Right Triangles and the Pythagorean Theorem

There is a good chance you have heard of the Pythagorean Theorem before, even if math is not your favorite subject. It shows up everywhere: in construction, navigation, video games, architecture, and even in measuring your television screen. This single relationship between the sides of a right triangle has been known for over 2,500 years, and people still use it every single day. Whether a carpenter is checking that a wall is perfectly square or a pilot is calculating the shortest flight path, the Pythagorean Theorem is quietly doing the work behind the scenes.

The beautiful thing about this theorem is how simple it is to state, yet how powerful it is in practice. Once you understand it, you will start noticing right triangles everywhere, and you will have the tools to work with them confidently.

Core Concepts

What Makes a Triangle “Right”?

A right triangle is simply a triangle that has one angle measuring exactly $90°$, a right angle. That is the corner you see on a piece of paper, a door frame, or the edges of your phone screen. The presence of this right angle gives the triangle special properties that do not apply to other triangles.

The two sides that form the right angle are called the legs of the triangle. These are often labeled $a$ and $b$. The side opposite the right angle, the one that does not touch the right angle at all, is called the hypotenuse. The hypotenuse is always the longest side of a right triangle, and it is typically labeled $c$.

Here is an easy way to remember: the hypotenuse is “across” from the right angle. It is never one of the sides that forms the $90°$ corner.

The Pythagorean Theorem

Now for the main event. The Pythagorean Theorem tells us that in any right triangle, there is a precise relationship between the three sides:

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

In words: if you square the length of each leg and add those squares together, you get exactly the square of the hypotenuse.

This is not just approximately true or usually true. It is exactly true for every right triangle that has ever existed or ever will exist. The ancient Babylonians knew this relationship over 4,000 years ago. The Greek mathematician Pythagoras (or his followers) gets the credit for providing a formal proof, which is why we call it the Pythagorean Theorem.

Why does this matter? Because if you know any two sides of a right triangle, you can always find the third. This is incredibly useful in countless real-world situations.

Pythagorean Triples

Some right triangles have the special property that all three sides are whole numbers. These sets of three integers that satisfy $a^2 + b^2 = c^2$ are called Pythagorean triples.

The most famous is 3-4-5. Let us check: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$. It works!

Here are some other common Pythagorean triples worth memorizing:

• 3, 4, 5: The classic. Carpenters love this one.
• 5, 12, 13: Another common one. Check: $25 + 144 = 169 = 13^2$.
• 8, 15, 17: Check: $64 + 225 = 289 = 17^2$.
• 7, 24, 25: Check: $49 + 576 = 625 = 25^2$.

Here is a useful fact: if you multiply all three numbers in a Pythagorean triple by the same value, you get another Pythagorean triple. For example, since 3-4-5 works, so does 6-8-10 (multiply by 2), and 9-12-15 (multiply by 3), and 30-40-50 (multiply by 10). This is why the 3-4-5 triple is especially handy; it scales to any size you need.

The Converse of the Pythagorean Theorem

The Pythagorean Theorem works in reverse, too. If you have a triangle and want to know whether it is a right triangle, just check whether $a^2 + b^2 = c^2$ (where $c$ is the longest side).

The Converse of the Pythagorean Theorem: If the sides of a triangle satisfy $a^2 + b^2 = c^2$, then the triangle is a right triangle, with the right angle opposite the longest side $c$.

This is incredibly useful for checking whether corners are “square” in construction or verifying that a shape has a right angle.

Classifying Triangles by Their Angles

The relationship between the sides can actually tell you what type of triangle you have:

• If $a^2 + b^2 = c^2$: The triangle is right (has a $90°$ angle)
• If $a^2 + b^2 > c^2$: The triangle is acute (all angles less than $90°$)
• If $a^2 + b^2 < c^2$: The triangle is obtuse (has an angle greater than $90°$)

Think of it this way: in a right triangle, the two shorter sides are “just right” to satisfy the equation. If those sides are “too big” (their squares sum to more than $c^2$), the angle opposite $c$ gets squeezed smaller than $90°$, making the triangle acute. If those sides are “too small,” the angle opposite $c$ opens up beyond $90°$, making the triangle obtuse.

Geometric Mean

Before we dive into special right triangles, there is one more concept that connects nicely with right triangles: the geometric mean.

The geometric mean of two positive numbers $a$ and $b$ is $\sqrt{ab}$.

Why does this matter for right triangles? When you draw the altitude (height) from the right angle to the hypotenuse in a right triangle, you create smaller triangles with some beautiful relationships:

• The altitude is the geometric mean of the two segments it creates on the hypotenuse.
• Each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg.

