Right Triangle Trigonometry Calculator

Calculate sides, angles, area, perimeter, and trig ratios of right triangles using SOH-CAH-TOA and Pythagorean theorem.

Angle Mode: Degrees
Units:
Two Sides
Side & Angle
Angle & Side (Hyp)
Area & Side
cm
Leg opposite to angle α
cm
Leg opposite to angle β
Quick Examples
3-4-5 Triangle
5-12-13 Triangle
7-24-25 Triangle
8-15-17 Triangle
45°-45°-90°
30°-60°-90°
SOH-CAH-TOA - Trigonometric Ratios
SOH
Sine = Opposite / Hypotenuse
CAH
Cosine = Adjacent / Hypotenuse
TOA
Tangent = Opposite / Adjacent
Pythagoras
a² + b² = c²

Right Triangle Trigonometry

Right triangle trigonometry deals with the relationships between the angles and sides of right triangles. The primary trigonometric functions are sine, cosine, and tangent, often remembered by the acronym SOH-CAH-TOA.

Right Triangle Properties:

  • One angle is exactly 90° (right angle)
  • The side opposite the right angle is called the hypotenuse (longest side)
  • The other two sides are called legs
  • The two acute angles are complementary (sum to 90°)

SOHCAHTOA Mnemonic:

SOH: Sin(θ) = Opposite / Hypotenuse

CAH: Cos(θ) = Adjacent / Hypotenuse

TOA: Tan(θ) = Opposite / Adjacent

Trigonometric Ratios Table

Angle (θ) sin(θ) cos(θ) tan(θ)
0° 0 1 0
30° 1/2 = 0.5 √3/2 ≈ 0.866 √3/3 ≈ 0.577
45° √2/2 ≈ 0.707 √2/2 ≈ 0.707 1
60° √3/2 ≈ 0.866 1/2 = 0.5 √3 ≈ 1.732
90° 1 0 ∞ (undefined)

Key Formulas

1

Pythagorean Theorem: a² + b² = c²

c = √(a² + b²), a = √(c² - b²), b = √(c² - a²)

2

Trigonometric Functions:

sin(A) = a/c, cos(A) = b/c, tan(A) = a/b

sin(B) = b/c, cos(B) = a/c, tan(B) = b/a

3

Angle Sum: A + B + C = 180°

Since C = 90°, then A + B = 90° (complementary angles)

4

Area: Area = ½ × a × b

Perimeter: Perimeter = a + b + c

Applications of Right Triangle Trigonometry

  • Architecture & Construction: Calculating roof slopes, stair angles, and structural loads
  • Surveying & Navigation: Determining distances and heights using angles
  • Physics & Engineering: Analyzing forces, vectors, and projectile motion
  • Astronomy: Measuring distances to stars and planets
  • Computer Graphics: Calculating angles and perspectives in 3D rendering

Calculator Features:

  • Three input modes: Two sides, Side & angle, or Any three values
  • Calculates all missing sides and angles automatically
  • Visualizes the triangle with labeled sides and angles
  • Shows calculation steps and formulas used
  • Calculates area and perimeter

Frequently Asked Questions

SOHCAHTOA is a mnemonic device to remember the three basic trigonometric ratios:
  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent
These ratios relate the angles of a right triangle to the lengths of its sides.

The hypotenuse is always:
  • The side opposite the right angle (90° angle)
  • The longest side of a right triangle
  • The side that does not form the right angle
In standard right triangle notation, side c is typically the hypotenuse, with sides a and b forming the right angle.

This calculator handles decimal values for sides and angles. You can enter values like 3.5, 12.75, or 45.5°. For fractional values, you can enter them as decimals (e.g., 1/2 = 0.5, 3/4 = 0.75). The calculator will provide results with decimal precision.

This calculator uses degrees for angle measurements, which is the most common unit for right triangle problems. If you have angles in radians, convert them to degrees first (1 radian ≈ 57.2958°). Most scientific calculators and programming languages have functions to convert between degrees and radians.

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the Pythagorean theorem a² + b² = c². Common examples include:
  • 3-4-5 (and its multiples like 6-8-10, 9-12-15)
  • 5-12-13
  • 8-15-17
  • 7-24-25
These triples are useful in construction and carpentry for creating right angles.

Trigonometric Ratios

Function Abbreviation Ratio Definition (for angle θ)
Sine sin Opposite/Hypotenuse sin(θ) = a/c
Cosine cos Adjacent/Hypotenuse cos(θ) = b/c
Tangent tan Opposite/Adjacent tan(θ) = a/b
Cosecant csc Hypotenuse/Opposite csc(θ) = c/a
Secant sec Hypotenuse/Adjacent sec(θ) = c/b
Cotangent cot Adjacent/Opposite cot(θ) = b/a

Common Pythagorean Triples

3-4-5 Triangle
3² + 4² = 5²

9 + 16 = 25

Angles: ≈36.87°, ≈53.13°, 90°

5-12-13 Triangle
5² + 12² = 13²

25 + 144 = 169

Angles: ≈22.62°, ≈67.38°, 90°

7-24-25 Triangle
7² + 24² = 25²

49 + 576 = 625

Angles: ≈16.26°, ≈73.74°, 90°

8-15-17 Triangle
8² + 15² = 17²

64 + 225 = 289

Angles: ≈28.07°, ≈61.93°, 90°