Visual Triangle Geometry

Plain English first

A triangle is three points connected by three straight lines. That's all.

What makes triangles powerful is that the three angles are never independent — they are locked together. If you know two angles, the third is forced. If you know two sides and an angle, the whole shape is determined.

The most important fact: the three inside angles of any triangle always add up to exactly 180°, for every triangle that has ever existed.

Standard math notation

Angle sum:
  A + B + C = 180°

Pythagorean theorem (right triangles only):
  a² + b² = c²
  where c is the hypotenuse (side opposite the 90° angle)

Area:
  area = (base × height) / 2

Exterior angle rule:
  exterior angle = sum of the two non-adjacent interior angles

Verbose Python with descriptive names

def compute_third_angle(first_angle_degrees, second_angle_degrees):
    """
    In any triangle, the three angles must add to 180°.
    If you know two, the third is determined.
    """
    third_angle_degrees = 180 - first_angle_degrees - second_angle_degrees
    return third_angle_degrees


def compute_triangle_area(base_length, perpendicular_height):
    """
    The area of a triangle is half the area of the rectangle
    that surrounds it (same base and height).

    IMPORTANT: height must be perpendicular (straight up/down)
    relative to the base — not the slant side length.
    """
    area = (base_length * perpendicular_height) / 2
    return area


def compute_hypotenuse(short_leg_length, long_leg_length):
    """
    In a right triangle (one 90° angle), the hypotenuse is the
    longest side — the one opposite the right angle.
    Pythagorean theorem: a² + b² = c²  →  c = √(a² + b²)
    """
    sum_of_squares = (short_leg_length ** 2) + (long_leg_length ** 2)
    hypotenuse_length = sum_of_squares ** 0.5  # square root without math library
    return hypotenuse_length


def compute_missing_leg(hypotenuse_length, known_leg_length):
    """
    If you know the hypotenuse and one leg, find the other leg.
    Rearranged Pythagorean theorem: b = √(c² - a²)
    """
    missing_leg_squared = (hypotenuse_length ** 2) - (known_leg_length ** 2)
    missing_leg_length  = missing_leg_squared ** 0.5
    return missing_leg_length


# Example: classic 3-4-5 right triangle
short_leg = 3
long_leg  = 4
hypotenuse = compute_hypotenuse(short_leg, long_leg)
print(hypotenuse)  # 5.0

Special right triangles (memorize these)

45-45-90 triangle:
  Two equal legs (x), hypotenuse = x × √2 ≈ x × 1.414
  Comes from cutting a square diagonally.

30-60-90 triangle:
  Short leg = x
  Long leg  = x × √3 ≈ x × 1.732
  Hypotenuse = 2x
  Comes from cutting an equilateral triangle in half.

SQRT2 = 1.4142135623730951  # √2
SQRT3 = 1.7320508075688772  # √3

def sides_of_45_45_90(leg_length):
    """Both legs equal, hypotenuse = leg × √2"""
    return leg_length, leg_length, leg_length * SQRT2

def sides_of_30_60_90(short_leg):
    """Short leg, long leg = short × √3, hypotenuse = short × 2"""
    return short_leg, short_leg * SQRT3, short_leg * 2

Triangle types reference

Common mistakes and exam traps

• Diagrams are not to scale. Use the given numbers, not your eyes.
• Height ≠ slant side. For area, the height must be perpendicular to the base. It can fall outside the triangle if the triangle is obtuse.
• Exterior angle trap: the exterior angle equals the sum of the two remote interior angles, not 180 minus the adjacent angle (though both happen to equal 180 − adjacent angle when you work it out — the point is to think "sum of two remote" directly).
• Isosceles trap: equal sides → equal opposite angles. Not adjacent angles.

See also

• Visual Unit Circle Sine and Cosine Without Math Libraries — trig starts with right triangles
• Visual Tangent Without Math Libraries
• Visual Geometry and Trigonometry — Table of Contents