These relationships are useful in more advanced problems and help explain why certain constructions work the way they do.

Special Right Triangles: 45-45-90

Some right triangles show up so often that they have their own names based on their angles. The first is the 45-45-90 triangle, also called an isosceles right triangle because the two legs are equal.

In a 45-45-90 triangle:

• The two legs are equal (both are some length $x$)
• The hypotenuse is $x\sqrt{2}$

$$\text{Legs: } x, x \qquad \text{Hypotenuse: } x\sqrt{2}$$

Where does $\sqrt{2}$ come from? Use the Pythagorean Theorem: if both legs are $x$, then $x^2 + x^2 = c^2$, which gives $2x^2 = c^2$, so $c = x\sqrt{2}$.

Think of it this way: the diagonal of a square is always $\sqrt{2}$ times the length of a side. If you cut a square in half diagonally, you get two 45-45-90 triangles.

Special Right Triangles: 30-60-90

The other special right triangle is the 30-60-90 triangle. This one comes from cutting an equilateral triangle in half.

In a 30-60-90 triangle:

• The side opposite the $30°$ angle (shortest leg) is $x$
• The side opposite the $60°$ angle (longer leg) is $x\sqrt{3}$
• The hypotenuse (opposite the $90°$ angle) is $2x$

$$\text{Shortest leg: } x \qquad \text{Longer leg: } x\sqrt{3} \qquad \text{Hypotenuse: } 2x$$

The ratio is always $1 : \sqrt{3} : 2$.

Here is why: start with an equilateral triangle where all sides have length $2x$. Draw the altitude from one vertex to the opposite side. This altitude bisects the opposite side (splitting it into two segments of length $x$) and bisects the vertex angle (creating two $30°$ angles). You now have two 30-60-90 triangles.

The altitude has length $x\sqrt{3}$ because by the Pythagorean Theorem: $(x)^2 + (\text{altitude})^2 = (2x)^2$, so $x^2 + (\text{altitude})^2 = 4x^2$, giving $(\text{altitude})^2 = 3x^2$, and altitude $= x\sqrt{3}$.

Notation and Terminology

Term	Meaning	Example
Right triangle	Triangle with one $90°$ angle	The corner of a book
Legs	The two sides that form the right angle	$a$ and $b$ in $a^2 + b^2 = c^2$
Hypotenuse	Side opposite the right angle (longest side)	$c$ in $a^2 + b^2 = c^2$
Pythagorean triple	Three integers satisfying $a^2 + b^2 = c^2$	3, 4, 5 or 5, 12, 13
Converse	The “reverse” statement of a theorem	If $a^2 + b^2 = c^2$, then right triangle
Acute triangle	All angles less than $90°$	Satisfied when $a^2 + b^2 > c^2$
Obtuse triangle	Has one angle greater than $90°$	Satisfied when $a^2 + b^2 < c^2$
Geometric mean	$\sqrt{ab}$ for two numbers $a$ and $b$	Geometric mean of 4 and 9 is 6
45-45-90	Isosceles right triangle	Sides: $x$, $x$, $x\sqrt{2}$
30-60-90	Half of an equilateral triangle	Sides: $x$, $x\sqrt{3}$, $2x$

Examples

Key Properties and Rules

The Pythagorean Theorem and Its Converse

Pythagorean Theorem: In a right triangle with legs $a$ and $b$ and hypotenuse $c$: $$a^2 + b^2 = c^2$$

Converse: If three sides of a triangle satisfy $a^2 + b^2 = c^2$ (where $c$ is the longest), then the triangle is a right triangle.

Triangle Classification by Sides

For a triangle with sides $a$, $b$, and $c$ (where $c$ is the longest):

$$a^2 + b^2 = c^2 \implies \text{Right triangle}$$ $$a^2 + b^2 > c^2 \implies \text{Acute triangle}$$ $$a^2 + b^2 < c^2 \implies \text{Obtuse triangle}$$

Common Pythagorean Triples

Triple	Verification
3, 4, 5	$9 + 16 = 25$
5, 12, 13	$25 + 144 = 169$
8, 15, 17	$64 + 225 = 289$
7, 24, 25	$49 + 576 = 625$

Scaling rule: Multiply any triple by $k$ to get another valid triple: $ka$, $kb$, $kc$.

