Geometry 30-60-90 Triangle Practice ), Special Right Triangles (Practice)

Side opposite the 30° angle: $x$

Side opposite the 60° angle: $x * √3$

Side opposite the 90° angle: $2x$


2, 2√3, 4


7, 7√3, 14


√3, 3, 2√3


(Why is the longer leg 3? In this triangle, the shortest leg ($x$) is $√3$, so for the longer leg, $x√3 = √3 * √3 = √9 = 3$. And the hypotenuse is 2 times the shortest leg, or $2√3$)

And so on.

Đang xem: Geometry 30-60-90 triangle practice

The side opposite the 30° angle is always the smallest, because 30 degrees is the smallest angle. The side opposite the 60° angle will be the middle length, because 60 degrees is the mid-sized degree angle in this triangle. And, finally, the side opposite the 90° angle will always be the largest side (the hypotenuse) because 90 degrees is the largest angle.

Example 1


The longer leg must, therefore, be opposite the 60° angle and measure $6 * √3$, or $6√3$.

Example 2


Since 18 is the measure opposite the 60° angle, it must be equal to $x√3$. The shortest leg must then measure $18/√3$.

(Note that the leg length will actually be $18/{√3} * {√3}/{√3} = {18√3}/3 = 6√3$ because a denominator cannot contain a radical/square root).

And the hypotenuse will be $2(18/√3)$

(Note that, again, you cannot have a radical in the denominator, so the final answer will really be 2 times the leg length of $6√3$ => $12√3$).

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Example 3



No need to consult the magic eight ball—these rules always work.

First, let”s forget about right triangles for a second and look at an equilateral triangle.


An equilateral triangle is a triangle that has all equal sides and all equal angles. Because a triangle”s interior angles always add up to 180° and $180/3 = 60$, an equilateral triangle will always have three 60° angles.


Now let”s drop down a height from the topmost angle to the base of the triangle.


We”ve now created two right angles and two congruent (equal) triangles.

How do we know they”re equal triangles? Because we dropped a height from an equilateral triangle, we”ve split the base exactly in half. The new triangles also share one side length (the height), and they each have the same hypotenuse length. Because they share three side lengths in common (SSS), this means the triangles are congruent.

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Note: not only are the two triangles congruent based on the principles of side-side-side lengths, or SSS, but also based on side-angle-side measures (SAS), angle-angle-side (AAS), and angle-side-angle (ASA). Basically? They”re most definitely congruent.

So let us call our original side length $x$ and our bisected length $x/2$.

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Now all that leaves us to do is to find our mid-side length that the two triangles share. To do this, we can simply use the Pythagorean theorem.


$a^2 + b^2 = c^2$

$(x/2)^2 + b^2 = x^2$

$b^2 = x^2 – ({x^2}/4)$

$b^2 = {4x^2}/4 – {x^2}/4$

$b^2 = {3x^2}/4$

$b = {√3x}/2$

So we”re left with: $x/2, {x√3}/2, x$

Now let”s multiply each measure by 2, just to make life easier and avoid all the fractions. That way, we”re left with:

$x$, $x√3$, $2x$



With the special triangle ratios, you can figure out missing triangle heights or leg lengths (without having to use the Pythagorean theorem), find the area of a triangle by using missing height or base length information, and quickly calculate perimeters.



Sine of 30° will always be $1/2$.

Cosine of 60° will always be $1/2$.

Though the other sines, cosines, and tangents are fairly simple, these are the two that are the easiest to memorize and are likely to show up on tests. So knowing these rules will allow you to find these trigonometry measurements as quickly as possible.

Some people memorize the ratio by thinking, “$i x$, $o 2 i x$, $i x o √ o3$,” because the “1, 2, 3” succession is typically easy to remember. The one precaution to using this technique is to remember that the longest side is actually the $2x$, not the $x$ times $√3$.

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Another way to remember your ratios is to use a mnemonic wordplay on the 1: root 3: 2 ratio in their proper order. For example, “Jackie Mitchell struck out Lou Gehrig and “won Ruthy too,””: one, root three, two. (And it”s a true baseball history fact to boot!)

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