◺ Right Triangle Calculator: Formulas, Solved Examples, and Free Tool

A right triangle has one 90 degree angle and two shorter sides called legs. This right triangle calculator finds every missing side, angle, and area in seconds. You only need two known values to solve the rest.

Maybe you know two sides and need the third one. Maybe you have one angle and one side instead. This page walks through the exact formulas first, then lets you plug in your own numbers. You will also find special triangle ratios, real examples, and answers to common questions people ask.

What Is a Right Triangle

A right triangle is a triangle with one angle that measures exactly 90 degrees. That 90 degree angle is called the right angle. The other two angles are always smaller than 90 degrees each.

Most people also call this shape a right angled triangle. Both names mean the same thing. You will see both terms used across textbooks, worksheets, and online tools.

Parts of a Right Triangle

Every right triangle has three sides with specific names.

• Leg a and leg b form the right angle itself.
• The hypotenuse is the longest side in the triangle.
• The hypotenuse always sits opposite the right angle.

Picture a triangle standing on one leg. The vertical side and the horizontal side are the legs. The slanted side connecting them is the hypotenuse.

Right Triangle vs Other Triangle Types

A right triangle is not the only triangle type you will run into.

• An acute triangle has three angles under 90 degrees.
• An obtuse triangle has one angle over 90 degrees.
• A right triangle sits exactly between those two types.

If your triangle has no right angle at all, this page will not give you the correct formulas. A general triangle calculator handles those other cases instead.

Properties and Elements of a Right Triangle

A right triangle carries more hidden measurements than most people expect. Beyond the three sides and three angles, several other points and lines sit inside every right triangle. These properties come up often in geometry problems and design work.

Most calculators skip this part completely. Knowing these values gives you a fuller picture of your triangle beyond just the basics.

Altitude to the Hypotenuse

The altitude is a line drawn from the right angle straight down to the hypotenuse. This line splits the original triangle into two smaller triangles.

The formula is simple. Altitude h equals leg a times leg b, divided by the hypotenuse c.

• h = (a × b) / c

Both smaller triangles created by this altitude are similar to the original triangle. This fact becomes useful later when working with geometric mean problems.

Median, Centroid, and Circumcenter

A median connects a vertex to the midpoint of the opposite side. In a right triangle, the median to the hypotenuse has a neat shortcut.

• The median to the hypotenuse always equals half the hypotenuse length.

The circumcenter is the point equally distant from all three vertices. In a right triangle, this point sits exactly at the midpoint of the hypotenuse. No calculation needed once you know the hypotenuse.

The orthocenter, where all three altitudes meet, lands right on the right angle vertex itself. This only happens in right triangles, not in other triangle types.

Inradius and Circumradius

Every triangle has a circle that fits perfectly inside it and one that passes through all three corners. The radius of each circle follows a set formula in a right triangle.

• Inradius: r = (a + b − c) / 2
• Circumradius: R = c / 2

The circumradius formula makes sense once you remember the circumcenter sits at the hypotenuse midpoint. Half the hypotenuse is simply the distance to that point.

Types of Right Triangles

Not every right triangle looks the same. The lengths of the legs decide what type of right triangle you are working with. Knowing the type helps you pick the fastest solving method.

Isosceles Right Triangle

An isosceles right triangle has two legs of equal length. Because the legs match, the two acute angles also match at 45 degrees each.

• Both legs are equal in length.
• Both acute angles measure exactly 45 degrees.
• The triangle has one line of symmetry down the middle.

This triangle type has a shortcut for area. Since both legs are equal, you only need one leg length to find the area.

Scalene Right Triangle

A scalene right triangle has three sides of different lengths. One angle is still fixed at 90 degrees, but the other two angles are never equal to each other.

Most right triangles you run into are scalene. A triangle with legs of 3 and 4 units is a common scalene example. Yes, a triangle can absolutely be both scalene and right at the same time.

Can a Right Triangle Be Equilateral or Isosceles

People often ask if a right triangle can also be equilateral. The answer is no.

An equilateral triangle needs three equal 60 degree angles. A right triangle already has one angle locked at 90 degrees. Those two conditions cannot exist together in the same shape.

Being isosceles is a different story. A right triangle can absolutely be isosceles, as shown above with the 45-45-90 case. Isosceles and equilateral are not the same requirement.

The Pythagorean Theorem

The Pythagorean theorem is the most useful formula for any right triangle. It connects the two legs and the hypotenuse in one simple equation.