Special Right Triangle Ratios

45-45-90 Triangle: $$x : x : x\sqrt{2}$$

• Legs are equal
• Hypotenuse = leg $\times \sqrt{2}$
• Leg = hypotenuse $\div \sqrt{2}$ = hypotenuse $\times \frac{\sqrt{2}}{2}$

30-60-90 Triangle: $$x : x\sqrt{3} : 2x$$

• Shortest leg (opposite $30°$) = $x$
• Longer leg (opposite $60°$) = $x\sqrt{3}$
• Hypotenuse (opposite $90°$) = $2x$
• Hypotenuse is always twice the shortest leg

Geometric Mean Relationships

In a right triangle with altitude $h$ drawn to hypotenuse $c$, which creates segments $p$ and $q$ on the hypotenuse:

$$h = \sqrt{pq}$$

$$a = \sqrt{cp} \quad \text{and} \quad b = \sqrt{cq}$$

where $a$ is the leg adjacent to segment $p$ and $b$ is the leg adjacent to segment $q$.

Real-World Applications

Construction: Checking for Square Corners

Carpenters and builders use the 3-4-5 rule constantly. To check if a corner is square:

1. Measure 3 feet along one edge from the corner
2. Measure 4 feet along the other edge from the corner
3. The diagonal between these two points should be exactly 5 feet

If the diagonal is exactly 5 feet, the corner is a perfect $90°$ angle. If not, the corner needs adjustment. This technique works at any scale: 6-8-10, 9-12-15, or even 30-40-50 for large-scale work.

Navigation: Finding Straight-Line Distance

Suppose you walk 3 miles east and then 4 miles north. How far are you from your starting point “as the crow flies”?

Your path forms two legs of a right triangle. The straight-line distance back is the hypotenuse: $$d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ miles}$$

Pilots, sailors, and hikers use this principle constantly when calculating direct distances between points.

Ladder Safety

Safety guidelines recommend that a ladder be placed so that the distance from the wall is about one-quarter of the ladder’s length. For a 20-foot ladder, that means placing the base 5 feet from the wall.

How high up the wall does the ladder reach? $$h = \sqrt{20^2 - 5^2} = \sqrt{400 - 25} = \sqrt{375} \approx 19.4 \text{ feet}$$

The Pythagorean Theorem helps you calculate safe ladder placements and working heights.

Screen Diagonal Measurements

Television and monitor sizes are measured diagonally. If you know a TV is 48 inches wide and 27 inches tall, you can find its diagonal: $$d = \sqrt{48^2 + 27^2} = \sqrt{2304 + 729} = \sqrt{3033} \approx 55 \text{ inches}$$

This is why it is called a “55-inch TV” even though neither the width nor height is 55 inches.

Sports: Field Distances

In baseball, the distance from home plate to first base is 90 feet. The distance from home plate to second base (diagonally across the diamond) is: $$d = \sqrt{90^2 + 90^2} = \sqrt{8100 + 8100} = \sqrt{16200} = 90\sqrt{2} \approx 127.3 \text{ feet}$$

This is exactly the 45-45-90 relationship in action!

Self-Test Problems

Summary

• A right triangle has one $90°$ angle. The two sides forming this angle are the legs ($a$ and $b$), and the side opposite the right angle is the hypotenuse ($c$), which is always the longest side.

• The Pythagorean Theorem states: $a^2 + b^2 = c^2$. This relationship holds for every right triangle and allows you to find any side when you know the other two.

• Pythagorean triples are sets of three positive integers that satisfy the theorem (3-4-5, 5-12-13, 8-15-17, etc.). Recognizing these can save you calculation time.

• The converse works too: if $a^2 + b^2 = c^2$ for a triangle’s sides, then the triangle must be a right triangle.

• You can classify triangles using this relationship:

  • $a^2 + b^2 = c^2$: right triangle
  • $a^2 + b^2 > c^2$: acute triangle
  • $a^2 + b^2 < c^2$: obtuse triangle

• 45-45-90 triangles have sides in the ratio $x : x : x\sqrt{2}$. They are isosceles right triangles, formed by cutting a square diagonally.

• 30-60-90 triangles have sides in the ratio $x : x\sqrt{3} : 2x$. They come from cutting an equilateral triangle in half.

• The Pythagorean Theorem appears everywhere in real life: construction (checking square corners), navigation (finding direct distances), ladder safety calculations, screen measurements, and countless other applications.

• The geometric mean connects to right triangles through the altitude to the hypotenuse, creating proportional relationships useful in more advanced problems.

When you truly understand the Pythagorean Theorem and special right triangles, you have tools that will serve you well in trigonometry, physics, engineering, and everyday problem-solving. This 2,500-year-old theorem remains one of the most practically useful pieces of mathematics ever discovered.