We will now also cover some useful equations for other polygons Rectangles We then calculate S by dividing perimeter by 2:įinally, we can use S in Heron's formula to get: First, we calculate the perimeter by adding the three side lengths: To further are explanation of polygons and to extend our analysis of triangles, there is a second formula to calculate area. Complex polygons on the other hand are ones that have sides which intersect, this also means that regular facts about polygons may not apply. Simple polygons are ones that have no intersecting sides. The third category we can place polygons in is simple or complex. Concave polygons are always irregular since not all of the internal angles can be equal however, not all irregular polygons are concave. We can say that all regular polygons are also convex, but not all convex polygons are regular. Convex polygons are polygons which have no angles greater than 180 degrees. Irregular polygons therefore are all polygons which are not regular.Ĭoncave polygons are polygons which have at least one internal angle greater than 180 degrees. Regular polygons are that which all sides are equal in length and all internal angles are equal size.

Polygons have properties which we can use to analyse them in problems, we will cover them below. The line segments must intersect with each other only at their endpoints. They do not have any curved surfaces, they should have at least three sides. Polygonal shapes are closed, two-dimensional shapes which have straight line segments. The last theorem states that angles in alternate segments to each other are equal.Īs you can see circle theorems are very useful during geometric calculations as they allow us to gain much more knowledge about a particular problem, when only having a small amount of info initially. This theorem states that any tangents which stem from the same point outside of a circle will be equal in length.Īngles in alternate segments are identical Tangents from a point outside the circle are equal in length This is because any tangent is perpendicular to the radius of a circle at its point of contact. The next theorem states that the angle between the radius and a tangent to a circle will be 90 degrees. The angle between the radius and a tangent is 90 degrees This theorem states that if there is a perpendicular line to a chord, the chord will be bisected by that line. Perpendicular line from centre to chord will bisect the chord When we have a quadrilateral shape within a circle, the opposite angles will add to 180 degrees, this is because the interior angles of a quadrilateral add to make 360 degrees. Opposite angles in a cyclic quadrilateral sum to 180 degrees This theorem means that if we have to find the angles within a semi circle, it will be formed from a right-angle triangle, and therefore an angle will equal 90 degrees.

With this theorem, we can say that angles located in the same segment from a common chord will be equal. The first theorem that we will look at states that the angle at the centre of the circle is double that of the angle at the circumference, this will be useful to us as it means with one of the two angles we are able to easily calculate the other.

Angle at the centre is twice that of the circumference We can use these common properties of circles on a Euclidean plane in order to retrieve much information and data about circles, as well as use it to solve problems later down the line. To further the discussion about circles in geometry, here we shall look at circle theorems and their functions. In this article, we shall further expand on this, covering topics such as circle theorems, polygonal shapes and more. In the last section, we covered the basics of geometry including points, lines, curves, and some information about 2D shapes like triangles and circles. In this article, we have covered Mathematics for Computational Geometry focused on 2D Shapes such as Circle properties/ theorems, Polygons and more.

Geometry theorems list