• a² + b² = c²

This formula only works because of the 90 degree angle between the legs. It will not give correct results on triangles without a right angle.

Finding the Hypotenuse

To find the hypotenuse, square both legs, add the results together, then take the square root.

• c = √(a² + b²)

Say leg a is 3 and leg b is 4. Square each one to get 9 and 16. Add them for 25, then take the square root to get 5.

Finding a Missing Leg

If you already know the hypotenuse and one leg, the same formula works in reverse. Subtract the known leg squared from the hypotenuse squared, then take the square root.

• a = √(c² − b²)

Using a hypotenuse of 5 and a leg of 4, subtract 16 from 25 to get 9. The square root of 9 is 3, your missing leg.

Pythagorean Triples Table

A Pythagorean triple is a set of three whole numbers that satisfy the theorem exactly. These show up often in geometry problems and construction work.

Leg a	Leg b	Hypotenuse c	Check
3	4	5	9 + 16 = 25
5	12	13	25 + 144 = 169
8	15	17	64 + 225 = 289
7	24	25	49 + 576 = 625
20	21	29	400 + 441 = 841
9	40	41	81 + 1600 = 1681

Any multiple of these numbers also forms a right triangle. Double every number in the 3-4-5 triple and you get 6, 8, 10, which still works perfectly.

Real Life Example, Squaring a Corner

Builders use the 3-4-5 triple to check if a corner is truly square. Measure 3 feet along one wall and 4 feet along the other wall.

If the diagonal distance between those two points measures exactly 5 feet, the corner is a true 90 degrees. This trick works on any job site without needing a protractor.

Need to solve the theorem on its own, without the full right triangle? Try the Pythagorean theorem calculator instead.

How to Tell If a Triangle Is a Right Triangle

Sometimes you only have three side lengths and need to check if the triangle is right. There is a fast way to confirm this without measuring any angles.

The Converse of the Pythagorean Theorem

If the square of the longest side equals the sum of the squares of the other two sides, the triangle is right. This works in reverse of the standard theorem.

• Check: does a² + b² = c²?
• If yes, the triangle has a 90 degree angle.
• If no, the triangle is either acute or obtuse instead.

This rule only applies to right triangles specifically. It cannot confirm angles in triangles that lack a 90 degree angle.

Quick Side Length Check

Try this with a triangle of sides 9, 40, and 41. Square the two smaller sides, 81 and 1600, then add them for 1681.

The square root of 1681 is 41, matching the longest side exactly. This confirms the triangle is right without measuring a single angle.

A triangle with sides 5, 6, and 8 fails this test. Squaring gives 25 and 36, adding up to 61, not 64. That triangle is not a right triangle.

Right Triangle Angles

Angles in a right triangle follow a predictable pattern once you know the basic rule. This makes solving for missing angles much easier than it sounds.

Complementary Angles in a Right Triangle

The two acute angles in a right triangle always add up to exactly 90 degrees. Mathematicians call this pair complementary angles.

If one acute angle measures 30 degrees, the other automatically measures 60 degrees. You never need extra information to find that second angle.

How Many Degrees Are in a Right Triangle

Every triangle, regardless of type, has angles that total 180 degrees. A right triangle fixes one of those angles at 90 degrees from the start.

• Total angle sum: 180 degrees
• Right angle: 90 degrees
• Remaining two angles: 90 degrees combined

This leaves only 90 degrees to split between the two acute angles, which is why they are always complementary.

Finding a Missing Angle

If you already know one acute angle, subtract it from 90 to get the other one instantly.

• Angle B = 90° − Angle A

When you only have side lengths and no angle at all, inverse trig functions step in. Arcsin, arccos, and arctan convert a side ratio back into an angle measurement.

For example, a triangle with legs 3 and 4 gives an angle of arctan(3/4), which works out to about 36.87 degrees.

SOHCAHTOA and Right Triangle Trigonometry

Trigonometry gives you another way to solve a right triangle when angles are involved. SOHCAHTOA is the memory trick most people learn first.

Sine, Cosine, and Tangent Ratios

Each letter in SOHCAHTOA points to one trig ratio and the sides it connects.

• SOH: sin(θ) = opposite / hypotenuse
• CAH: cos(θ) = adjacent / hypotenuse
• TOA: tan(θ) = opposite / adjacent

Opposite means the side across from the angle you are working with. Adjacent means the side touching that angle, not counting the hypotenuse.

