Dr Duncan James > UPmaths > Geometry

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Geometry is an important topic in mathematics. It is often studied just using numbers, although it can also include algebra. The mathematics of geometry leads to creating buildings, bridges and mobile phones. It is also important for computer games, aeroplanes and other things that rely on understanding how things move. This is an important topic!


Compare the direction of two lines. 360 degrees represent all the way around. Parallel lines are also covered here.

Interior, Alternate, Vertically Opposite and Corresponding Angle Definitions (YouTube)


The distance/length around the outside of a shape.


The flat space that could be covered in paint, covered in paper, built on, stood on, slept on, covered in grass-seed and so-on.

Many area problems can be used with symmetry or ideas based on the pattern.

A three-dimensional shape like a box has a surface area. This is the total area found on the outside:

There is a simplified approach that will deal with most two-dimensional shapes. Just split the shape up into triangles, rectangles and circles. Then deal with each bit separately.


Triangles are a topic on their own. A key reason for this is that with just three sides and three angles predictable, easy-to-describe patterns have been found that do not exist for shapes with more sides.

How to get Started with Different Triangle Questions (UP3) (YouTube)

Fast Summary of Triangle Question Approaches (UP3) (YouTube)

Five Fast Triangle Questions with Worked Answers (UP3) (YouTube)

How to Find the Area of a Triangle 5 Different Ways (UP3) (YouTube)


Common questions on this topic include the total of all the interior angles and the interior/exterior angle of regular polygons.

Regular and Irregular Polygons Two Useful Techniques (YouTube)

Circle Theorems

We can extend our knowledge of circles with what are often called circle theorems.

There are a number of symmetrical/geometrical patterns in circles that are often called "circle theorems".

Other patterns in the geometry of circles are also commonly used in circle geometry.

Intersecting Chord Theorem (YouTube)

Intersecting Secant Theorem (YouTube)

Mixed Two-Dimensional Questions

Problems can be created that use perimeter, area, angles and all the other two-dimensional skills.

Parallelogram Trisector Discussion (YouTube)


Volume is the amount of space inside a box or other shape. It might represent the amount of breathable air in a spaceship, the space in juice carton, the space in a box for posting something and so-on.

Mixed Geometry Questions

Real-life geometry problems require you to deal with mixtures of different geometry skills.

More 3D Geometry

There are more skills for working in three dimensions including a three dimensional version of Pythagora's Theorem.

Coordinate Geometry

Core skills here include straight lines, gradients and the distance, midpoint and gradient between two points.

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