Polygons, special quadrilaterals and 3D plans

Geometry and measures • Properties and constructions

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Pyramid (n-sided base) - face/edge/vertex counts

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A pyramid with an n-sided base has n + 1 faces, 2n edges and n + 1 vertices, because lateral faces meet at a single apex.

Key concepts

What you'll likely be quizzed about

Polygons and interior angles

A polygon is a closed plane figure formed by a finite number of straight line segments meeting only at their endpoints. A polygon with n sides has an interior-angle sum equal to (n − 2) × 180° because the polygon can be partitioned into (n − 2) triangles by drawing diagonals from one vertex. A regular polygon has all sides and all interior angles equal. The measure of each interior angle in a regular n-sided polygon equals ((n − 2) × 180°) ÷ n, which follows directly from the interior-angle sum formula.

Special quadrilaterals: definitions and limiting factors

Square: all four sides equal and all interior angles 90°, which causes diagonals to be equal, perpendicular, and to bisect angles. Rectangle: opposite sides equal and all angles 90°, which causes diagonals to be equal and to bisect each other. Parallelogram: opposite sides equal and parallel, which causes opposite angles to be equal and diagonals to bisect each other. Rhombus: all four sides equal and opposite sides parallel, which causes diagonals to be perpendicular and to bisect opposite angles. Kite: two pairs of adjacent equal sides and exactly one pair of opposite angles equal, which causes diagonals to be perpendicular with one diagonal bisecting the other. Trapezium: exactly one pair of parallel sides, which limits parallel-side reasoning to that single pair and prevents full parallelogram properties from applying.

Diagonals and consequences

Diagonal properties follow from side and angle constraints and enable angle and length deductions. In a parallelogram, diagonals bisect each other; therefore, each diagonal divides the parallelogram into two congruent triangles, which causes corresponding sides and angles to match. In a rhombus, diagonals are perpendicular and bisect the interior angles, which causes right-angled triangle relationships within the rhombus useful for length calculations. In rectangles and squares, equal diagonals create isosceles triangles on either side of each diagonal, which causes equal base angles and enables use of symmetry in constructions and proofs.

Triangles and other plane figures

Triangle angle sum equals 180° because the three interior angles form a straight line when a parallel is drawn to one side from the opposite vertex. Classification by sides (equilateral, isosceles, scalene) or by angles (acute, right, obtuse) dictates symmetry and congruence properties. An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has three equal sides and three 60° angles. Other plane figures (e.g., regular polygons, sectors) follow the same partition and symmetry principles. Angle-chasing and triangle congruence (SSS, SAS, ASA, RHS) provide methods to derive unknown measures from given constraints.

Plans and elevations (orthographic projection)

A plan is a view from above showing the top outline and positions of features. An elevation is a view from the front or side showing vertical faces and heights. Orthographic projection treats views independently and aligns corresponding points using projection lines perpendicular to the projection plane; therefore, a feature that is vertically aligned in 3D appears at the same horizontal position in plan and front elevation. Construction of plans and elevations requires projecting key vertices and edges onto the chosen view planes. Interpretation of existing views requires cross-referencing matching dimensions and shapes in the plan and elevations to reconstruct possible 3D arrangements.

Properties of 3D shapes: faces, edges, vertices

Cube: six square faces, twelve edges, eight vertices; all faces congruent and all edges equal, which causes strong symmetry and identical orthographic projections along principal axes. Cuboid: six rectangular faces, twelve edges, eight vertices; opposite faces are equal, which causes pairwise congruent projections. Prism with an n-sided polygon base: n + 2 faces (including two congruent polygonal faces and n rectangular lateral faces), 3n edges, and 2n vertices; lateral faces are rectangles if the prism is right. Cylinder: two circular faces and one curved surface, with no true edges or vertices in the polyhedral sense; the curved surface causes a continuous lateral boundary. Pyramid with an n-sided base: n + 1 faces (n triangular lateral faces and one base), 2n edges, and n + 1 vertices; lateral faces meet at a single apex. Cone: one circular face and one curved surface with a single apex and no vertices in the polyhedral sense beyond the apex; the curved surface meets the base along a continuous rim. Sphere: one curved surface only, with no faces, edges or vertices.

Key notes

Important points to keep in mind

Use (n − 2) × 180° to find the interior-angle sum of any n-sided polygon.

A shape must meet all defining conditions to be classified as a special quadrilateral (e.g., all sides equal and all angles 90° required for a square).

Diagonals provide congruent triangles or right angles; check diagonal properties to deduce lengths and angles.

Match plan and elevation features by projecting corresponding vertices with perpendicular lines between view planes.

Count faces, edges and vertices for prisms and pyramids using the base polygon size: prism edges = 3n, vertices = 2n; pyramid vertices = n + 1.

Distinguish flat faces from curved surfaces: cylinders, cones and spheres include curved surfaces and lack polyhedral edges.

State whether a property is necessary, sufficient, or both (e.g., all sides equal is necessary for a rhombus but not sufficient alone for a square).

Draw sketches of plan and front/side elevations with consistent dimensions to avoid mismatched reconstructions.

Use symmetry to simplify angle and length calculations in regular polygons and special quadrilaterals.

Label vertices and projection lines when constructing orthographic views to maintain clarity and prevent misalignment.