Cotangent and Other Ratios

Cotangent is simply the flipped version of tangent. It divides the adjacent side by the opposite side instead of the other way around.

• cot(θ) = adjacent / opposite

SOHCAHTOA works only on right triangles, since it relies on that fixed 90 degree angle. Other triangle types need the law of sines or law of cosines instead.

Using Inverse Trig to Find an Angle

When two sides are known but the angle is not, inverse trig functions solve for it directly.

• Use arcsin when you know the opposite side and hypotenuse.
• Use arccos when you know the adjacent side and hypotenuse.
• Use arctan when you know both legs but not the hypotenuse.

These functions undo the regular trig ratios, turning a fraction back into a degree measurement.

Worked Trigonometry Example

Take a right triangle with angle A at 36.87 degrees and a hypotenuse of 5. Solve for both legs using sine and cosine.

• a = c × sin(A) = 5 × sin(36.87°) = 3
• b = c × cos(A) = 5 × cos(36.87°) = 4
• B = 90° − 36.87° = 53.13°

All three sides and both angles are now solved from just one angle and one side.

How to Solve a Right Triangle

Solving a right triangle means finding every missing side and angle at once. You only need two known values, plus the right angle itself, to solve the rest.

The method changes depending on what you already know. Match your known values to one of the four approaches below.

Solving With Two Known Sides

If you know two sides, the Pythagorean theorem finds the third one first. Once all three sides are known, use inverse trig to find both angles.

• Two legs known: find the hypotenuse, then both angles.
• One leg and hypotenuse known: find the other leg, then both angles.

Solving With One Angle and One Side

If you know one angle and one side, trig ratios solve the rest directly. The ratio you use depends on which side you already have.

• Angle and hypotenuse: use sine and cosine for both legs.
• Angle and one leg: use tangent, sine, or cosine as needed.

Solving With Area and One Leg

Sometimes area is given instead of a second side. Double the area, then divide by the known leg to get the other leg.

• Other leg = (2 × Area) / known leg

Once both legs are known, the Pythagorean theorem finds the hypotenuse the same way as before.

Solving With Only One Known Side

A single side alone is never enough on its own. You need one more clue, either an angle or confirmation that the triangle is a special type.

If the triangle is a 30-60-90 or 45-45-90 triangle, fixed ratios let you solve everything from that one side. Otherwise, look for a second measurement first.

Right Triangle Calculator

Now that you understand the formulas, the right triangle calculator below does the work for you. Enter any two known values and it solves the rest instantly.

This right angle triangle calculator covers every method described above in one tool. No manual formula switching required.

What the Calculator Solves

The calculator offers several input modes, so you can match it to whatever values you already have.

• Leg a and leg b together.
• Leg a and the hypotenuse together.
• Leg b and the hypotenuse together.
• One leg paired with angle A.
• The hypotenuse paired with angle A.

Beyond the missing side and angle, it also returns area, perimeter, altitude, inradius, and circumradius in one pass. Special triangle and SOHCAHTOA modes are built in as separate tabs.

How to Read Your Results

Once you hit solve, the results panel fills in every side, every angle, and the extra properties covered earlier in this guide. A step by step box beneath it shows exactly how each value was calculated.

Switch to the SOHCAHTOA tab if you prefer working directly from an angle and a single side. The special triangles tab handles 30-60-90 and 45-45-90 cases using their fixed ratios, without needing the full solver.

Area of a Right Triangle

Area tells you how much space sits inside the triangle. A right triangle makes this calculation easier than most other triangle types.

Standard Area Formula

The two legs act as base and height, so no extra measuring is needed.

• Area = ½ × a × b

For legs of 3 and 4, multiply them together for 12, then take half for an area of 6. This works every time, regardless of the triangle’s size.

Area Using the Hypotenuse and Altitude

If you already found the altitude drawn to the hypotenuse, there is a second way to calculate area.

• Area = ½ × c × h

This formula gives the same result as the standard one. It comes in handy when the altitude is already known but one leg is missing.

Area of an Isosceles Right Triangle

Since both legs are equal in an isosceles right triangle, the area formula simplifies further.

• Area = ½ × leg²

Only one leg length is needed here instead of two separate values. A leg of 5 gives an area of 12.5 using this shortcut.

Perimeter and Semiperimeter of a Right Triangle

Perimeter measures the total distance around the outside of the triangle. It has one of the simplest formulas on this page.

• Perimeter = a + b + c

Add all three sides together and you have your answer. A triangle with sides 3, 4, and 5 has a perimeter of 12.

