Geometrical terms Revision Notes

A point represents an exact position in space with no length, width, or height. For example, the corner of a sheet of paper can be considered a point.

A vertex is a point where two or more lines or edges meet, such as the corner of a triangle.

A line extends infinitely in both directions and has no thickness. For instance, a straight railway track can be modelled as a line segment, but mathematically a line continues without end in both directions beyond the segment shown in a diagram.

• Plane: a flat, two-dimensional surface extending infinitely, like an idealised tabletop.
• Parallel lines: two lines in the same plane that never meet, e.g., opposite edges of a rectangular table.
• Perpendicular lines: lines that intersect at a right angle (90°), such as the corner edges of a book.
• Perpendicular bisector: a line that cuts a segment into two equal parts at 90°, for example, the line dividing a segment AB into two equal halves at a right angle.

Angles measure the amount of turn between two lines meeting at a point (vertex).

A right angle measures exactly 90°. For example, the corner of a square is a right angle.

An acute angle is less than 90°, such as 45° in an equilateral triangle's altitude when split appropriately (though all angles in an equilateral triangle are 60°, which is acute).

• Obtuse angle: greater than 90° but less than 180°, e.g., 120° in some irregular quadrilaterals.
• Reflex angle: greater than 180° but less than 360°, such as 270° in a clock showing 9 o'clock.
• Interior angles: angles inside a polygon, e.g., the angles inside a triangle.
• Exterior angles: angles formed outside a polygon when a side is extended.
• Bearing: a direction measured in degrees clockwise from north (0°). For example, a bearing of 045° points northeast.

Triangles are classified by side lengths and angles:

Equilateral: all three sides equal and all angles 60°.

Isosceles: two sides equal and two angles equal, e.g., a triangle with sides 5 cm, 5 cm, and 8 cm.

• Scalene: all sides and angles different, such as a triangle with sides 3 cm, 4 cm, and 5 cm.
• Right-angled: one angle exactly 90°, like the classic 3-4-5 triangle.
• Special quadrilaterals include:
• Square: four equal sides and four right angles.
• Rectangle: opposite sides equal and four right angles.
• Kite: two pairs of adjacent equal sides.
• Rhombus: four equal sides with opposite equal angles.
• Parallelogram: opposite sides parallel and equal in length, opposite angles equal, e.g., a leaning rectangle shape without right angles but with 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).

For example, a regular hexagon has six equal sides and six equal interior angles of 120° each.

• The sum of interior angles of an n-sided polygon is (n-2) × 180°. For a pentagon, sum = (5-2)×180° = 540°.
• Regular polygons have each interior angle equal to [(n-2)×180°]/n. For a regular octagon, each interior angle = (8-2)×180°/8 = 135°.

Solids have length, width, and height. Common solids include:

Cube: six square faces, all edges equal, e.g., a dice.

Cuboid: six rectangular faces, edges can differ, like a shoebox.

• Prism: solid with two identical polygonal faces parallel and other faces rectangular, e.g., a triangular prism.
• Cylinder: two parallel circular faces connected by a curved surface.
• Pyramid: a base polygon with triangular faces meeting at a vertex, e.g., a square pyramid.
• Cone: circular base tapering to a point.
• Sphere: perfectly round 3D shape, like a ball.
• Hemisphere: half of a sphere.
• Frustum: a pyramid or cone with the top cut off parallel to the base.
• Faces: flat surfaces of a solid, e.g., a cube has 6 faces; edges: where two faces meet; surfaces: all faces combined.

A circle is the set of points equidistant from a centre point.

Key parts include: radius (distance from centre to circumference), diameter (twice the radius), and circumference (distance around the circle).

For example, if radius r = 5 cm, diameter d = 2 × 5 = 10 cm, circumference C = 2πr ≈ 31.42 cm.

• Semicircle: half of a circle formed by a diameter.
• Chord: a line segment joining two points on the circle, not necessarily through the centre.
• Tangent: a line touching the circle at exactly one point, perpendicular to the radius at the point of contact.
• Major arc: the longer arc between two points on the circle.
• Minor arc: the shorter arc between two points.
• Sector: the 'pie-slice' area bounded by two radii and the arc between them.
• Segment: the area between a chord and the corresponding arc.