Height and Distance
Standard Triangle Values Used in Height and Distance
A triangle marked with angles 75° and 15° with sides:
- Vertical side: √3 − 1
- Base: √3 + 1
- Slant side: 2√2
Angle Setup Sketch
Angles shown: 45°, 30°, 60°, 15° with length markings of 1, 2, and √3 − 1. This is a shortcut configuration for height-distance questions.
Combined Triangle Sketch
- Upper slant side: 2√3
- Left side: √3
- Internal angles: 30°, 60°, 30°
- Bottom total horizontal distance: 3
- Internal parts: 1 and 2
Triangle Formulas
General Triangle
A triangle split into three parts p1, p2, p3:
h = p1 + p2 + p3
Isosceles Triangle
Equal sides a, base b split into b/2 and b/2:
Area = (b/4) √(4a² − b²)
Height = √(a² − b²/4) = (1/2) √(4a² − b²)
Right Triangle
Where p = perpendicular, b = base, h = hypotenuse:
r = (p + b − h) / 2
R = h / 2
Parallelogram and Rhombus
Parallelogram Diagonals
d1² + d2² = 2(a² + b²)
Rhombus
4a² = d1² + d2²
side = (1/2) √(d1² + d2²)
Regular Polygon Notes
Area of Regular Hexagon
Area = 6 × (√3/4) × a² = (3√3/2) × a²
Polygon Angle Formulas
- Each exterior angle = 360 / n
- Each interior angle = 180(n − 2) / n
- Sum of interior angles = 180(n − 2)
- For hexagon: exterior angle = 60°
Path Inside a Rectangle
- Path made outside = 2x(L + B + 2x)
- Path made inside = 2x(L + B − 2x)
- Middle coverage = x(L + B − x)