Semiperimeter is exactly half of that total. It shows up in other formulas, including the inradius calculation covered earlier.

• Semiperimeter (s) = (a + b + c) / 2

Using the same 3-4-5 triangle, the semiperimeter works out to 6. This single number becomes useful once you move into more advanced area and radius formulas.

Special Right Triangles

Special right triangles have fixed angle measures, so their sides always follow the same set ratio. This makes solving them faster than a regular right triangle.

30-60-90 Triangle Ratios

A 30-60-90 triangle always keeps its three sides in the same fixed ratio.

• Short leg (opposite 30°) : long leg (opposite 60°) : hypotenuse
• Ratio: 1 : √3 : 2

If the short leg is 5, multiply by √3 for a long leg of about 8.66. Double the short leg for a hypotenuse of 10.

45-45-90 Triangle Ratios

A 45-45-90 triangle is also an isosceles right triangle, with both legs equal.

• Leg : leg : hypotenuse
• Ratio: 1 : 1 : √2

If each leg measures 5, multiply by √2 for a hypotenuse of about 7.07. Both acute angles stay fixed at 45 degrees no matter the size.

Solving Special Triangles From One Side

Because the ratios never change, one known side unlocks the entire triangle.

• 30-60-90 example: short leg of 4 gives a long leg of 4√3 and a hypotenuse of 8.
• 45-45-90 example: hypotenuse of 6 gives each leg a length of 6 ÷ √2, about 4.24.

No angle measurement is needed for either case, since the angles are already fixed by definition.

Congruence, Similarity, and Geometric Mean

Two right triangles can sometimes match each other exactly, or share the same shape at different sizes. These ideas come up often in proofs and design work.

Congruence Rules for Right Triangles

Right triangles have a shortcut congruence rule not available to other triangle types. It is called the HL rule, short for hypotenuse leg.

• If the hypotenuse and one leg match between two right triangles, they are congruent.
• This works even without checking any angle measurements directly.

Similar Right Triangles and Proportions

Two right triangles are similar when their angles match, even if their sizes differ. Every side stays in the same proportion between the two triangles.

• Matching angles mean matching side ratios throughout the triangle.
• A larger and smaller right triangle can share this relationship easily.

Geometric Mean and the Altitude Rule

Remember the altitude drawn to the hypotenuse from earlier in this guide. That altitude splits the original triangle into two smaller triangles.

Both smaller triangles are similar to each other and to the original triangle. This relationship lets you solve for missing lengths using the geometric mean, without needing full trig calculations.

Right Triangles in Real Life

Right triangle math shows up far outside the classroom. These formulas solve real problems in several everyday fields.

• Carpentry: roof pitch, stair stringers, and rafter length all rely on right triangle ratios.
• Architecture: slope and angle of inclination for ramps and roof design.
• Navigation: finding bearing and distance between two points using right angle offsets.
• Physics: breaking a force or velocity into two perpendicular components.
• Surveying: measuring distances that cannot be walked directly using triangulation.
• Everyday math: the diagonal screen size of a phone or TV is a hypotenuse.

Common Mistakes When Solving Right Triangles

A few small errors trip people up again and again when working through these problems.

• Mislabeling the hypotenuse. It is always opposite the right angle and always the longest side.
• Mixing degrees and radians. Check your calculator mode before running any trig function.
• Using the wrong trig ratio. Confirm which side is opposite and which is adjacent first.
• Forgetting units. Area comes out in square units, while sides stay in linear units.

Double checking these four points before finishing a problem saves most people from small, avoidable errors.

Right Triangle vs Non Right Triangle

Every formula on this page depends on that fixed 90 degree angle between the legs. Without it, none of these shortcuts apply.

A non right triangle needs different tools entirely, like the law of sines or law of cosines. Side and angle relationships work differently once that right angle disappears.

If your triangle does not have a 90 degree angle, this page will not give accurate results. A general triangle tool handles SSS, SAS, ASA, and AAS cases instead, covering any triangle shape.

Frequently Asked Questions

Final Thoughts

Solving a right triangle comes down to matching your known values to the right formula. Two sides call for the Pythagorean theorem. An angle and a side call for SOHCAHTOA.

Use the right triangle calculator above whenever you want an instant answer instead of working through the math by hand. Enter your known values, choose the matching mode, and every side, angle, and property fills in at once.

Bookmark this page if you work with right triangles often, whether for schoolwork, carpentry, or design projects.