A parabola is a type of curve that is defined as the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). It is a conic section, meaning it's formed by the intersection of a plane and a right circular cone.

The shape of a parabola resembles that of a symmetric curve. It can open upwards, downwards, to the left, or to the right. The basic equation of a parabola in a Cartesian coordinate system is typically represented as:

• Vertical axis of symmetry (opens upward or downward): y=ax²+bx+c where aa, bb, and cc are constants and a≠0.  This equation represents a parabola that either opens upwards or downwards.
• Horizontal axis of symmetry (opens to the left or right): x=ay²+by+c.  This equation represents a parabola that opens to the left or right.

The vertex is the lowest or highest point on the parabola depending on its orientation. The focus and directrix play essential roles in defining a parabola: all points on the parabola are equidistant from the focus and the directrix.

Parabolas have numerous applications in mathematics, physics, engineering, and other fields due to their unique properties and appearance. They are commonly seen in projectile motion, optics, satellite dishes, and the design of certain architectural structures.

Parabola Calculators

• Parabola Formula:  This computes the y coordinate of a parabola in the form y = a•x²+b•x+c
• Parabolic Area: This computes the area within a section of a parabola
• Parabolic Area (Concave): This computes the outer area of a section of a parabola.
• Parabolic Arc Length: This computes the length a long a segment of a parabola.
• Paraboloid Volume: This is the volume of a parabola rotated around an axis (i.e. paraboloid)
• Paraboloid Surface Area : This is the surface area of a paraboloid.
• Paraboloid Weight: This is the weight or mass of a paraboloid.
• Ballistic Flight Parabolic Equation: This provides the formula of the parabola that matches a ballistic flight.