Understanding Hypotenuse, Adjacent, and Opposite Sides

Important: Adjacent and opposite are sides of a right triangle, defined relative to a specific acute angle. The hypotenuse is fixed.

1. The Hypotenuse

Definition

The longest side of a right triangle.

Location

It is always the side opposite the right angle (90° angle).

Key Fact

It never changes for a given triangle and is always the hypotenuse, no matter which acute angle you're using as your reference.

2. Adjacent Side (Relative to a chosen angle)

Definition

The leg that forms the chosen acute angle, along with the hypotenuse.

Memory Aid

The side touching or next to the angle (other than the hypotenuse).

3. Opposite Side (Relative to a chosen angle)

Definition

The leg that is across from the chosen acute angle. It does not form the angle.

Memory Aid

The side facing the angle.

Visual Explanation

View from Angle θ (Theta)

θ

Hypotenuse

Adjacent (to θ)

Opposite (to θ)

• Hypotenuse: The slanted side (always)
• Adjacent: The bottom horizontal leg (touching θ)
• Opposite: The vertical leg (across from θ)

View from Angle α (Alpha)

α

Hypotenuse

Adjacent (to α)

Opposite (to α)

• Hypotenuse: The same slanted side (unchanged)
• Adjacent: The vertical leg (now touching α)
• Opposite: The bottom horizontal leg (now across from α)

Notice: The Opposite side for θ is the Adjacent side for α, and vice-versa. The sides swap roles when you change reference angles!

Connection to Trigonometry

This naming convention is the foundation of the three primary trigonometric ratios:

Function	Ratio	Explanation
Sine (sin)	Opposite / Hypotenuse	Compares the side opposite the angle to the hypotenuse
Cosine (cos)	Adjacent / Hypotenuse	Compares the side adjacent to the angle to the hypotenuse
Tangent (tan)	Opposite / Adjacent	Compares the side opposite to the side adjacent to the angle

Example: In the first triangle above, for angle θ:

• sin θ = (Opposite to θ) / Hypotenuse
• cos θ = (Adjacent to θ) / Hypotenuse
• tan θ = (Opposite to θ) / (Adjacent to θ)

SOH-CAH-TOA
(The classic mnemonic for remembering trigonometric ratios)

SOH
Sine = Opposite / Hypotenuse

CAH
Cosine = Adjacent / Hypotenuse

TOA
Tangent = Opposite / Adjacent

Key Takeaway

Always ask: "Which acute angle am I using as my reference point?" Once you pick the angle:

• Hypotenuse is fixed (opposite the right angle).
• Opposite is the side directly across from your chosen angle.
• Adjacent is the side next to your angle that isn't the hypotenuse.

This HTML page visually explains the concepts of hypotenuse, adjacent, and opposite sides in right triangles.