Mathematics 5 min read
Equilateral Triangles: The Perfect Triangle Explained
Equilateral triangle: three equal sides, three 60° angles. Height (√3/2)s, area (√3/4)s², the 30-60-90 link, a table of common sizes and worked problems.
Enter Known Values
Sides
Angles
Results
Reference Guide
Example: 3-4-5 right triangle
Enter any known values above to calculate the rest
Side A
-
Side B
-
Side C
-
Angle A
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Angle B
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Area
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Perimeter
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How to Use This Calculator
Enter any two known values (sides or angles) of your right triangle, then click the Calculate button to compute all the remaining values. The calculator uses the Pythagorean theorem and trigonometric functions to solve for unknown sides and angles. You can also press Enter after typing a value for quick calculation.
Formulas Used:
- Pythagorean Theorem: a² + b² = c²
- Sine: sin(angle) = opposite / hypotenuse
- Cosine: cos(angle) = adjacent / hypotenuse
- Tangent: tan(angle) = opposite / adjacent
- Area: ½ × base × height
- Perimeter: a + b + c
Why Use This Calculator?
Who Benefits
- • Students learning trigonometry and geometry
- • Teachers preparing lessons and examples
- • Engineers & architects doing quick calculations
- • DIYers planning home improvement projects
- • Construction workers measuring cuts and angles
Problems Solved
- • Quick verification of manual calculations
- • Visual understanding of triangle relationships
- • Unit conversion between feet and inches
- • Finding missing sides when you know angles
- • Learning aid with step-by-step formulas
Accuracy & Methodology
Our calculator uses IEEE 754 double-precision floating-point arithmetic and standard trigonometric formulas (Pythagorean theorem, sine, cosine, tangent) to deliver accurate results. All calculations happen directly in your browser for instant feedback. Results are displayed to appropriate precision, and we recommend verification for critical professional applications.
Calculating Right Triangles
a² + b² = c²
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of squares of the other two sides. For more on this, see the Khan Academy right triangle trigonometry course. Enter any two sides in the calculator above to find the third.
Triangle Area
Area = ½ × b × h
The area of a right triangle is half the product of its base and height. The calculator above computes it when you enter two sides.
Finding Angles
sin θ = opp/hyp
cos θ = adj/hyp
tan θ = opp/adj
Trigonometric functions help find angles from sides. Learn more about finding angles in right triangles at Math is Fun. Enter any two values in the calculator above, and it will instantly compute all angles using sine, cosine, and tangent.
Frequently Asked Questions
How do I find the hypotenuse of a right triangle?
Use the Pythagorean theorem: c = √(a² + b²). Enter the two legs (sides a and b) in our calculator, and it will calculate the hypotenuse (side c) automatically.
What is the relationship between the angles in a right triangle?
The three angles in any triangle always sum to 180°. Since a right triangle has one 90° angle, the other two angles must sum to 90°. These are called complementary angles.
How do I calculate the area of a right triangle?
The area of a right triangle is: Area = (1/2) × base × height. For right triangles, the two legs serve as the base and height, making the formula: Area = (1/2) × a × b.
How accurate are the calculations?
Our calculator uses IEEE 754 double-precision floating-point arithmetic, which provides approximately 15-17 significant decimal digits of precision. Results are rounded to a practical number of decimal places for display. While highly accurate for educational and most practical purposes, we recommend verifying calculations for mission-critical professional applications.
Can I use this for professional or construction work?
Yes. This calculator was built by someone with over a decade of experience using trigonometry in the construction field. It's great for quick calculations, verifying measurements, and planning cuts. However, for critical structural work or engineering applications, we recommend double-checking results with professional tools or a licensed professional.
What formulas does the calculator use?
The calculator uses these standard trigonometric formulas:
- Pythagorean Theorem: a² + b² = c² (finding sides)
- Inverse Tangent: angle = arctan(opposite/adjacent)
- Sine/Cosine: Used to find sides when an angle is known
- Area: ½ × base × height
- Perimeter: Sum of all three sides
Do you store my input data?
