Semi-Minor Axis

Definition

The semi-minor axis is one of the defining features of an ellipse, which is a closed, two-dimensional shape with a curved boundary. It represents the length of the shorter of the two principal axes that intersect at the center of the ellipse, perpendicular to the semi-major axis.

5 Must Know Facts For Your Next Test

1. The semi-minor axis is always less than or equal to the semi-major axis, with the ratio between the two defining the eccentricity of the ellipse.
2. The semi-minor axis, along with the semi-major axis, determines the overall shape and proportions of the ellipse.
3. The area of an ellipse is calculated as $\pi$ multiplied by the semi-major axis and the semi-minor axis.
4. The semi-minor axis is one of the key parameters used in the equation of an ellipse, which takes the form $(x/a)^2 + (y/b)^2 = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.
5. The semi-minor axis is an important consideration in applications involving ellipses, such as the design of lenses, the paths of planetary orbits, and the cross-sections of certain architectural structures.

Review Questions

• Explain the relationship between the semi-minor axis and the semi-major axis of an ellipse.
  • The semi-minor axis and the semi-major axis are the two defining axes of an ellipse. The semi-minor axis represents the length of the shorter of the two principal axes, while the semi-major axis represents the length of the longer axis. The ratio between the semi-minor axis and the semi-major axis determines the eccentricity of the ellipse, with a perfect circle having a ratio of 1 and more elongated ellipses having smaller ratios.
• Describe how the semi-minor axis is used in the equation of an ellipse.
  • The equation of an ellipse takes the form $(x/a)^2 + (y/b)^2 = 1$, where $a$ represents the semi-major axis and $b$ represents the semi-minor axis. This equation is used to define the shape and dimensions of the ellipse, with the semi-minor axis being one of the key parameters that determines the overall size and proportions of the shape. The semi-minor axis, along with the semi-major axis, is crucial in calculating the area of an ellipse, which is given by the formula $\pi ab$, where $a$ is the semi-major axis and $b$ is the semi-minor axis.
• Analyze the importance of the semi-minor axis in various applications involving ellipses.
  • The semi-minor axis is a critical parameter in many real-world applications that utilize elliptical shapes. In the design of lenses, the semi-minor axis determines the curvature and focusing properties of the lens. In the study of planetary orbits, the semi-minor axis, along with the semi-major axis, defines the shape and size of the elliptical path of a planet around the Sun. In architecture, the semi-minor axis is an important consideration in the design of certain structures, such as domes and arches, where the elliptical cross-section is an important design element. Understanding the role of the semi-minor axis is essential in these and other applications that involve the use of ellipses.

Related terms

Ellipse:

A closed, two-dimensional shape with a curved boundary that is defined by two principal axes: the major axis and the minor axis.

Semi-Major Axis:

The longer of the two principal axes that intersect at the center of an ellipse, defining the maximum width of the shape.

Eccentricity:

A measure of how much an ellipse deviates from being a perfect circle, calculated as the ratio of the distance between the foci to the length of the major axis.