Geometrical terms

Points, lines and vertices

Geometry questions are written in a precise vocabulary, and a mark is often lost not through bad arithmetic but because a word was misread. This lesson collects the words the syllabus lists.

• A point marks a position. It has no size. Points are named with capital letters: the point $A$.
• A line is straight. Strictly, a line goes on for ever in both directions; the piece between two points $A$ and $B$ is a line segment, written $AB$. In exam questions "the line $AB$" usually means that segment.
• A vertex is a corner: a point where two sides of a shape meet, or where edges of a solid meet. The plural is vertices.

Angles are named with three letters, and the middle letter is the vertex. Angle $ABC$, written $\angle ABC$, is the angle at $B$ between the lines $BA$ and $BC$. Reading $\angle ABC$ as "the angle at $A$" is one of the most common slips in the whole paper.

Parallel and perpendicular lines

• Parallel lines go in exactly the same direction, so they never meet however far they are extended. They stay the same distance apart. On a diagram parallel lines are marked with matching arrows. We write $AB\parallel CD$.
• Perpendicular lines meet at a right angle, ${90}^{°}$. A right angle is marked with a small square in the corner. We write $AB\perp CD$.

On a square grid you can check both without measuring. Describe each line by its step: "4 across and 2 up". A parallel line has the same step, or a multiple of it ("2 across and 1 up" is the same direction). A perpendicular line has the step turned through a quarter turn: swap the two numbers and change one sign, so "4 across, 2 up" becomes "2 across, 4 down" or "2 back, 4 up". The reason is that turning a right-angled step triangle through ${90}^{°}$ swaps its across and up sides.

Types of angle

An angle measures an amount of turn. A full turn is ${360}^{°}$, a half turn (a straight line) is ${180}^{°}$, and a quarter turn is a right angle, ${90}^{°}$.

• An acute angle is less than ${90}^{°}$.
• A right angle is exactly ${90}^{°}$.
• An obtuse angle is more than ${90}^{°}$ but less than ${180}^{°}$.
• A reflex angle is more than ${180}^{°}$ but less than ${360}^{°}$.

Notice that ${90}^{°}$ is neither acute nor obtuse, and ${180}^{°}$ is neither obtuse nor reflex: those are the boundaries.

Wherever two lines meet at a point there are two angles between them, one on each side, and together they make a full turn. So if the smaller angle is ${70}^{°}$, the reflex angle on the other side is ${360}^{°}-{70}^{°}={290}^{°}$. This is why you never need a protractor that reads past ${180}^{°}$.

Interior and exterior angles

In a polygon, an interior angle is the angle inside the shape at a vertex, between two sides.

An exterior angle is made by extending one side beyond the vertex: it is the angle between that extension and the next side. It is outside the shape.

Because the side and its extension form a straight line, the interior and exterior angles at the same vertex always add up to ${180}^{°}$:

$$\text{interior angle}+\text{exterior angle}={180}^{°}$$

So an interior angle of ${110}^{°}$ goes with an exterior angle of ${70}^{°}$. The angle properties of polygons, such as the sum of the exterior angles, are studied in 4.6; here you need to recognise which angle is which.

Bearings: the word

A bearing is a direction, given as an angle measured clockwise from north, and always written with three figures: ${045}^{°}$, not ${45}^{°}$. Due east is ${090}^{°}$, due south ${180}^{°}$ and due west ${270}^{°}$. Bearings are used fully in 4.3, Scale drawings; for now, recognise the word and the three-figure form.

Congruent, similar and scale factor

• Two shapes are congruent if they are exactly the same shape and the same size. One could be placed exactly on top of the other, possibly after turning it over or round. Congruent shapes have all corresponding sides equal and all corresponding angles equal.
• Two shapes are similar if one is an enlargement of the other. They are the same shape, but not necessarily the same size: all corresponding angles are equal and all corresponding lengths are in the same ratio.
• The scale factor is that ratio: $\text{scale factor}=\frac{\text{length on the new shape}}{\text{corresponding length on the original}}$.

In the grid below, B is similar to A with scale factor 2: each length of A doubles. C is congruent to A; it has been turned, but it is still 2 by 1. D is 3 by 2, and $\frac{3}{2}\ne \frac{2}{1}$, so D is not similar to A, even though it is also a rectangle. All rectangles have the same angles, so for rectangles it is the side ratio that decides.

