Geometry is the mathematics of shape, size and position — points and lines, angles and triangles, circles and solids, and the way they move and fit together. It is the oldest branch of mathematics, and the most visual: nearly every result is something you can see, and prove with a careful picture.

The engine of geometry is the theorem: a fact, true for every figure of its kind, that you justify once and then use forever. This course builds them up in order — each new theorem leaning on the ones before — and lets you practise each by chasing angles and lengths right on the diagram.

The shape of the journey

Seven stages, from a single angle to circles, solids and transformations.

• Stage 1 — Lines & angles. The alphabet: angles on a line, around a point, and the angles parallel lines make.
• Stage 2 — Triangles. The angle sum, the exterior angle, congruence, similarity and Pythagoras.
• Stage 3 — Quadrilaterals & polygons. Four-sided shapes and the angle rules for any polygon.
• Stage 4 — Circles. The circle theorems — angles at the centre, in a segment, and in a semicircle.
• Stage 5 — Area, perimeter & volume. Measuring flat shapes and solids.
• Stage 6 — Coordinate geometry. Shape meets algebra on the grid.
• Stage 7 — Transformations & symmetry. Sliding, turning, flipping and scaling figures.

Stage 1 — Lines & angles

1. Points, Lines and Planes
2. Measuring Angles
3. Angles on a Line and at a Point
4. Vertically Opposite Angles
5. Parallel Lines
6. Corresponding Angles
7. Alternate Interior Angles
8. Co-Interior Angles
9. Angle Chasing (practice)

Stage 2 — Triangles

1. Types of Triangle
2. The Angle Sum of a Triangle
3. The Exterior Angle of a Triangle
4. Isosceles Triangles
5. Congruent Triangles
6. Similar Triangles
7. The Pythagorean Theorem

Stage 3 — Quadrilaterals & polygons

1. Types of Quadrilateral
2. Properties of Parallelograms
3. The Angle Sum of a Polygon
4. Exterior Angles of a Polygon
5. Regular Polygons

Stage 4 — Circles

1. Parts of a Circle
2. The Angle at the Centre
3. Angles in the Same Segment
4. The Angle in a Semicircle
5. Cyclic Quadrilaterals
6. Tangents to a Circle

Stage 5 — Area, perimeter & volume

1. Perimeter and Area
2. Area of Parallelograms and Trapezia
3. Circumference and Area of a Circle
4. Surface Area of Solids
5. Volume of Prisms and Cylinders
6. Volume of Pyramids, Cones and Spheres

Stage 6 — Coordinate geometry

1. Distance Between Two Points
2. The Midpoint of a Segment
3. The Gradient of a Line
4. The Equation of a Straight Line

Stage 7 — Transformations & symmetry

1. Reflection
2. Rotation
3. Translation
4. Enlargement
5. Symmetry

Let's get started

We begin where geometry itself begins — with parallel lines and the angles a crossing line makes. From there every triangle theorem follows.

Let's get started → Parallel Lines