The semi-minor axis is one of the defining characteristics of an ellipse, which is a closed, two-dimensional shape that resembles an elongated circle. The semi-minor axis is the shorter of the two axes that determine the size and shape of the ellipse.
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The semi-minor axis is always perpendicular to the major axis and intersects the center of the ellipse.
The length of the semi-minor axis, along with the length of the major axis, determines the overall size and shape of the ellipse.
As the semi-minor axis increases in length, the ellipse becomes more circular, while a shorter semi-minor axis results in a more elongated ellipse.
The semi-minor axis, along with the major axis, can be used to calculate the area of an ellipse using the formula: $\pi \times a \times b$, where $a$ is the length of the major axis and $b$ is the length of the semi-minor axis.
The eccentricity of an ellipse is inversely related to the ratio of the semi-minor axis to the major axis, with a higher eccentricity indicating a more elongated shape.
Review Questions
Explain the relationship between the semi-minor axis and the overall shape of an ellipse.
The length of the semi-minor axis is directly related to the shape of the ellipse. A longer semi-minor axis results in a more circular ellipse, while a shorter semi-minor axis creates a more elongated, or flattened, ellipse. The ratio of the semi-minor axis to the major axis determines the eccentricity of the ellipse, with a higher eccentricity indicating a more elongated shape.
Describe how the semi-minor axis is used to calculate the area of an ellipse.
The semi-minor axis, along with the major axis, can be used to calculate the area of an ellipse using the formula: $\pi \times a \times b$, where $a$ is the length of the major axis and $b$ is the length of the semi-minor axis. This formula takes into account the two defining axes of the ellipse to determine its total area. The semi-minor axis is a crucial component of this calculation, as it directly impacts the overall size and shape of the ellipse.
Analyze the role of the semi-minor axis in the context of the broader properties of an ellipse.
The semi-minor axis is a fundamental characteristic of an ellipse that, along with the major axis, defines the shape and size of the two-dimensional figure. The semi-minor axis is perpendicular to the major axis and intersects the center of the ellipse. The ratio of the semi-minor axis to the major axis determines the eccentricity of the ellipse, which is a measure of how elongated or flattened the shape is. Understanding the semi-minor axis and its relationship to the other properties of an ellipse is crucial for accurately describing, analyzing, and working with this important geometric shape.
A closed, two-dimensional shape that is defined by two axes: the major axis and the minor axis. The semi-minor axis is half the length of the minor axis.