Hyperbola Calculator

What this tool does

A hyperbola has two branches that never quite reach two straight lines — the asymptotes. Their slope is b ÷ a, and that single ratio decides the whole shape: when a and b are equal the asymptotes cross at right angles and the hyperbola is called rectangular. The eccentricity of a hyperbola is always greater than one, which is what separates it from an ellipse.

Formula

x²/a² − y²/b² = 1 · c = √(a² + b²) · e = c ÷ a · asymptotes y = ± (b ÷ a) x · latus rectum = 2b² ÷ a

Variables

Symbol	Meaning	Unit
`a`	a, the semi-transverse axis	m
`b`	b, the semi-conjugate axis	m
`CC`	c, from the centre to a focus	m
`EC`	Eccentricity	—
`AS`	Slope of the asymptotes	—
`AA`	Angle of the asymptotes	°
`VX`	Vertices at x = ±	m
`FX`	Foci at x = ±	m
`TA`	Transverse axis, the full width	m
`CA`	Conjugate axis, the full height	m
`LR`	Latus rectum, the width at the focus	m
`DR`	Directrix, the line x =	m
`RE`	Is it rectangular (a = b)?	—

Worked example

a, the semi-transverse axis3 m
b, the semi-conjugate axis4 m

c, from the centre to a focus5.000000 m
Eccentricity1.666667
Slope of the asymptotes1.333333
Angle of the asymptotes53.130102 °
Vertices at x = ±3.000000 m
Foci at x = ±5.000000 m
Transverse axis, the full width6.000000 m
Conjugate axis, the full height8.000000 m
Latus rectum, the width at the focus10.666667 m
Directrix, the line x =1.800000 m
Is it rectangular (a = b)?No

Limitations

The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.