key term - Semi-Minor Axis

Definition

The semi-minor axis is one of the two principal axes of an ellipse, the other being the semi-major axis. It represents the shorter of the two axes and is essential in defining the shape and size of an ellipse.

5 Must Know Facts For Your Next Test

1. The semi-minor axis of an ellipse is always less than or equal to the semi-major axis, with the two being equal in the case of a circle.
2. The semi-minor axis, along with the semi-major axis, determines the area of an ellipse, which is calculated as $\pi$ times the product of the two axes.
3. The ratio of the semi-minor axis to the semi-major axis is known as the aspect ratio, which is a key parameter in the visual representation of an ellipse.
4. The semi-minor axis is a critical factor in the calculation of the eccentricity of an ellipse, which in turn affects the shape and properties of the curve.
5. The semi-minor axis, together with the semi-major axis, defines the foci of the ellipse, which are the two points that determine the shape of the curve.

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 principal axes that define the size and shape of an ellipse. The semi-minor axis represents the shorter of the two axes, while the semi-major axis represents the longer axis. The ratio of the semi-minor axis to the semi-major axis is known as the aspect ratio, and it determines the eccentricity of the ellipse. In the case of a circle, the semi-minor and semi-major axes are equal, resulting in an eccentricity of 0.
• Describe how the semi-minor axis is used to calculate the area of an ellipse.
  • The area of an ellipse is calculated as $\pi$ times the product of the semi-major axis and the semi-minor axis. The semi-minor axis is a crucial component of this formula, as it directly affects the size and shape of the ellipse. A larger semi-minor axis will result in a larger area, while a smaller semi-minor axis will lead to a smaller area. The semi-minor axis, along with the semi-major axis, provides a complete description of the ellipse's dimensions and allows for the accurate calculation of its area.
• Analyze the role of the semi-minor axis in determining the eccentricity and focal points of an ellipse.
  • The semi-minor axis, in combination with the semi-major axis, plays a significant role in determining the eccentricity and focal points of an ellipse. Eccentricity is a measure of how much an ellipse deviates from a perfect circle, and it is calculated as the ratio of the semi-minor axis to the semi-major axis. The closer the eccentricity is to 0, the more circular the ellipse becomes. Additionally, the semi-minor axis, along with the semi-major axis, defines the location of the focal points within the ellipse. These focal points are essential in understanding the properties and behavior of the elliptical curve.

Related terms

Semi-Major Axis:

The semi-major axis is the longer of the two principal axes of an ellipse, and it represents the maximum distance from the center to the edge of the ellipse.

Eccentricity:

Eccentricity is a measure of how much an ellipse deviates from a perfect circle, with a value between 0 and 1. It is calculated as the ratio of the semi-minor axis to the semi-major axis.

Focal Points: The focal points of an ellipse are two points within the ellipse that define its shape. The distance between the focal points and the center of the ellipse is determined by the semi-major and semi-minor axes.