Calculate all properties of right triangles including sides, angles, area, and perimeter. Enter any two values to find all remaining triangle measurements with step-by-step solutions and visual representation.
A right triangle is a special type of triangle that contains one 90-degree angle. This unique property creates specific relationships between the sides and angles that can be used to calculate all triangle properties when only two values are known. Right triangles are fundamental to trigonometry, geometry, and many practical applications.
Right triangle calculations are essential in construction, navigation, engineering, architecture, and physics. Understanding these relationships helps solve real-world problems involving slopes, distances, heights, and angular measurements in various professional and academic contexts.
Fundamental relationship between the three sides
Relationships between angles and sides
Geometric properties of the triangle
| Triangle Type | Angles | Side Ratios | Common Uses |
|---|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 | Isosceles right triangles, squares |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | Equilateral triangle halves |
| 3-4-5 | 36.87°, 53.13°, 90° | 3 : 4 : 5 | Construction, carpentry |
| 5-12-13 | 22.62°, 67.38°, 90° | 5 : 12 : 13 | Surveying, large construction |
Sine (sin): Ratio of the opposite side to the hypotenuse. sin(α) = opposite/hypotenuse = a/c. Used to find sides when an angle and hypotenuse are known.
Cosine (cos): Ratio of the adjacent side to the hypotenuse. cos(α) = adjacent/hypotenuse = b/c. Used to find the side next to a known angle.
Tangent (tan): Ratio of the opposite side to the adjacent side. tan(α) = opposite/adjacent = a/b. Used when working with two legs of the triangle.
Inverse Functions: Use arcsin, arccos, and arctan to find angles when sides are known. For example, α = arcsin(a/c) finds the angle when opposite side and hypotenuse are known.
Two Sides Known: Use the Pythagorean theorem to find the third side, then use trigonometric ratios to find the angles. This is the most straightforward approach.
One Side and One Angle: Use trigonometric functions to find the other sides. For example, if you know angle α and side c, then a = c×sin(α) and b = c×cos(α).
Two Angles: Since the angles in a triangle sum to 180°, knowing two angles gives you the third. You need at least one side length to determine the triangle's size.
Area and One Side: If you know the area and one side, you can find other measurements using area formulas and geometric relationships.
Construction and Architecture: Calculating roof slopes, ramp inclinations, stair dimensions, and ensuring structures are level and square. The 3-4-5 rule is commonly used to verify right angles.
Navigation and Surveying: Determining distances and directions, calculating elevations, and mapping terrain. GPS systems use triangulation principles based on right triangle mathematics.
Engineering and Physics: Analyzing forces in structures, calculating component vectors, designing mechanical systems, and solving problems involving motion and energy.
Computer Graphics and Gaming: Calculating distances, rotations, and transformations in 2D and 3D space. Right triangle principles are fundamental to computer graphics algorithms.
Draw a Diagram: Always sketch the triangle and label known values. This visual approach helps identify which formulas to use and prevents calculation errors.
Identify the Right Angle: Clearly mark the 90-degree angle and identify the hypotenuse. This orientation is crucial for applying trigonometric functions correctly.
Choose Appropriate Functions: Select trigonometric functions based on which sides and angles are known. Use SOH-CAH-TOA (Sine-Opposite-Hypotenuse, Cosine-Adjacent-Hypotenuse, Tangent-Opposite-Adjacent) as a memory aid.
Check Your Work: Verify results using the Pythagorean theorem and ensure that angles sum to 180 degrees. Check that the hypotenuse is the longest side.
Confusing Opposite and Adjacent: Always identify sides relative to the angle you're working with. The opposite side is across from the angle, adjacent is next to it.
Degree vs. Radian Mode: Ensure your calculator is in the correct angle mode. Most practical problems use degrees, but some mathematical contexts require radians.
Assuming Wrong Triangle Type: Verify that you actually have a right triangle before applying these formulas. For non-right triangles, use the law of sines or cosines.
Rounding Too Early: Keep extra decimal places during intermediate calculations and round only the final answer to avoid cumulative rounding errors.
Unit Circle Relationships: Right triangles inscribed in the unit circle help understand how trigonometric functions relate to coordinates and angles beyond 90 degrees.
Similar Triangles: Right triangles with the same angles are similar, meaning their sides are proportional. This principle enables scaling and proportion calculations.
Vector Components: Right triangles help decompose vectors into horizontal and vertical components, essential for physics and engineering calculations.