triangle,
.
Also, the base is the smallest side times , so in our case it is .
The height is just the smallest side, .
Substituting these values into the formula given for area of a triangle, we obtain the answer .
First, we know that in a 30-60-90 triangle,
Also, the base is the smallest side times
The height is just the smallest side,
Substituting these values into the formula given for area of a triangle, we obtain the answer
In a triangle, the side opposite the degree angle is . How long is the side opposite the degree angle?
In a
Tap to see back →
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
Based on the 30-60-90 identity, the measure of the side opposite the 30 degree angle is doubled to get the hypotenuse.
Therefore,
In a 30-60-90 triangle, the length of the side opposite the angle is . What is the length of the hypotenuse?
In a 30-60-90 triangle, the length of the side opposite the
Tap to see back →
By definition, the length of the hypotenuse is twice the length of the side opposite the angle.
Recall that the hypotenuse is the side opposite the angle.
Thus, using the equation below, where ss represents the short side (that opposite the angle) we get:
Plugging in our values for the short side we find the hypotenuse as follows:
By definition, the length of the hypotenuse is twice the length of the side opposite the
Recall that the hypotenuse is the side opposite the
Thus, using the equation below, where ss represents the short side (that opposite the
Plugging in our values for the short side we find the hypotenuse as follows:
A triangle has three angles , and such that and . The side opposite to measures units in length. How long is the side opposite of ?
A triangle has three angles
Tap to see back →
A triangle with a angle relation is a , , degree triangle. The side opposite the smallest angle of a triangle is the shortest side, of length . The side opposite the largest angle is the longest side, measuring twice the length of the shortest side for this triangle, units.
Therefore, to make the above statement true .
A triangle with a
Therefore, to make the above statement true
Triangle is equilateral with a side length of .
What is the height of the triangle?
Triangle
What is the height of the triangle?
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An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..
An equilateral triangle has internal angles of 60°, so the sin of one of those angles is equivalent to the height of the triangle divided by the side length,
so..