No. All calculations are performed directly in your web browser using JavaScript. The values you enter never leave your device—nothing is sent to our servers. We use privacy-friendly analytics (Plausible) that doesn't track individual users or collect personal data. Your calculations are completely private.
How do I calculate with feet and inches?
Use the unit selector dropdown in the Results panel. Select "Feet" to see results displayed in feet and fractional inches (e.g., 5' 3-1/2"). When Feet is selected, enter your values in decimal feet (e.g., 5.5 for 5'6"). The calculator will convert results to feet and inches automatically, approximating to the nearest 1/16" for practical use.
Learn More About Trigonometry
View all articlesDeepen your understanding of right triangles and trigonometry with our guides and tutorials:
Tutorial 8 min read
Trigonometry Word Problems: A Step-by-Step Solving Framework
Trigonometry word problems step by step: a five-step method (draw, record, identify, choose the ratio, solve and check) with worked examples and practice.
Applications 7 min read
Why Are Triangles the Strongest Shape? The Engineering Secret Hidden in Geometry
Triangles are the strongest shape in engineering: three fixed sides lock the angles. The geometry, a worked force example, and how trusses use it.
Tutorial 8 min read
How to Calculate Triangle Area: 5 Methods Explained
Triangle area formula: A = ½ × base × height, so a 12 cm base and 8 cm height give 48 cm². Without the height, use ½ab sin C, Heron's formula or coordinates.
Mathematics 8 min read
Law of Sines Explained: When and How to Use It
The Law of Sines, a/sin A = b/sin B = c/sin C, solves non-right triangles in AAS, ASA and SSA cases. When to use it, worked examples, the ambiguous case.
Tutorial 7 min read
How to Check Your Triangle Answers: Verification Techniques That Catch Mistakes
How to check your triangle answers: the 180° angle sum, Pythagorean and triangle inequality tests, Law of Sines ratios, area checks and a worked example.
Understanding Trigonometry
The math behind every structure, screen, and signal
Trigonometry studies the relationships between the angles and sides of triangles. Engineers use those relationships to design bridges, surveyors to measure land they cannot walk across, game developers to move characters across a screen, and electricians to calculate voltage in alternating current circuits.
The word itself comes from the Greek trigonon (triangle) and metron (measure). People have worked with triangle ratios for nearly 4,000 years: a Babylonian clay tablet from around 1800 BCE (Plimpton 322) lists Pythagorean triples, the side lengths of whole-number right triangles. The same ratios are used today to position robot arms and to generate tones in audio synthesis.
This site does those calculations quickly and explains them. Each calculator on MyTrigCalculator.com gives the answer along with the formula behind it, so you can understand the math and check the result. It is used by students working through homework, carpenters squaring up floor joists, and engineers verifying load calculations.
Sine, Cosine & Tangent
SOH — CAH — TOA
sin θ
Opp / Hyp
cos θ
Adj / Hyp
tan θ
Opp / Adj
Every trigonometric calculation ultimately rests on three fundamental functions — sine, cosine, and tangent. These functions define the ratio between specific sides of a right triangle relative to a given angle.
For example, when a surveyor aims a theodolite at the top of a building and reads a 32° angle of elevation from 50 meters away, it's the tangent function that converts that angle into the building's height: height = 50 × tan(32°) = 31.24 meters.
The three functions also have reciprocal counterparts — cosecant (csc), secant (sec), and cotangent (cot). The ratios depend only on the angle. The sine of 30° is always exactly 0.5, regardless of how large or small the triangle is. That is why angles measured on a scale drawing or a software model hold for the full-size structure.
The Unit Circle
The unit circle is a circle with a radius of exactly 1, centered at the origin of a coordinate plane. It extends sine and cosine from the acute angles of a right triangle to any angle in a full 360° turn, which is what lets them describe repeating waves such as sound and alternating current.
