Right Triangle Calculator
Solve from two sides, side and angle, area and one side, or a square-corner check. Live diagram, step-by-step working, and trade contexts.
Solve any right triangle from two known values. Square-corner check, draggable diagram, trade contexts, and copyable working.
A right triangle calculator solves a triangle with one 90° corner when you supply any two independent facts, such as two sides, one side and an acute angle, or area plus one side. It returns all three sides, both acute angles, area, perimeter, altitude to the hypotenuse, slope, and roof pitch, with step-by-step formulas using your numbers.
Students use it for homework; builders use it to check rafters, stringers, ladders, and ramps before cut lists go to the saw. You get a live diagram, unit switching, radian expressions like pi/4, and a square-corner mode made for slabs and deck frames. Once your measurements check out, Invoice Mama makes it simple to turn them into a professional quote or invoice.
How to Use the Right Triangle Calculator
Four solve modes, one results panel
- 1
Pick a solve mode
Two sides, side plus angle, area plus one side, or square-corner check. The mode cards use plain English so you are not guessing math jargon on a jobsite.
- 2
Enter values and units
Choose length units and decimal places. Angles accept degrees or radians. Marked fields show whether you typed a value or the tool calculated it.
- 3
Read, copy, or share
Every output has a copy button. Step-by-step working is collapsible. Bookmark the URL to send the same triangle to a crew member.
Right Triangle Formulas
Quick reference for legs, hypotenuse, and area
Quantity | Formula |
|---|---|
| Hypotenuse c | √(a² + b²) |
| Leg a | √(c² − b²) |
| Leg b | √(c² − a²) |
| Angle α (from sides) | atan(a ÷ b) |
| Angle β | 90° − α |
| Area | (a × b) ÷ 2 |
| Perimeter | a + b + c |
| Altitude h to c | (a × b) ÷ c |
| Segments p, q on c | p = a² ÷ c, q = b² ÷ c |
| Inradius | (a + b − c) ÷ 2 |
| Circumradius | c ÷ 2 |
How to Find the Missing Side of a Right Triangle
Legs, hypotenuse, and when c must be longest
If you know both legs, the hypotenuse is always √(a² + b²). If you know one leg and the hypotenuse, subtract the square of the leg from the square of c and take the square root for the other leg. The hypotenuse must be longer than either leg; if your tape says otherwise, remeasure or check which side is actually c.
How to Find the Angles of a Right Triangle
SOH CAH TOA and arctangent
Ratio | Definition | With our labels (α at bottom-left) | Use when |
|---|---|---|---|
| sin(α) | opposite ÷ hypotenuse | a ÷ c | Find a from c and α |
| cos(α) | adjacent ÷ hypotenuse | b ÷ c | Find b from c and α |
| tan(α) | opposite ÷ adjacent | a ÷ b | Find a from b and α |
From two sides alone, α = atan(a ÷ b). That is the same as sin⁻¹(a ÷ c) or cos⁻¹(b ÷ c) when all three sides are known. On paper, label rise (vertical) and run (horizontal) on separate axes so you do not swap legs when you type values. Results show both acute angles and keep the right angle fixed at 90°.
Pythagorean Triples and Special Right Triangles
3-4-5 squaring, 30-60-90, and 45-45-90
Triple | ×2 | ×3 | Notes |
|---|---|---|---|
| 3 - 4 - 5 | 6 - 8 - 10 | 9 - 12 - 15 | Classic framing square |
| 5 - 12 - 13 | 10 - 24 - 26 | 15 - 36 - 39 | Larger decks and layouts |
| 8 - 15 - 17 | 16 - 30 - 34 | 24 - 45 - 51 | Longer diagonals |
| 7 - 24 - 25 | 14 - 48 - 50 | 21 - 72 - 75 | Wide slabs |
| 20 - 21 - 29 | 40 - 42 - 58 | 60 - 63 - 87 | Near-square rectangles |
Triangle | Side ratios | Angle tie-in | Example |
|---|---|---|---|
| 30° – 60° – 90° | 1 : √3 : 2 | a : b : c if α = 30° | 5 – 8.66 – 10 (scaled) |
| 45° – 45° – 90° | 1 : 1 : √2 | Isosceles right triangle | 7 – 7 – 9.90 (scaled) |
Right Triangles on the Job Site
Worked examples with real numbers
Squaring a 12 ft × 16 ft slab frame: expect diagonal √(12² + 16²) = √(144 + 256) = 20 ft. Measure the diagonal; if you are within your tolerance, the corner is square. A 3-4-5 check at 9 ft, 12 ft, and 15 ft on the same layout tells the same story on a smaller triangle.
Roof rafter: 14 ft horizontal run and 7 ft rise give c = √(14² + 7²) = √245 ≈ 15.65 ft rafter length (before overhang). Pitch is 6:12. Stair stringer: 9 ft total rise and 11 ft total run → c ≈ 14.21 ft along the stringer board. Pair with our decking calculator when you move from layout to material takeoff.
Ladder safety: wall height 20 ft and base 5 ft from the wall → ladder length √(20² + 5²) ≈ 20.62 ft. The 4:1 rule wants at least 5 ft base for 20 ft climb; you are right on the line. Ramp: 24 in rise and 20 ft run → slope about 10%, steeper than typical 1:12 accessible ramps; ramp length √(2² + 20²) ≈ 20.10 ft along the surface.
