Geometrical terms Revision Notes

Geometry begins with fundamental terms: a point represents a precise location with no size; a vertex is a corner where two or more lines meet; a line extends infinitely in both directions with no thickness.

Lines can be parallel (always the same distance apart, never meeting) or perpendicular (intersecting at right angles). Bearings describe direction using degrees clockwise from north, e.g., a bearing of 045° points northeast.

Angles measure the turn between two lines meeting at a vertex. Key angle types include: right angle (exactly 90°), acute angle (less than 90°), obtuse angle (between 90° and 180°), and reflex angle (between 180° and 360°). Interior angles lie inside a shape, exterior angles lie outside.

• Point: exact location, no size.
• Vertex: corner point where lines meet.
• Line: infinite length, no thickness.
• Parallel lines: never meet, equal distance apart.
• Perpendicular lines: meet at 90°.
• Bearing: direction measured clockwise from north.
• Right angle: 90°.
• Acute angle: <90°; Obtuse angle: >90° and <180°; Reflex angle: >180° and <360°; Interior and exterior angles defined by position relative to shape.

Triangles are three-sided polygons classified by side lengths and angles. Equilateral triangles have three equal sides and angles; isosceles have two equal sides; scalene have all sides different. Right-angled triangles have one 90° angle.

Special quadrilaterals have four sides with distinctive properties: squares have four equal sides and four right angles; rectangles have opposite sides equal and four right angles; kites have two pairs of adjacent equal sides; rhombuses have four equal sides but angles are not necessarily right angles; parallelograms have opposite sides parallel and equal; trapeziums have only one pair of parallel sides.

• Equilateral triangle: 3 equal sides and angles.
• Isosceles triangle: 2 equal sides.
• Scalene triangle: all sides different.
• Right-angled triangle: one 90° angle.
• Square: 4 equal sides, 4 right angles.
• Rectangle: opposite sides equal, 4 right angles.
• Kite: 2 pairs of adjacent equal sides.
• Rhombus: 4 equal sides, opposite angles equal, sides parallel in pairs (parallelogram). Parallelogram: opposite sides equal and parallel. Trapezium: one pair of parallel sides.

Polygons are closed shapes with straight sides. Regular polygons have all sides and angles equal; irregular polygons do not. Common polygons include pentagon (5 sides), hexagon (6 sides), octagon (8 sides), and decagon (10 sides).

The sum of interior angles of a polygon with n sides is (n-2) × 180°. For example, a pentagon's interior angles sum to (5-2)×180° = 540°.

• Polygon: closed shape with straight sides.
• Regular polygon: all sides and angles equal.
• Irregular polygon: sides and/or angles not equal.
• Pentagon: 5 sides; Hexagon: 6 sides; Octagon: 8 sides; Decagon: 10 sides.
• Sum of interior angles = (n-2) × 180°, where n = number of sides.

Simple solids are three-dimensional shapes. A cube has six equal square faces; a cuboid has rectangular faces with opposite faces equal; a prism has two identical polygonal ends connected by rectangular faces; a cylinder has circular ends connected by a curved surface; a pyramid has a polygonal base and triangular faces meeting at a vertex; a cone has a circular base and a curved surface tapering to a point; a sphere is perfectly round with all points on the surface equidistant from the centre.

Note: The term 'hemisphere' is not required for this syllabus.

• Cube: 6 equal square faces.
• Cuboid: rectangular faces, opposite equal.
• Prism: two identical polygonal ends, rectangular sides.
• Cylinder: two circular ends, curved surface.
• Pyramid: polygon base, triangular faces meeting at apex.
• Cone: circular base, curved surface tapering to point.
• Sphere: all surface points equidistant from centre.

A circle is defined by its centre point and radius, the distance from the centre to any point on the circle. The diameter is twice the radius, passing through the centre.

The circumference is the distance around the circle, calculated by C = 2πr, where r is the radius.

A semicircle is half a circle formed by a diameter and the arc between its endpoints. A chord is a line segment joining two points on the circle, not necessarily passing through the centre. A tangent touches the circle at exactly one point and is perpendicular to the radius at the point of contact. An arc is a part of the circle's circumference. A sector is the area enclosed by two radii and the arc between them. A segment is the area between a chord and the corresponding arc.

• Centre: fixed point equidistant from all points on circle.
• Radius (r): distance from centre to circle.
• Diameter (d): twice the radius, passes through centre.
• Circumference (C): perimeter of circle, C = 2πr.
• Semicircle: half circle formed by diameter and arc.
• Chord: line segment joining two points on circle.
• Tangent: line touching circle at one point, perpendicular to radius.
• Arc: part of circumference between two points on circle's edge. Sector: area between two radii and arc. Segment: area between chord and arc.