Contemporary geometry has many sub fields

  • Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry.Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics.

  • Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics, including in general relativity.

  • Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, such as connectedness and compactness.

  • Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.

  • Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in many areas, including cryptography and string theory.

  • Discrete geometry is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. It shares many methods and principles with combinatorics.

Pythagoras

Pythagoras’ theorem states that for all right-angled triangles, ‘The square on the hypotenuse is equal to the sum of the squares on the other two sides’. The hypotenuse is the longest side and it’s always opposite the right angle.

Diagram of a right-angled triangle

In this triangle  a2 = b2 + c2 and angle A is a right angle.

Pythagoras’ theorem only works for right-angled triangles, so you can use it to test whether a triangle has a right angle or not.

In the triangle above, if  a2 < b2 + c2  the angle A is acute.

In the triangle above, if  a2 > b2 + cthe angle A is obtuse.