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Mach Cone Angle Calculator

Enter a supersonic Mach number to find the Mach cone half-angle.

Result

23.58°
0.4115 rad
Supersonic (M>1): cone exists

How it is calculated

The Mach number M is the object's speed divided by the local speed of sound. For a body moving faster than sound, the shock wave front makes a half-angle with the flight direction given by μ = arcsin(1 ÷ M). The tool computes this in radians first, then converts to degrees with degrees = radians × 180 ÷ π. It shows the angle in degrees to two decimals, in radians to four decimals, and a note that the cone exists. Only M > 1 is accepted: at exactly 1 or below the arcsine formula has no cone to describe (the argument 1/M would reach or exceed 1), so the tool returns no result. The angle is measured between the cone's surface and the axis of motion, so the full opening is twice the value.

Worked example

Take a Mach number of 2.5. First, 1 ÷ 2.5 = 0.4. The arcsine of 0.4 is 0.4115 radians. In degrees, 0.4115 × 180 ÷ π = 23.58°. So the cone half-angle is 23.58°, the radians output is 0.4115 rad, and the whole cone opens to about 47.2°. The tool reports 23.58° with the note that the flow is supersonic and a cone exists.

Try this example →

Common mistakes

  • Entering speed in km/h or m/s instead of the Mach number. The tool expects a ratio with no units, so divide your speed by the local speed of sound first.
  • Entering a subsonic or Mach 1 value and expecting an angle. Below Mach 1 there is no cone, and the tool gives no result at exactly 1.
  • Taking the displayed angle as the full opening of the cone. It is the half-angle, so double it for the total width.
  • Using the sea-level speed of sound at altitude. Mach is relative to the local sound speed, which falls with colder air, so the same ground speed gives a different Mach number.
  • Expecting the angle to grow with speed. It shrinks as Mach rises: 2.5 gives 23.58° but 3 gives 19.47°.

About this calculator

When an object moves faster than the speed of sound, it outruns the pressure waves it generates, which pile up into a cone-shaped shockwave trailing behind it — the Mach cone, named after physicist Ernst Mach. The half-angle of that cone, called the Mach angle (μ), is given by μ = arcsin(1/M), where M is the object's Mach number (speed divided by the local speed of sound).

The cone exists only for supersonic speeds, so enter a Mach number greater than 1; at Mach 1 or below there is no cone and the calculator gives no result. Just above Mach 1 the angle approaches 90° (the cone flattens toward a plane wavefront), and it narrows as speed increases — at Mach 2 it's about 30°, at Mach 3 about 19.5°. Aerospace engineers use this relationship to predict shockwave geometry around supersonic aircraft and missiles, and it is also what determines the geometry of the audible sonic boom that sweeps across the ground beneath a supersonic flight path.

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