Geometrical Terms

The words that fix a position or a direction

A point marks a position and has no size; it is named with a capital letter, so point A means the exact spot labelled A. A vertex (plural vertices) is a point where two straight sides or edges meet - the corner of a shape. Every triangle has three vertices and every quadrilateral has four.

A line is straight and goes on for ever in both directions. The straight piece between two named points is a line segment, and that is the one with a measurable length. A plane is a flat surface that goes on for ever in every direction; the page in front of you is a model of one, and Extended candidates are expected to use the word.

Two lines are parallel if they are always the same distance apart and never meet, however far they are drawn. On a diagram they are marked with matching arrowheads. Two lines are perpendicular if they meet at a right angle, marked with a small square in the corner. A perpendicular bisector cuts a line segment in half and at a right angle, and it is the Extended column's name for the set of points that are the same distance from each end of the segment.

A bearing gives a direction as three figures, measured clockwise from north, so due north is 000°, due east is 090° and due south is 180°. Bearings let a question send you from one point to another without printing a compass on the diagram.

Angles: right, acute, obtuse and reflex

An angle is measured in degrees and is named either with three letters, angle ABC with B at the corner, or with a single letter or number. A right angle is exactly 90°. An acute angle is less than 90°. An obtuse angle is more than 90° but less than 180°. A reflex angle is more than 180° and less than 360° - it is the angle on the outside of the turn, not the one that sits inside the corner.

A straight line is an angle of 180° and a full turn is 360°. Two angles that add to 180° are supplementary, and two that add to 90° are complementary.

Interior angles are the ones inside a shape, where two of its sides meet. Exterior angles are the ones outside, formed by a side and the line extended from the next side. An interior angle and the exterior angle beside it sit on a straight line, so they add to 180°. The full angle work - angle sums, parallel lines and the angles of polygons - belongs to the Angles topic; what you need here are the names and the sizes that go with them.

The triangle family: four names

A triangle is named by its sides or by its angles, and each name brings properties that the papers expect you to use without being told. An equilateral triangle has three equal sides and three equal angles of 60°. An isosceles triangle has two equal sides, and the two angles opposite those sides are equal; the equal sides are marked with matching ticks. A scalene triangle has three sides of different lengths and three angles of different sizes. A right-angled triangle has one angle of 90° and can carry one of the other names as well: a right-angled isosceles triangle has angles 90°, 45° and 45°.

The three angles of any triangle add to 180°. That is why a triangle can have at most one right angle and at most one obtuse angle, and why the two equal angles of an isosceles triangle must be acute.

The quadrilateral family: six names

A quadrilateral has four straight sides and four vertices, and its interior angles add to 360°. Cambridge names six special quadrilaterals, and the differences between them come down to which sides are equal, which pairs are parallel, and which angles are right angles.

A square has four equal sides and four right angles. A rectangle has four right angles and two pairs of equal sides. A kite has two pairs of adjacent equal sides - the equal pairs meet at a point - and one pair of equal opposite angles. A rhombus has four equal sides but no right angles. A parallelogram has two pairs of parallel sides, with opposite sides equal and opposite angles equal. A trapezium has exactly one pair of parallel sides.

The names overlap rather than exclude each other. Every square is a rectangle, a rhombus and a parallelogram; every rhombus and every rectangle is a parallelogram. A trapezium is never a parallelogram, because exactly one pair of parallel sides rules out two pairs.

Polygons: regular, irregular, and the names up to ten sides

A polygon is a closed shape whose sides are all straight. It is regular when every side is the same length and every interior angle is the same size; otherwise it is irregular. A regular polygon is the only kind that has as many lines of symmetry as it has sides.

Cambridge names four polygons in particular: the pentagon (5 sides), the hexagon (6 sides), the octagon (8 sides) and the decagon (10 sides). The pattern continues in the same way, so a 7-sided polygon is a heptagon and a 9-sided one is a nonagon, but those two are not on the syllabus list - recognise them if they appear, do not expect to be asked for them.

A triangle is a 3-sided polygon and a quadrilateral is a 4-sided one, so these names cover them too. The interior angles of an n-sided polygon add to $(n-2)\times 180$°, which is why a pentagon's angles add to 540° and a hexagon's to 720°. The Angles topic does that arithmetic; here what matters is the name and the fact that a regular polygon looks the same all the way round.

The parts of a circle

The centre is the point in the middle, and every point on the circle is the same distance from it. That distance is the radius (plural radii), and a straight line drawn from the centre to the circle is also called a radius. The diameter is a straight line through the centre joining two points on the circle, so the diameter is twice the radius.

The circumference is the distance all the way round the circle. A chord is any straight line joining two points on the circle, which makes the diameter the longest chord there is. A tangent touches the circle at exactly one point and is perpendicular to the radius drawn to that point. A semicircle is half of the circle, cut off by a diameter.

An arc is part of the circumference. The longer of the two arcs that a chord cuts off is the major arc and the shorter is the minor arc, and the Extended column names both. A sector is the region between two radii and the arc joining them - a slice of cake. A segment is the region between a chord and the arc cut off by that chord. The one-line test is the number of straight edges: a sector has two, a segment has one.

Solids: faces, surfaces and edges

A face is a flat surface of a solid. A surface is any outer boundary, flat or curved, so a cylinder has three surfaces, two flat faces and one curved. An edge is where two faces meet, and a vertex is a corner where edges meet.

The solids you must be able to name are the cube (six equal square faces), the cuboid (six rectangular faces), the prism (the same cross-section all the way along, with flat ends), the cylinder (a prism with a circular cross-section), the pyramid (a polygon base with triangular faces meeting at a point), the cone (a circular base and a curved surface meeting at a point) and the sphere (every point of its surface the same distance from the centre). A prism is named by its cross-section, so a hexagonal prism has a hexagon at each end.

The Extended column adds the hemisphere, half a sphere, and the frustum, the solid left when the top of a cone or a pyramid is sliced off by a plane parallel to the base - the shape of a flowerpot.

Comparing shapes, and the terms only the Extended column asks for

Two shapes are congruent if they are exactly the same shape and the same size, so one could be picked up and placed on the other. They are similar if they are the same shape but not necessarily the same size: every angle matches and every length in one shape is multiplied by the same number to give the matching length in the other. That number is the scale factor. Congruent shapes are similar shapes with a scale factor of 1.

A net is a flat shape that folds up to make a solid. A cross of six squares folds into a cube, and a rectangle with a circle at each end folds into a cylinder. When a question gives you a net, the work is seeing which edges meet when it folds.

The Extended column is longer than the Core one, and the extra words are worth learning because a question can hinge on one of them. A plane is a flat surface going on for ever. A perpendicular bisector cuts a line segment in half at a right angle. The major arc is the longer of the two arcs a chord cuts off and the minor arc is the shorter. A hemisphere is half a sphere, and a frustum is what is left when the top is cut off a cone or a pyramid.

• Congruent: same shape, same size. Similar: same shape, every length multiplied by the same scale factor
• A scale factor of 3 turns a side of 5 cm into 15 cm, and a scale factor of $\frac{1}{2}$ turns it into 2.5 cm
• The diameter is the longest chord of a circle, so any other chord is shorter than it
• A prism is named by its cross-section: a pentagonal prism has a pentagon at each end