One-page revision sheet

• A polygon is a closed shape with straight sides. It is regular only if the sides and the angles are all equal.
• The interior angles of an n-sided polygon total 180(n − 2)° — it splits into n − 2 triangles.
• For a regular polygon, each interior angle is 180(n − 2) ÷ n.
• The exterior angles of any polygon total 360°, whatever the number of sides.
• So each exterior angle of a regular n-gon is 360 ÷ n.
• An interior angle and its exterior angle lie on a straight line, so they add to 180°.
• Going through the exterior angle is usually the quicker route: one fixed total and one division.
• If 360 ÷ (exterior angle) is not a whole number, no such regular polygon exists.

Key words: polygon  ·  regular polygon  ·  vertex  ·  interior angle  ·  exterior angle  ·  angle sum  ·  pentagon  ·  hexagon  ·  octagon

• The centre is the middle; the radius reaches from it to the edge.
• The diameter goes right across through the centre, so d = 2r and r = d ÷ 2.
• The circumference is the distance all the way round — the perimeter of a circle.
• A chord joins two points on the circle. The diameter is the longest chord.
• An arc is a part of the circumference.
• A circle has infinitely many lines of symmetry — every line through the centre.
• Its order of rotational symmetry is infinite: any turn about the centre leaves it unchanged.
• Read the question carefully: “how wide across” means the diameter, “how far from the middle” means the radius.

Key words: centre  ·  radius  ·  diameter  ·  circumference  ·  chord  ·  arc

• A line of symmetry is a fold line: the two halves match exactly.
• The order of rotational symmetry is how many times a shape looks the same in one full turn.
• The smallest turn that works is 360 ÷ order.
• The smallest possible order is 1, not 0 — every shape returns to itself after a full turn.
• A regular n-gon has exactly n lines of symmetry and order n.
• The two kinds of symmetry are independent: a parallelogram has no lines but order 2.
• A circle has infinitely many lines and infinite order.
• Check a fold line by imagining the paper folded — “it looks about right” is not enough.

Key words: line of symmetry  ·  reflective symmetry  ·  rotational symmetry  ·  order of rotational symmetry  ·  centre of rotation

• A face is a flat surface, an edge is where two faces meet, a vertex is a corner.
• A net is the flat shape that folds up into the solid — a box cut open and laid out.
• A prism has the same cross-section all the way along; a pyramid rises to a single apex.
• A prism on an n-sided shape has n + 2 faces, 3n edges and 2n vertices.
• A pyramid on an n-sided base has n + 1 faces, 2n edges and n + 1 vertices.
• Euler's rule: faces + vertices − edges = 2 for any solid with flat faces and no holes.
• Use Euler's rule as a check on every count you make.
• A cube has eleven different nets — there is rarely only one right answer.

Key words: face  ·  edge  ·  vertex  ·  net  ·  prism  ·  pyramid  ·  cross-section