Congruent shapes are similar with scale factor 1. You are not expected to prove that two shapes are congruent, but you must recognise congruent and similar shapes and find a scale factor.

Triangles

• Equilateral: all three sides equal, and all three angles ${60}^{°}$.
• Isosceles: two sides equal, and the two angles opposite them (the base angles) are equal.
• Scalene: no sides equal and no angles equal.
• Right-angled: one angle is ${90}^{°}$. The side opposite it is the longest side, the hypotenuse.

The names describe different things, so a triangle can have two of them: a triangle with angles ${90}^{°}$, ${45}^{°}$ and ${45}^{°}$ is a right-angled isosceles triangle. On diagrams, equal sides are marked with matching dashes (ticks).

Special quadrilaterals

A quadrilateral has four straight sides. Six special ones must be known by their properties, because questions often describe a shape in words and ask for its name.

The families overlap. A square is a special rectangle (its sides are also equal) and a special rhombus (its angles are also ${90}^{°}$). Every rectangle and every rhombus is a parallelogram. So when a question asks for "the mathematical name", give the most precise name the facts allow, and check every property you were told.

Polygons

A polygon is a flat, closed shape made of straight sides. A regular polygon has all its sides equal and all its angles equal. An irregular polygon does not.

Both conditions matter. A rhombus has four equal sides but its angles are not all equal, so it is not regular. A rectangle has four equal angles but its sides are not all equal, so it is not regular either. The square is the regular quadrilateral, and the equilateral triangle is the regular triangle.

The names to know:

• triangle: 3 sides;
• quadrilateral: 4 sides;
• pentagon: 5 sides;
• hexagon: 6 sides;
• octagon: 8 sides;
• decagon: 10 sides.

A polygon has the same number of vertices as sides. ("Oct" means 8, as in octopus; "dec" means 10, as in decimal.)

Solids, faces, edges and vertices

A solid is a three-dimensional shape.

• A face is a flat surface of a solid. A surface can also be curved: a cylinder has two flat faces and one curved surface.
• An edge is a line where two faces meet.
• A vertex is a corner, where edges meet.

The solids to know:

• cube: six congruent square faces;
• cuboid: six rectangular faces, in three pairs of congruent opposite faces;
• prism: the same cross-section all the way through, with two congruent end faces joined by rectangles. A triangular prism has triangular ends;
• cylinder: a prism-like solid with circular ends: two flat faces and one curved surface;
• pyramid: a polygon base with triangular faces meeting at one vertex, the apex. A square-based pyramid has 5 faces;
• cone: a circular base and a curved surface rising to one vertex;
• sphere: one curved surface, with no edges and no vertices, like a ball.

You can count faces, edges and vertices without a picture. A prism whose ends have $n$ sides has $n+2$ faces (the $n$ rectangles and the two ends), $3n$ edges ($n$ round each end and $n$ joining them) and $2n$ vertices ($n$ at each end). A pyramid on an $n$-sided base has $n+1$ faces, $2n$ edges and $n+1$ vertices.

Nets

A net is a flat shape that folds up to make a solid. The net of a cube is six squares joined edge to edge in an arrangement that folds without overlapping. You will draw and use nets in 4.2; the word to know here is that a net is the solid "unfolded", so the area of a net is the surface area of the solid.

Parts of a circle

• The centre is the point in the middle, the same distance from every point on the circle.
• A radius is a line from the centre to the circle. The plural is radii. All radii of a circle are equal.
• A diameter is a line from one side of the circle to the other through the centre. It is two radii long: $d=2r$.
• The circumference is the distance all the way round the circle: its perimeter.
• A semicircle is half a circle, cut off by a diameter.
• A chord is a straight line joining two points on the circle. A diameter is the longest chord.
• A tangent is a straight line that touches the circle at exactly one point, without crossing it.
• An arc is part of the circumference.
• A sector is the region between two radii and an arc, shaped like a slice of pizza.
• A segment is the region between a chord and an arc.

The pairs to keep apart: a sector is bounded by two radii, a segment by a chord; a chord crosses the circle, a tangent only touches it.