Topic 04 · 18 lessons

Geometry

Geometry is the maths of shape, size and position. It began with surveyors measuring fields and builders setting out right angles, and it still underpins architecture, engineering, computer graphics and navigation.

Our geometry lessons focus on the facts you will use most often, such as angle rules, triangle properties, circle measurements and area, and on the reasons those facts are true.

Lessons in this topic

4.1Shapes All AroundFlat shapes have sides and corners. Solid shapes have faces, edges and vertices. Counting them tells you what a shape is.
4.2Position, Direction and TurnsPosition says where something is. Direction says which way to go. A turn changes the way you face. Words and counted steps are enough to describe all three.
4.3Recognising 3D ShapesA 3D shape takes up space. Learn to name the common ones, match them to real objects and count their faces, edges and vertices.
4.4Nets, Faces, Edges and VerticesA solid has flat faces, straight edges and corner vertices. Unfold it and you get a net, a flat pattern that folds back into the solid.
4.5Lines, Segments, Parallel and PerpendicularA line goes on forever, a ray has one end and a segment has two. Parallel lines never meet, and perpendicular lines meet at a right angle.
4.6AnglesAn angle measures an amount of turn. Full, half and quarter turns are 360, 180 and 90 degrees, and the size of an angle gives it a name.
4.7Measuring and Drawing AnglesTo measure an angle with a protractor, line up the vertex and one arm with zero, then read the scale that starts at zero. Estimate first to catch mistakes.
4.8Angle Rules on Lines and Around a PointAngles on a straight line add to 180 degrees, angles around a point add to 360, and vertically opposite angles are equal.
4.9TrianglesThree sides, three angles that always add to 180 degrees. Equilateral, isosceles and scalene triangles are named by their sides.
4.10Quadrilaterals and Their PropertiesA quadrilateral has four straight sides and four angles that add to 360 degrees. Each type has its own sides and angles.
4.11Line and Rotational SymmetryA shape has line symmetry if a mirror line splits it into two matching halves. It has rotational symmetry if it fits onto itself before a full turn.
4.12Polygons and Interior AnglesA polygon with n sides splits into n minus 2 triangles, so its interior angles add to (n minus 2) times 180 degrees. A regular polygon shares that total equally.
4.13Area and PerimeterPerimeter is the distance around a shape and area is the space inside it. For a rectangle, perimeter is 2 times (length plus width) and area is length times width.
4.14Area of Triangles, Parallelograms and TrapeziaA parallelogram has area base times height. A triangle is half of one, and a trapezium is half of two joined together. Always use the perpendicular height.
4.15CirclesEvery circle, whatever its size, shares one ratio: pi. Circumference and area both follow from it.
4.16Pythagoras' TheoremIn a right-angled triangle, the square on the longest side equals the sum of the squares on the other two sides: a squared plus b squared equals c squared.
4.17TransformationsFour ways to move or resize a shape. Three keep it the same size; enlargement scales it.
4.18Congruence and SimilarityCongruent shapes are identical in size and shape. Similar shapes have the same shape but a different size, with all lengths in the same ratio.

The big ideas

a)Angles measure turn: A full turn is $360^\circ$ , a straight line is $180^\circ$ and a right angle is $90^\circ$ . Most angle problems come back to these three facts.
b)Triangles are rigid: Three fixed side lengths can only make one triangle. That is why triangles appear in bridges, roofs and bicycle frames.
c)π links round and straight: For every circle, the distance around divided by the distance across is the same number: $\pi \approx 3.14159$ .
d)Moving without changing: Translations, rotations and reflections move a shape but keep its size and angles, so the image is congruent to the original.

Worked example

The missing angle

A triangle has angles of $48^\circ$ and $67^\circ$ . Find the third angle.

The angles inside any triangle add up to $180^\circ$ .
Add the known angles: $48^\circ + 67^\circ = 115^\circ$ .
Subtract from the total: $180^\circ - 115^\circ = 65^\circ$ .

Answer: The third angle is $65^\circ$ .

Where you will meet it

Working out how much paint a wall needs, checking that a shelf is square, plotting a route on a map or designing a logo all lean on geometry.

The wider field

Geometry also covers these ideas, which build on the lessons above:

Polygons and quadrilaterals
3D solids and nets
Symmetry
Coordinate geometry
Vectors
Trigonometry