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Chapter 30: Problem 822

Find the area of an ellipse with semi-axes a and b.

Short Answer

Expert verified
The area A of an ellipse with semi-axes a and b is calculated using the formula: \(A = \pi * a * b\). Given the lengths of semi-axes a and b, simply plug the values into the formula and compute the product of \(\pi\), a, and b to find the area of the ellipse.

Step by step solution

01

Identify the semi-axes a and b

Given an ellipse with known lengths of semi-axes a and b. In general, an ellipse is represented by an equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Where (x, y) are the coordinates of any point on the ellipse, and a and b are the semi-major and semi-minor axes, respectively. A semi-major axis (a) is the longest distance from the center to the edge of the ellipse, while a semi-minor axis (b) is the shortest distance from the center to the edge of the ellipse.
02

Use the formula for the area of an ellipse

The formula for the area A of an ellipse with semi-axes a and b is: A = 蟺 * a * b
03

Calculate the area A

Plug the given values of a and b into the formula and compute the area A: A = 蟺 * a * b By computing the product of 蟺, a, and b, you will find the area of the ellipse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Semi-major Axis
When discussing an ellipse, the semi-major axis is a critical part of its geometric structure. It is the longest radius that stretches from the center of the ellipse to its perimeter. If you imagine an ellipse akin to a flattened circle, the semi-major axis represents its largest spread.

The semi-major axis is denoted by the letter "a." This axis is pivotal in defining the shape and size of the ellipse. Its length dictates how elongated or circular the ellipse appears. A larger semi-major axis compared to the semi-minor axis indicates more elongation.
  • "a" is always positive since it represents length.
  • It lies along the major diameter, which is twice the length of the semi-major axis.
  • The endpoints of this axis are the farthest points from the ellipse's center when compared to any other point on the curve.
Understanding the semi-major axis helps in comprehending how an ellipse maintains its form and dimensions.
Semi-minor Axis
The semi-minor axis is the shorter radius extending from the center of the ellipse to its boundary. It plays a vital role in balancing the dimensions of the ellipse, creating a unique shape distinct from a circle.

Represented by "b," the semi-minor axis influences the "roundness" of the ellipse. It determines how much the ellipse is squeezed compared to a perfect circle.
  • "b" is always less than or equal to "a" in the context of an ellipse.
  • It lies along the minor diameter, which is twice the length of the semi-minor axis.
  • The semi-minor axis extends perpendicular to the semi-major axis.
This axis is essential in shaping the internal structure and symmetry of the ellipse.
Ellipse Equation
The equation of an ellipse succinctly encapsulates the relationship between its semi-major and semi-minor axes. The standard form of the equation is:

\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]

Here, "a" is the semi-major axis, and "b" is the semi-minor axis. This equation shows that every point \( (x, y) \) lying on the ellipse satisfies this specific relationship.
  • The denominator \( a^2 \) corresponds to the semi-major axis.
  • The denominator \( b^2 \) corresponds to the semi-minor axis.
  • The equation is derived based on the ellipse's geometrical properties and symmetry.
Understanding this equation helps in calculating not only the position of points on the ellipse but also in deriving other properties such as area. This symmetry and balance exhibited by the equation underpin the fundamental nature of an ellipse across mathematics.

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Most popular questions from this chapter

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