Ellipse

Properties

• It has two lines of symmetry, along its two axes.
• When a equals b, the ellipse is a circle.
• It has two special points called foci. For every point on the curve, the two distances to the foci add up to the same total, 2a.
• The planets move round the Sun in ellipses.

Why the formulas work

An ellipse is a circle of radius b stretched sideways by the factor a/b. Stretching in one direction multiplies every area by the same factor, so the area is πb² × a/b = πab. The perimeter is harder. It has no simple exact formula, so a very accurate approximation found by the Indian mathematician Srinivasa Ramanujan is used.

Worked example

Take a ellipse with semi-major axis a 10 cm and semi-minor axis b 6 cm.

1. Area. Area = πab. Here: π × 10 × 6 = 188.5 cm² (with π = 22/7: 188.57 cm²).
2. Perimeter (approx.). Perimeter ≈ π [3(a + b) − √((3a + b)(a + 3b))]. Here: π × [3 × (10 + 6) − √((3 × 10 + 6) × (10 + 3 × 6))] = 51.05 cm (with π = 22/7: 51.07 cm).

Finding a missing measurement

From the area and one semi-axis, the other is A ÷ (π × the known one). An area of 132 cm² with a = 7 cm gives b = 6 cm, taking π = 22/7.

Mistakes to avoid

• Using the full axes in place of the semi-axes. Halve them first.
• Expecting an exact perimeter. Every simple perimeter formula for an ellipse is an approximation.

Where you see it

Running tracks seen from above (roughly), the outline of an egg (roughly), a circle seen at a slant such as the top of a glass, and the orbits of planets are ellipses.

Try these

Work each one out, then tap to check.

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