Special Right Triangles

13 min read

Cut a square along its diagonal and you get a triangle whose three sides are exact: 1, 1, and $\sqrt{2}$. Cut an equilateral triangle straight down the middle and you get another one: 1, $\sqrt{3}$, and 2. Those two cuts are the only pictures behind this whole page. Once you have them, you can write the side of any 30, 45, or 60 degree triangle without ever reaching for a calculator.

By the end of this page you can:

• Name a triangle as 45-45-90 or 30-60-90 from its angles, and state its exact side ratio.
• Find every side of either triangle from one given side, forward or backward.
• Derive each ratio from the Pythagorean theorem instead of recalling it cold.

The floor. Memorize two ratios: $1 : 1 : \sqrt{2}$ for the 45-45-90 triangle and $1 : \sqrt{3} : 2$ for the 30-60-90 triangle. Everything else on this page is a way to use them or rebuild them when memory fails.

Quick Reference

The primary text is Stewart 7e, Section 6.2 Trigonometry of Right Triangles, which develops the 30-60-90 and 45-45-90 triangles and the exact values they produce. The open alternate is OpenStax Precalculus 2e, Section 5.4. Page numbers are pending faculty verification and are not printed here.

Before You Start

Prerequisite Hub

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        B["Pythagorean<br/>Theorem"]
    end

    subgraph ThisSkill["This Skill"]
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Builds on:

Unlocks:

There are no cross-course prerequisites for this skill.

The Idea

Both triangles come from shapes you already know, and both give side lengths that are exact instead of rounded. That is why they get a special name: the sides can be written with square roots and whole numbers, not decimals off a calculator.

Once the angles of a right triangle are 30, 45, or 60 degrees, its side ratio is locked. Scaling the triangle up or down keeps every angle the same, so the ratio of the sides never changes. That fixed ratio is what lets you read an exact answer off one diagram.

Official Definitions

45-45-90 triangle. An isosceles right triangle whose legs are equal; if each leg has length 1, the hypotenuse has length $\sqrt{2}$, giving the exact ratios used to evaluate trig functions at 45 degrees ($\pi/4$).

Source: OpenStax Precalculus 2e, Section 5.4 Right Triangle Trigonometry.

30-60-90 triangle. A right triangle with angles 30, 60, and 90 degrees; with the short leg 1, the longer leg is $\sqrt{3}$ and the hypotenuse is 2, giving the exact ratios for 30 degrees ($\pi/6$) and 60 degrees ($\pi/3$).

Source: OpenStax Precalculus 2e, Section 5.4 Right Triangle Trigonometry.

The scope record lists no separately named theorem. The two definitions carry the exact side ratios, each derived from the Pythagorean theorem.

Quick Reference

The longer side is always opposite the larger angle. In the 30-60-90 triangle, the side opposite 60 degrees beats the side opposite 30 degrees.

The Two Special Triangles

Each triangle is built once from a picture, and the ratio is fixed from then on.

The 45-45-90 Triangle

Cut the unit square along its diagonal.

        |\
        | \
      1 |  \  hypotenuse
        |   \
        |____\
          1

Both legs are 1 unit, and the two acute angles are equal, so each is 45 degrees. The hypotenuse comes straight from the Pythagorean theorem. With each leg $x$:

\[ \text{hypotenuse} = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2} \]

So the ratio of leg to leg to hypotenuse is $1 : 1 : \sqrt{2}$.

Predict then check. The diagonal of a unit square should be a little longer than the side. Since $\sqrt{2} \approx 1.414$, it is about 1.41 units: longer than 1, shorter than 2, as predicted.

The 30-60-90 Triangle

Start with an equilateral triangle of side $2x$ and drop a line from the top vertex to the middle of the base.

         /|\
        / | \
    2x /  |  \ 2x
      / 60| 60\
     /____|____\
       x     x

Cutting down the middle halves the base, so each half-base is $x$. It also splits the top 60 degree angle into two 30 degree angles. One half is a right triangle with a 30 degree angle at the top, 60 degrees at the bottom, and 90 degrees at the cut.

This half has a short leg of $x$ (the half-base) and a hypotenuse of $2x$ (a full side of the equilateral triangle). The remaining leg comes from the Pythagorean theorem:

\[ \text{long leg} = \sqrt{(2x)^2 - x^2} = \sqrt{4x^2 - x^2} = \sqrt{3x^2} = x\sqrt{3} \]

So the ratio of short leg to long leg to hypotenuse is $1 : \sqrt{3} : 2$. The short leg sits opposite 30 degrees, the long leg $x\sqrt{3}$ opposite 60 degrees, and the hypotenuse $2x$ opposite 90 degrees.

Predict then check. The half-base was predicted to be half of a full side. A full side is $2x$ and the half-base came out to $x$, exactly half.

Both Triangles Side by Side

One triangle comes from a square, the other from an equilateral triangle, and both give side lengths you can write exactly.

Worked Examples

Example 1: Scaling a 45-45-90 Triangle

A 45-45-90 triangle has legs of 5. Find the exact hypotenuse.

