Circles, Ellipses, Parabolas and Hyperbolas
Our page on Polygons covers shapes made with straight lines, also known as ‘plane shapes’. This page explains more about shapes with curves, especially two-dimensional ones.
Two-dimensional curved shapes include circles, ellipses, parabolas, and hyperbolas, as well as arcs, sectors and segments. Three-dimensional curved shapes, including spheres, cylinders and cones, are covered on our page on Three-Dimensional Shapes.
Two-Dimensional Curved Shapes
Probably the most common two-dimensional curved shape is a circle.
In order to work with circles (and other curved shapes) in geometry it is important to understand the key properties of a circle:
- A line straight across the centre of a circle is the diameter.
- Half of the diameter is the radius.
- The line around the edge of a circle is the circumference.
Any point on the circumference of a circle is exactly the same distance away from the centre of the circle as any other point on the circumference.
Introducing π (pi)
π or pi is a Greek letter. In mathematics, it is used to represent a particular constant, which is also an irrational or infinite number (see our page on Special Numbers for more).
π has a value of 3.142 (although as it is infinite, this is an approximation of its exact value).
π is important because it is used to calculate the circumference and the area of a circle.
The circumference of a circle is equal to π x diameter, or 2 × π × radius (abbreviated to 2πr).
The area of a circle is equal to π × radius2. This formula is usually abbreviated to πr2
For more about area, see our page Calculating Area.
Sectors and Segments
Sectors and segments are 'slices' of a circle.
Sectors are shaped like a slice of pizza, with a curved edge and each straight side the same length as the radius of the circle, or pizza, from which it was cut. Pie charts are made up of a number of sectors relating in size to the data they show.
A sector can be any size, however a sector that is half a circle (180°) is called a semicircle, while a quarter circle sector (90°) is called a quadrant.
A segment is the curved part of a sector, the part that is left if you remove the triangle from a sector. Segments are made up of two lines. The arc (a section of the circumference of the circle - see below) and a chord - the straight line joining the two ends of the arc.
A sector is a fraction of a circle and therefore its area is a fraction of the area of the whole circle. To calculate the area of a sector you need to know its central angle, θ and the radius.
The area of a sector can then be calculated by using the following formula:
πr2 × (θ ÷ 360)
A section of the circumference of a circle is called an arc.
To calculate the length of an arc between points A and B, you need to know the angle at the centre between points A and B. θ (theta) is the symbol used to represent this angle subtended by A and B. In our example, we are using degrees for θ, but it is also possible to use radians.
You also need to know the radius (r) of the arc.
As there are 360° in the whole circle, the length of the arc is equal to the central angle (θ) divided by 360, then multiplied by the circumference of the whole circle (2πr).
2πr × (θ ÷ 360)
r = 10cm, θ = 88°, π = 3.14
Arc Length = 2 x 3.14 x 10 x (88 ÷ 360) = 62.8 × 0.24 = 15.07cm.
An ellipse is a curve on a plane (or flat surface) surrounding two focal points. A straight line drawn from one focal point to any point on the curve and then to the other focal point has the same length for every point on the curve.
Ellipses are very important in astronomy and physics, since every planet has an elliptical orbit with the sun as one of the focal points.
A circle is a specific form of ellipse, where the two focal points are in the same place (at the circle's centre). Ellipses may also be described as ‘oval’, but the word ‘oval’ is much less precise in maths, and simply means ‘broadly egg-shaped’.
Properties of an ellipse:
An ellipse has two main axes, and is symmetrical around them.
The longer axis is called the major axis; the shorter axis is the minor axis.
The four points where the axes cross the circumference are called the vertices.
The two focal points (or foci) are both on the major axis, and equal distances away from the centre.
The distance from one focal point to the circumference, and back to the other (the blue dotted line in our diagram) is the same as the length of the major axis.
The length of an ellipse (ie. how elongated a circle it is) is called the eccentricity. The eccentricity is defined as the ratio of the distance between focal points to the length of the ellipse. The linear eccentricity is defined as the distance from the focal point to the centre.
The area of an ellipse is calculated as π (½ x minor axis)(½ x major axis).
Parabolas and Hyperbolas
A parabola is a symmetrical curve, roughly u-shaped. A parabola is a section of a cone, cut parallel to the cone’s side. It has a single axis of symmetry, splitting it in half from top to bottom. The lowest point of the parabola's u-shape is called the vertex.
All parabolas are the same shape, no matter how big they are. Although they are infinite, meaning that the arms will never close up, the arms will eventually become parallel.
A hyperbola is also a u-shaped curve with a single central axis of symmetry. It is another type of conic section. Hyperbolas are cut almost vertically down, almost parallel to the cone’s central axis of symmetry.
Unlike parabolas, hyperbolas can be different shapes, because the angle of the cut can vary widely. Although, like a parabola, a hyperbola is infinite, its arms never become parallel.
The Relationship between Curved Shapes
All the shapes described on this page are known as conic sections, which means that they can be made by cutting across a cone.
This diagram shows the four different shapes described on this page as conic sections:
Skills You Need?
Circles are part of basic geometry, and you really need to know how to calculate basic properties of them.
It is, however, probably unlikely that you would need to do more than be aware of the existence of the other shapes unless you wished to get seriously into engineering, physics, or astronomy.
That said, you may find that you appreciate knowing that the concave curves of a power station cooling tower, or the light from a downward-pointing halogen lamp, are in the shape of a hyperbola.