Place a right triangle inside the unit circle so that the hypotenuse is the radius (length 1). In this configuration, the cosine of the angle equals the x-coordinate and the sine equals the y-coordinate. As the angle rotates from 0° to 360°, these coordinates trace out the familiar sine and cosine wave patterns.
This is why trigonometry appears in fields that seem to have nothing to do with triangles. Alternating current follows a sine wave. Sound waves are combinations of sine waves. Computer graphics engines use unit circle math to rotate objects and simulate camera movement.
The most commonly referenced angles are 0°, 30°, 45°, 60°, and 90°. Learn these reference angles and you can solve most standard trigonometry problems without a calculator.
Real-World Applications
From construction sites to outer space
Construction & Architecture
Every roof rafter and staircase is a right triangle problem. Roof pitch, stair stringers, and the 3-4-5 method for squaring corners all rely on trig.
Engineering & Physics
Force vectors decompose into components using sine and cosine. A single cable at 40° from horizontal supporting 500 lbs needs 500 ÷ sin(40°) = 777.9 lbs of tension.
Navigation & GPS
GPS relies on trilateration: the receiver measures its distance to several satellites and solves for the one position that fits all of them. Navigation still uses bearings rooted in trig.
Games & Computer Graphics
3D engines perform millions of trig operations per second for line-of-sight, camera rotation, and ray tracing. Trig brings video games to life.
Astronomy & Space
Trigonometric parallax has measured the distances to more than a billion stars. NASA uses trig for orbital trajectories and antenna pointing for deep space communication.
DIY & Home Improvement
Ladder safety angles, deck layouts, and fence grades all come down to right triangle math you can solve in seconds.
The Right Triangle
Everything in trigonometry connects back to the right triangle — a triangle containing one 90° angle. The Pythagorean theorem (a² + b² = c²) governs the relationship between its sides.
Knowing any two measurements (two sides, or one side and one angle) is enough to determine everything else — all three sides, all three angles, the area, and the perimeter. This is the principle behind every calculator on this site.
Non-right triangles use extensions of the same principles. The Law of Sines and Law of Cosines together can solve any triangle from three known parts, as long as at least one is a side. Surveying, navigation, and structural analysis rely on both.
Common Trig Formulas
| Formula | Expression | Use case |
|---|---|---|
| Pythagorean | a² + b² = c² | Third side of a right triangle |
| Sine | sin(θ) = opp / hyp | Heights, vertical components |
| Cosine | cos(θ) = adj / hyp | Distances, projections |
| Tangent | tan(θ) = opp / adj | Slopes, elevation angles |
| Law of Sines | a/sin A = b/sin B | Non-right triangles (AAS, ASA) |
| Law of Cosines | c² = a² + b² − 2ab cos C | Non-right triangles (SAS, SSS) |
| Area (base-height) | A = ½ × b × h | When height is known |
| Area (SAS) | A = ½ ab sin C | Two sides + included angle |
| Heron's formula | √[s(s−a)(s−b)(s−c)] | All three sides known |
| Deg → Rad | rad = deg × π / 180 | Converting angle units |
Why Accuracy Matters
All calculators on MyTrigCalculator.com perform computations using IEEE 754 double-precision floating-point arithmetic — the same standard used by scientific computing software, engineering applications, and financial systems. This provides approximately 15–17 significant decimal digits of precision, which exceeds the requirements of virtually any practical application.
The calculators also include construction-specific features like feet-and-inches mode with fractional inch precision down to 1/16" — a feature built from real-world experience on job sites where decimal feet are impractical and tape measures read in fractions.
Every calculation runs entirely in your browser. No data is sent to any server, no inputs are stored, and no personal information is collected. The site uses Plausible Analytics — a privacy-friendly, cookieless analytics platform — so there are no tracking cookies to worry about.