GCSE Maths: Constructions, Loci and 3D Problems

Constructions and loci questions ask you to draw accurately and to describe sets of points in words; 3D questions ask you to picture a solid from different directions and, on the Higher tier, to find lengths and angles inside it. This material covers these parts of the geometry content in the GCSE subject content for England: the standard ruler-and-compass constructions, loci, the properties of 3D solids, plans and elevations, exact trigonometric values, and Pythagoras' theorem and trigonometry in three dimensions.

The quiz has thirteen questions, all described in words so no drawing is needed to answer them. You identify the loci of points at a fixed distance from a point and from a line segment, equidistant from two points and from two intersecting lines, recognise the arcs used to bisect an angle, and explain why the shortest distance from a point to a line is the perpendicular one. Further questions count the edges of a prism and a pyramid, match an upright cylinder to its plan and front elevation, and identify a solid from its plan and two elevations. The Higher-tier questions find the space diagonal of a cuboid and the angle between that diagonal and the base, and one question checks exact values such as $\sin {30}^{\circ }$ and $\tan {30}^{\circ }$. The wrong options come from typical errors: a rectangle instead of the rounded 'stadium' locus, a single bisector where there are two, and using one edge of the base instead of the base diagonal in a 3D triangle.

The flashcards summarise the loci that match each construction, the plan and elevation definitions, Euler's relation between faces, vertices and edges as a quick check, the 3D form of Pythagoras' theorem, and a table of exact sine, cosine and tangent values.

The printable written work has eight tasks to answer in words and working: the steps of the perpendicular bisector construction and of a perpendicular from a point to a line, a loci problem in a rectangular garden, the grazing area of a tethered goat, the views of a cylinder lying on its side, faces, edges and vertices of two further solids, the height and edge angle of a pyramid, and an exact-value calculation without a calculator.

Constructions, loci, plans and elevations and exact trigonometric values are common to both tiers; three-dimensional Pythagoras and trigonometry are Higher tier and are labelled. In the exam, constructions must be drawn with a ruler and compasses and the arcs left visible, so use the written work to practise describing each step clearly. The material suits Year 10 and Year 11 students.

• Describe and carry out the standard ruler-and-compass constructions
• Identify and describe loci, including regions that satisfy several conditions
• Count faces, edges and vertices of prisms and pyramids
• Draw and interpret plans and elevations of 3D solids
• Recall exact values of sine, cosine and tangent for key angles
• Use Pythagoras' theorem and trigonometry in 3D shapes (Higher)

Practice material written by Zestly, based on the DfE GCSE mathematics subject content (DFE-00233-2013), Geometry and measures G1, G2, G12, G13, G20 and G21.