Every geometry question is written in a precise vocabulary: reflex, perpendicular, rhombus, frustum, segment. Learn the words once and the questions start to read themselves.
Name a shape by the properties it MUST have, not by how it looks: a trapezium has exactly one pair of parallel sides, a rhombus has four equal sides, and a square is a special rectangle AND a special rhombus.
Fig. 1: the parts of a circle with centre O. A sector is cut off by two radii; a segment is cut off by a chord
Section 01
Points, lines and angles.
Open the lab and drag the arm of the angle slowly from 0° all the way round to 360°. Watch the name change: acute, right, obtuse, straight, reflex. Stop at 210° and notice that the same two arms also make a 150° angle on the other side.
Point, vertex, line: a point marks a position; a vertex is a corner where lines meet (plural vertices); a line is straight and extends in both directions, a line segment has two ends.
Plane (Extended): a flat surface, such as a face of a cuboid or a sheet of paper, extending in every direction.
Parallel lines never meet and are always the same distance apart (marked with matching arrows). Perpendicular lines meet at a right angle, 90°.
Perpendicular bisector (Extended): the line that cuts a line segment in half at 90°.
Angle types: acute \(0^\circ < x < 90^\circ\); right \(x = 90^\circ\); obtuse \(90^\circ < x < 180^\circ\); reflex \(180^\circ < x < 360^\circ\).
Interior and exterior angles: an interior angle is inside a polygon at a vertex; the exterior angle is between one side and the next side extended outwards. At each vertex, interior + exterior = 180°.
Congruent shapes are identical in shape and size. Similar shapes have the same shape but one is an enlargement of the other, by a scale factor.
Worked example Naming angles
Classify the angles \(35^\circ\), \(90^\circ\), \(128^\circ\) and \(215^\circ\). Then find the reflex angle that goes with \(128^\circ\).
\(35^\circ\) is between 0° and 90°, so it is acute.
\(90^\circ\) is a right angle.
\(128^\circ\) is between 90° and 180°, so it is obtuse.
\(215^\circ\) is between 180° and 360°, so it is reflex.
The two angles between the same arms make a full turn, so the reflex partner of \(128^\circ\) is \(360^\circ - 128^\circ = 232^\circ\).
A reflex angle is NOT just a big obtuse angle. \(170^\circ\) is obtuse; anything over \(180^\circ\) is reflex. When asked for a reflex angle, subtract the inside angle from \(360^\circ\), not from \(180^\circ\).
Section 02
Triangles and quadrilaterals.
The special quadrilaterals form a family. Each one is defined by a small set of properties; the others follow from them.
Shape
Sides
Angles and diagonals
Square
4 equal; 2 pairs parallel
all 90°; diagonals equal, bisect at 90°
Rectangle
opposite sides equal and parallel
all 90°; diagonals equal, bisect each other
Rhombus
4 equal; 2 pairs parallel
opposite angles equal; diagonals bisect at 90°
Parallelogram
opposite sides equal and parallel
opposite angles equal; diagonals bisect each other
Kite
2 pairs of equal ADJACENT sides
one pair of equal opposite angles; diagonals cross at 90°
Trapezium
exactly 1 pair of parallel sides
co-interior angles between the parallel sides add to 180°
Triangles: equilateral (3 equal sides, all angles 60°), isosceles (2 equal sides, 2 equal base angles), scalene (no equal sides), right-angled (one angle of 90°).
Worked example Name the quadrilateral
A quadrilateral has four equal sides, but none of its angles is a right angle. Its diagonals cross at \(90^\circ\). Name it, and say why it is not a square.
Four equal sides: it is a rhombus or a square.
A square must have four right angles, and this shape has none.
Diagonals crossing at 90° is a property every rhombus has, which agrees.
Answer: A rhombus. It is not a square because its angles are not 90°.
Your turn
A triangle has angles \(45^\circ\), \(45^\circ\) and \(90^\circ\). Use two words to describe it.
Two equal angles means two equal sides, so it is isosceles; one angle is 90°, so it is right-angled.
Examiner trap
A kite's equal sides are next to each other (adjacent); a parallelogram's equal sides are opposite. And a trapezium has exactly ONE pair of parallel sides: a parallelogram is not a trapezium in IGCSE.
Stage 1 · Learn
Check your understanding
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An angle measures \(200^\circ\). What type of angle is it?
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Which quadrilateral has exactly one pair of parallel sides?
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A straight line from the centre of a circle to its circumference is a
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The region between a chord and an arc is called a
Section 03
Polygons, solids and nets.
A polygon is a flat shape with straight sides; it is regular when all its sides AND all its angles are equal. A solid is described by its faces (flat surfaces), edges (where two faces meet) and vertices (corners).
Curved solids: cylinder, cone, sphere; hemisphere (half a sphere) and frustum (a cone or pyramid with its top sliced off parallel to the base) are Extended terms.
A prism has the same cross-section all the way through; a pyramid rises from a base to a single vertex.
A net is the flat pattern that folds up to make a solid. A cube has 11 different nets.
Worked example Counting faces, edges and vertices
A prism has a hexagon as its cross-section. Find its number of faces, edges and vertices.
Faces: 2 hexagonal ends plus 1 rectangle for each of the 6 sides, so \(2 + 6 = 8\).
Edges: 6 round each end, plus 6 joining the ends, so \(6 + 6 + 6 = 18\).
Vertices: 6 on each end, so \(6 + 6 = 12\).
Check with \(F + V - E = 2\): \(8 + 12 - 18 = 2\). It works.
Answer: 8 faces, 18 edges, 12 vertices
Your turn
How many faces, edges and vertices does a pentagonal pyramid have?
Faces: 1 pentagon base + 5 triangles = 6. Edges: 5 round the base + 5 up to the apex = 10. Vertices: 5 on the base + 1 apex = 6.
Examiner trap
Do not count the same edge twice. Every edge is shared by exactly two faces, so counting the edges of each face and adding gives double the true number.
Section 04
The parts of a circle.
In this lab tab, click each label in turn and watch the matching part of the circle light up. Pay special attention to the difference between a sector and a segment, which Fig. 1 also shows.
Radius (plural radii): centre to circumference. Diameter: a chord through the centre, twice the radius.
Circumference: the perimeter of the circle. An arc is part of the circumference; the shorter one is the minor arc, the longer one the major arc (Extended).
Chord: a straight line joining two points on the circumference. Tangent: a straight line that touches the circle at exactly one point.
Sector: the region between two radii and an arc (a pizza slice). Segment: the region between a chord and an arc. Semicircle: half a circle, cut off by a diameter.
Examiner trap
Sector and segment are the classic mix-up. A sector always reaches the centre (two radii); a segment never does unless the chord is a diameter.
Stage 2 · Exam
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How many edges does a square-based pyramid have?
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Find the reflex angle between the hands of a clock at exactly 5 o'clock.
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A triangle has two sides of 6 cm with a right angle between them. Which description is correct?
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A solid has 7 faces: 2 pentagons and 5 rectangles. How many vertices does it have?
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