A 55 in 16:9 TV is a relatable check: screen width ≈ 47.9 in and height ≈ 27.0 in, so diagonal √(47.9² + 27.0²) ≈ 55 in, matching the size on the box. School classic: legs 5 and 12 → c = 13, α ≈ 22.62°, area = 30 square units. Area 50 with leg 10 → other leg 10, a 45-45-90 triangle.
Right Triangle Reference Tables
Slope, pitch, and unit conversions
Angle | Slope % | Ratio 1:X | Roof pitch | Notes |
|---|---|---|---|---|
| 5° | 8.75% | 1:11.4 | 1.05:12 | Very low roof |
| 9.5° | 16.7% | 1:6 | 2:12 | Minimum shingles (check manufacturer) |
| 18.4° | 33.3% | 1:3 | 4:12 | Moderate pitch |
| 26.6° | 50% | 1:2 | 6:12 | Common residential |
| 45° | 100% | 1:1 | 12:12 | Steep / 45-45-90 |
| 60° | 173% | 1:0.58 | 20.8:12 | Very steep (rare roofs) |
Unit | To meters | Example |
|---|---|---|
| Millimeters | ÷ 1,000 | 1,000 mm = 1 m |
| Centimeters | ÷ 100 | 250 cm = 2.5 m |
| Meters | × 1 | 10 m = 10 m |
| Inches | ÷ 39.3701 | 39.37 in ≈ 1 m |
| Feet | ÷ 3.28084 | 10 ft ≈ 3.05 m |
| Yards | ÷ 1.09361 | 1 yd ≈ 0.91 m |
Frequently Asked Questions
Clear answers on sides, angles, area, and job-site layout
How do I find the missing side of a right triangle?
Use the Pythagorean theorem: c = √(a² + b²) for the hypotenuse, or a = √(c² − b²) for a leg. Enter any two sides you know in our two-sides mode and the calculator solves the third instantly, with unit conversion and a diagram you can drag to sanity-check the shape.
How do I find the angles of a right triangle with two sides?
With legs a and b, the acute angle α at the bottom-left is atan(a ÷ b) in degrees. The other acute angle β is 90° minus α. Enter both legs here and you get α, β, slope percent, and roof pitch without hunting for an arctangent button on a basic calculator.
Can you solve a right triangle with only one side?
No. One side length does not fix the shape; the triangle could be tall and skinny or short and wide. You need a second fact: another side, an acute angle, the area, or a measured diagonal for a square-corner check. The tool explains that in plain English if you stop at one entry.
What is the formula for the area of a right triangle?
Area = (leg a × leg b) ÷ 2 because the legs are perpendicular. If you only know area and one leg, the other leg is 2 × area ÷ that leg. Area plus hypotenuse mode solves both legs when the area is not larger than c² ÷ 4.
How do I find the hypotenuse of a right triangle?
Square each leg, add them, and take the square root: c = √(a² + b²). Example: 3 and 4 give 9 + 16 = 25, so c = 5. Enter any two sides and the calculator returns c with step-by-step substitution using your real numbers.
What is a 3-4-5 triangle and how is it used to square a corner?
Sides in a 3:4:5 ratio form a right angle. Carpenters measure 3 ft along one edge, 4 ft along the other, and 5 ft on the diagonal; if the diagonal matches, the corner is square. Use multiples (6-8-10) for larger slabs. Our square-corner mode compares your measured diagonal to the math.
What are the side ratios of a 30-60-90 triangle?
Short leg : long leg : hypotenuse = 1 : √3 : 2. If the short leg is 5 units, the long leg is about 8.66 and the hypotenuse is 10. The calculator flags 30-60-90 and 45-45-90 shapes when your angles and sides match within a small tolerance.
What is the altitude of a right triangle and how do I calculate it?
The altitude h to the hypotenuse is the perpendicular height from the right-angle vertex down to side c. Formula: h = (a × b) ÷ c. It splits the hypotenuse into segments p and q where p = a² ÷ c and q = b² ÷ c. Toggle h on the live diagram to see it drawn.
How do I find the length of a roof rafter or stair stringer?
Treat total rise as leg a and total run as leg b. Rafter or stringer length is c = √(rise² + run²). For a 6:12 pitch roof with 12 ft run, rise is 6 ft and rafter length is √(6² + 12²) ≈ 13.42 ft. Switch context to roof or stair for labeled fields and hints.
Can a right triangle have two equal sides?
Yes. A 45-45-90 triangle has two equal legs and hypotenuse = leg × √2. It is the only right triangle with two equal sides because the other acute angle must be 45°. Enter equal legs here and you will see the 45-45-90 note in results.
How do I use sine, cosine, and tangent to solve a right triangle?
Label the sides opposite, adjacent, and hypotenuse relative to angle α. Then sin(α) = opposite ÷ hypotenuse, cos(α) = adjacent ÷ hypotenuse, and tan(α) = opposite ÷ adjacent. Side-plus-angle mode applies these automatically, for example a = c × sin(α) when you know the hypotenuse and α.
Related Calculators & Tools
More free measurement and quoting tools
- Slope CalculatorPercent grade and rise over run when you already know the triangle legs.
- Roof Pitch Calculatorx:12 pitch presets and degree conversions for rafter layout.
- Square Footage CalculatorArea of rooms and slabs before you check diagonals.
- Concrete CalculatorCubic yards for slabs once layout is square.
- Decking CalculatorBoard counts after stringer and frame math checks out.
- Free Estimate MakerSend a construction or landscaping quote from your phone.
- Invoice GeneratorBill the job when the work is done.
Invoice Mama helps contractors and trades turn layout math into professional estimates and invoices without rebuilding the numbers in a spreadsheet.