If the plane cuts the cone at an angle between these two, such that it maintains contact with the sides of the cone in all locations, then an ellipse is formed (bottom left). If the plane cuts the cone parallel to the side of the cone, then a parabola is formed (centre).
perpendicular to its vertical axis), then a circle is formed (top left). If the plane cuts the cone at an angle parallel to the base of the cone (i.e. The diagram below shows a double cone, rather like a sand-timer. The characteristics of conic sections have been studied for millennia and were a subject of interest for ancient Greek mathematicians such as Euclid and Archimedes. shapes that are formed by slicing through a cone with a flat plane. They are closely related to each other and to circles and ellipses, because they are all conic sections, i.e. Parabolas and Hyperbolas are more forms of curved shapes, but they are more complicated to define than circles and ellipses. Parabolas, Hyperbolas and the Relationship Between Curved Shapes The area of an ellipse is calculated as π (½ x minor axis)(½ x major axis). When the distance from the centre to the focal point is the same as the distance from the centre to the vertex, then the ellipse has become a straight line and its eccentricity is equal to 1. The eccentricity increases as the ellipse becomes longer, but is always less than 1. The distance from the centre to the focal point is therefore zero. The eccentricity of a circle is zero, because the focal points are in exactly the same place (the centre) (we also say that they are coincident). The formula for calculating the eccentricity is: Eccentricity =ĭistance from centre to vertex on the major axis The extent to which an ellipse is elongated is defined by its eccentricity. The distance from one focal point to any point on the circumference, and back to the other focal point (the blue dotted line in our diagram) is the same as the length between the vertices on the major axis. The two focal points (or foci, sometimes called locus or loci) are both on the major axis, and equal distances away from the centre. The two points where the minor axis crosses the circumference are called the co-vertices.
The four points where the axes cross the circumference are called the vertices (singular vertex). The longer axis is called the major axis the shorter axis is the minor axis. 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.Īn ellipse has two main axes, and is symmetrical around them. Pie charts are made up of a number of sectors relating in size to the data they show.Ī 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.Ī segment is the curved part of a sector, the part that is left if you remove the triangle from a sector. 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. Sectors and segments are 'slices' of a circle. This formula is usually abbreviated to πr 2 For more about area, see our page Calculating Area. The area of a circle is equal to π × radius 2. The circumference of a circle is equal to π x diameter, or 2 × π × radius (abbreviated to 2πr). Π is important because it is used to calculate the circumference and the area of a circle. Π has a value of 3.142 (although as it is infinite, this is an approximation of its exact value). 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). Understanding Statistical Distributions.Area, Surface Area and Volume Reference Sheet.Simple Transformations of 2-Dimensional Shapes.Polar, Cylindrical and Spherical Coordinates.Introduction to Cartesian Coordinate Systems.Introduction to Geometry: Points, Lines and Planes.Percentage Change | Increase and Decrease.Mental Arithmetic – Basic Mental Maths Hacks.Ordering Mathematical Operations - BODMAS.Common Mathematical Symbols and Terminology.Special Numbers and Mathematical Concepts.How Good Are Your Numeracy Skills? Numeracy Quiz.
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