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Major Axis of an Ellipse — Definition, Formula & Examples

Major Axis of an Ellipse

The line passing through the foci, center, and vertices of an ellipse. It is also the principle axis of symmetry.

 

Ellipse with horizontal "major axis" labeled, shown as a line with arrows extending through the center to both ends.

 

 

See also

Minor axis of an ellipse, major diameter of an ellipse

Key Formula

Length of major axis=2a\text{Length of major axis} = 2a
Where:
  • aa = The semi-major axis — the distance from the center of the ellipse to either vertex along the major axis.
  • 2a2a = The full length of the major axis, spanning from one vertex to the other through the center.

Worked Example

Problem: Find the length of the major axis and identify the endpoints (vertices) for the ellipse given by the equation: x225+y29=1\frac{x^2}{25} + \frac{y^2}{9} = 1
Step 1: Identify the standard form. The equation is already in standard form x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. Compare to find a2a^2 and b2b^2.
a2=25,b2=9a^2 = 25, \quad b^2 = 9
Step 2: Determine which denominator is larger. Since 25>925 > 9, the larger value is under x2x^2, so the major axis is horizontal (along the xx-axis).
a2=25  ⟹  a=5a^2 = 25 \implies a = 5
Step 3: Calculate the length of the major axis using the formula 2a2a.
Length=2a=2(5)=10\text{Length} = 2a = 2(5) = 10
Step 4: Identify the vertices. Since the major axis is horizontal and the center is at the origin, the vertices are at (±a,0)(\pm a, 0).
Vertices: (−5,0) and (5,0)\text{Vertices: } (-5, 0) \text{ and } (5, 0)
Answer: The major axis has a length of 10 units, lies along the xx-axis, and has vertices at (−5,0)(-5, 0) and (5,0)(5, 0).

Another Example

This example differs because the ellipse has a shifted center (not at the origin) and its major axis is vertical rather than horizontal, showing that the larger denominator determines the direction of the major axis regardless of which variable it's under.

Problem: Find the length of the major axis and its direction for the ellipse: (x−3)24+(y+1)216=1\frac{(x - 3)^2}{4} + \frac{(y + 1)^2}{16} = 1
Step 1: Identify the center of the ellipse from the shifted standard form (x−h)2A+(y−k)2B=1\frac{(x-h)^2}{A} + \frac{(y-k)^2}{B} = 1.
Center: (h,k)=(3,−1)\text{Center: } (h, k) = (3, -1)
Step 2: Compare the two denominators. Here 44 is under (x−3)2(x-3)^2 and 1616 is under (y+1)2(y+1)^2. Since 16>416 > 4, the larger denominator is associated with yy, so the major axis is vertical.
a2=16  ⟹  a=4a^2 = 16 \implies a = 4
Step 3: Calculate the length of the major axis.
Length=2a=2(4)=8\text{Length} = 2a = 2(4) = 8
Step 4: Find the vertices. Since the major axis is vertical, move aa units up and down from the center.
Vertices: (3,−1+4)=(3,3) and (3,−1−4)=(3,−5)\text{Vertices: } (3, -1 + 4) = (3, 3) \text{ and } (3, -1 - 4) = (3, -5)
Answer: The major axis has a length of 8 units, is vertical, and runs from (3,−5)(3, -5) to (3,3)(3, 3).

Frequently Asked Questions

What is the difference between the major axis and the semi-major axis of an ellipse?
The major axis is the full line segment from one vertex to the other, passing through the center. Its length is 2a2a. The semi-major axis is half of that — just the distance from the center to one vertex, with length aa. Think of 'semi' as meaning 'half.'
How do you tell if the major axis is horizontal or vertical?
In the standard form x2A+y2B=1\frac{x^2}{A} + \frac{y^2}{B} = 1, compare the two denominators. If A>BA > B, the major axis is horizontal (along the xx-axis). If B>AB > A, the major axis is vertical (along the yy-axis). The larger denominator always corresponds to a2a^2.
How are the foci related to the major axis?
Both foci always lie on the major axis, equally spaced from the center. The distance from the center to each focus is cc, where c2=a2−b2c^2 = a^2 - b^2. Since c<ac < a, the foci are always located between the center and the vertices along the major axis.

Major Axis vs. Minor Axis

Major AxisMinor Axis
DefinitionThe longest diameter of the ellipse, passing through both fociThe shortest diameter of the ellipse, perpendicular to the major axis
Length formula2a2a (where aa is the larger value)2b2b (where bb is the smaller value)
Passes through foci?Yes — both foci lie on the major axisNo — the minor axis is perpendicular to the line containing the foci
EndpointsThe two vertices of the ellipseThe two co-vertices of the ellipse
RelationshipAlways a≥ba \geq bAlways b≤ab \leq a; when a=ba = b the shape is a circle

Why It Matters

The major axis is essential when graphing ellipses in algebra and precalculus — it tells you the ellipse's orientation and maximum width. In physics, planetary orbits are ellipses, and the major axis determines the size of the orbit and is directly linked to the orbital period through Kepler's third law. You will also encounter it in engineering applications like satellite dish design and optics, where the reflective properties of an ellipse depend on the positions of the foci along the major axis.

Common Mistakes

Mistake: Confusing aa with a2a^2: students read the denominator in x225\frac{x^2}{25} and call a=25a = 25 instead of a=5a = 5.
Correction: Remember that the denominators in the standard form represent a2a^2 and b2b^2, not aa and bb directly. Always take the square root: if a2=25a^2 = 25, then a=5a = 5, and the major axis length is 2(5)=102(5) = 10.
Mistake: Assuming the major axis is always horizontal (along the xx-axis).
Correction: The major axis is horizontal only when the larger denominator is under x2x^2. If the larger denominator is under y2y^2, the major axis is vertical. Always compare the two denominators before deciding the orientation.

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