Chapter 32 – Geometrical terms

A point marks a position and has no dimensions (0-D). A line has length but no width (1-D). A plane is a flat 2-D surface with length and width. A solid has length, width and height (3-D).

A drawing is made on flat paper, but it can represent a 3-D object. Use the vocabulary of point, line, plane, surface and solid precisely in written explanations.

Angles measure a turn about a vertex. A full turn is \(360^\circ\), a half turn \(180^\circ\), and a quarter turn \(90^\circ\). An acute angle is smaller than \(90^\circ\); a right angle is exactly \(90^\circ\); an obtuse angle lies between \(90^\circ\) and \(180^\circ\); and a reflex angle lies between \(180^\circ\) and \(360^\circ\).

When measuring an angle with a protractor, place its centre on the vertex, align its zero line with one arm, and read the correct scale. A straight angle is \(180^\circ\).

Parallel lines have the same direction and never meet. Perpendicular lines intersect at \(90^\circ\). A perpendicular bisector cuts a given line segment into two equal parts at right angles.

The arrows drawn on corresponding lines show that they are parallel. A small square marks a right angle. When a problem asks for a perpendicular bisector, both conditions—90° and equal halves—must be satisfied.

A bearing states a direction, measured clockwise from north at the starting point. Write it using three digits, such as \(045^\circ\), \(090^\circ\) and \(270^\circ\). Draw north lines at the correct point.

To reverse a bearing between two places, add or subtract \(180^\circ\) so that the answer lies between \(000^\circ\) and \(360^\circ\). Beware: the reverse direction starts at the other location.

Triangles are named by their sides and angles. An equilateral triangle has three equal sides and three \(60^\circ\) angles. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has three unequal sides; a right-angled triangle has one \(90^\circ\) angle.

Angle classification and side classification describe different properties. For example, a right-angled triangle may also be isosceles.

A quadrilateral has four sides. Know the defining properties of a square, rectangle, parallelogram, rhombus, trapezium, kite, and isosceles trapezium. The textbook describes a trapezium as having one pair of parallel sides.

A square has four equal sides and four right angles. A rectangle has four right angles and equal opposite sides. A rhombus has four equal sides. In a parallelogram, both pairs of opposite sides are parallel. A kite has two pairs of adjacent equal sides; an isosceles trapezium has one pair of parallel sides and equal non-parallel sides.

Diagonals give useful clues: square and rhombus diagonals meet at right angles; rectangle diagonals are equal; parallelogram diagonals bisect one another. A diagonal connects opposite corners.

A polygon is a closed 2-D shape made from straight line segments. Common names: pentagon (5 sides), hexagon (6), octagon (8), decagon (10). A regular polygon has equal sides and equal interior angles; other polygons are irregular.

An interior angle lies inside the polygon. An exterior angle is made by extending a side. A diagonal joins two non-adjacent vertices; a side is not a diagonal.

A solid is 3-D. An edge is where faces meet and a vertex is a corner; the plural is vertices. A face is a flat surface. The cube has six square faces; the cuboid has rectangular faces. A sphere has one curved surface, while a hemisphere, cylinder and cone have flat and curved surfaces.

A prism has the same cross-section along its length; triangular and rectangular prisms are examples. A pyramid has a polygonal base and triangular side faces meeting at a point. A frustum is the portion left after a cone or pyramid is cut parallel to its base.

A net is a flat arrangement of connected faces that folds to form a 3-D solid. Before deciding whether a net works, identify the required faces, check how they are joined, and imagine folding along every common edge.

A cube is made from six squares; a cuboid from three pairs of matching rectangles; a triangular prism from two congruent triangular ends and three rectangular side faces; and a triangular pyramid from four triangular faces. Here the emphasis is on polygon-faced solids.

Congruent figures have exactly the same shape and size: their corresponding side lengths and angles are equal. They remain congruent if one is turned, moved or reflected.

Similar figures have equal corresponding angles and corresponding side lengths in a constant ratio. The factor multiplying each length is the scale factor. Congruent figures are similar with scale factor 1.

The centre is equidistant from the circumference; the radius runs from centre to circumference, and the diameter runs through the centre from one side to the other. Thus \(d=2r\). The circumference is the boundary of the circle.

A chord joins two points on the circumference. A tangent touches the circle at one point. An arc is part of the circumference; a sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc. The longer and shorter parts are called major and minor.