Predict. The hypotenuse is the longest side, so the answer is more than 5; since $\sqrt{2} \approx 1.41$, expect near 7.

Solve. Multiply the ratio $1 : 1 : \sqrt{2}$ by 5 to get $5 : 5 : 5\sqrt{2}$. The hypotenuse is $5\sqrt{2}$.

Check. $5\sqrt{2} \approx 7.07$, near 7 and longer than the legs.

Example 2: Scaling a 30-60-90 Triangle From the Short Leg

A 30-60-90 triangle has its short leg (opposite 30 degrees) equal to 4. Find the other two sides.

Predict. The short leg is smallest, so the other two are both larger than 4.

Solve. Multiply $1 : \sqrt{3} : 2$ by 4 to get $4 : 4\sqrt{3} : 8$. The long leg (opposite 60 degrees) is $4\sqrt{3}$ and the hypotenuse is $8$.

Check. $4\sqrt{3} \approx 6.93$ and the hypotenuse is 8, both larger than 4, with the hypotenuse largest.

Example 3: Working Backward From the Hypotenuse

A 30-60-90 triangle has a hypotenuse of 10. Find the two legs.

Predict. The hypotenuse matches the 2 in $1 : \sqrt{3} : 2$, not the 1, so the scaling factor is not 10 itself.

One way. The hypotenuse 10 corresponds to the ratio entry 2, so multiply the whole ratio by 5: $5 : 5\sqrt{3} : 10$. The short leg is $5$, the long leg $5\sqrt{3}$.

Another way. The short leg is always half the hypotenuse, since it matches 1 and the hypotenuse matches 2. Half of 10 is 5, so the short leg is 5; the long leg is $5 \times \sqrt{3} = 5\sqrt{3}$. Both routes agree. The first scales the whole ratio at once; the second is faster when the hypotenuse is given.

Check. $5\sqrt{3} \approx 8.66$, between the short leg 5 and the hypotenuse 10.

Example 4: Reading an Exact Trig Value Off the Triangle

Use the 45-45-90 triangle to find the ratio of the side opposite 45 degrees to the hypotenuse.

Predict. Both legs are shorter than the hypotenuse, so the ratio is less than 1.

Solve. With a leg $x$ and hypotenuse $x\sqrt{2}$:

\[ \frac{x}{x\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

Check. $\dfrac{\sqrt{2}}{2} \approx 0.707$, less than 1 as predicted. This is the exact value of $\sin(45^\circ)$, which the next skill develops in full.

Common Misconceptions

Leveled Practice

Level 1: Direct Application

Problem 1. A 45-45-90 triangle has legs of 3. Find the exact hypotenuse.

Problem 2. A 30-60-90 triangle has a short leg of 5. Find the exact hypotenuse.

Level 2: Scaling Both Triangles

Problem 3. A 45-45-90 triangle has legs of 9. Find the exact hypotenuse.

Problem 4. A 30-60-90 triangle has a short leg of 7. Find the exact long leg and hypotenuse.

Level 3: Working Backward

Problem 5. A 45-45-90 triangle has hypotenuse $6\sqrt{2}$. Find the exact legs.

Problem 6. A 30-60-90 triangle has hypotenuse 18. Find the exact legs.

Level 4: Long Leg Given, Rationalize the Result

Problem 7. A 30-60-90 triangle has its long leg (opposite 60 degrees) equal to 6. Find the exact short leg and hypotenuse, with no square root in any denominator.

Level 5: Justify and Generalize

Problem 8. Explain why the diagonal of a square is always $\sqrt{2}$ times its side, and connect this to the 45-45-90 triangle. How would you convince a classmate it holds for a square of any size?

Check Yourself

Go Deeper (optional)

Mastery Checklist

Novice (Level 1-2):

• I can name a triangle as 45-45-90 or 30-60-90 from its angles or a pair of equal legs
• I can state $1 : 1 : \sqrt{2}$ for the 45-45-90 and $1 : \sqrt{3} : 2$ for the 30-60-90
• I can find the hypotenuse of a 45-45-90 triangle from its legs

Competent (Level 3-4):

• I can find all three sides of a 30-60-90 triangle from the short leg
• I can work backward from the hypotenuse to find both legs
• I can identify which side sits opposite each angle

Proficient (Level 5):

• I can find all sides when the long leg is given, and rationalize the result
• I can derive each ratio from the Pythagorean theorem rather than recalling it
• I can explain why scaling is multiplicative, not additive, and convince a classmate it holds at any size

Connections

Looking back:

• Angles and Degree Measure gives the angle reading that names each triangle.
• Pythagorean Theorem supplies the relationship that produces every exact side length.

Looking ahead:

• Right Triangle Trigonometry turns these fixed ratios into the exact sine, cosine, and tangent of 30, 45, and 60 degrees.

Resources

The scope record lists no Dr. Lynch video for this skill, so none is linked here.

Back to MATH 142 Skills | Previous: Pythagorean Theorem | Next: Right Triangle Trigonometry

Last updated: 2026-06-24