GCSE Foundation Maths: Constructions and Loci (Non-Calculator)

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GCSE Foundation Maths: Constructions and Loci (Non-Calculator): 10 questions, marked instantly with every answer explained. Free, no sign-up, no adverts in the test. This test is for GCSE (Years 10 and 11, ages 14–16). In the US, that's about Grades 9 and 10.

Before you start

10 questions on constructing triangles, perpendicular bisectors, angle bisectors and loci, picked at random from a bank of 27, so every try is different. Non-calculator: work these out without a calculator. Type a number or a short answer, or choose an answer. To draw a shape or an angle, drag the corners with the on-screen ruler and protractor, or use the arrow keys. Questions with units tell you which unit to use. Choose Practice to see each answer explained as you go, or Test to see your score and explanations at the end.

10 questions – a new mix each time you try
Drawing questions work with a mouse, a finger or the keyboard
Plus written questions for your teacher or a parent to check

Choose how to do it Test – answers and explanations at the end; no time limit Practice – see the answer after each question; no time limit

No sign-up and no adverts. Your answers stay on this device. Nothing you type in the test is sent to us – unless you choose to report a problem with a question.

How it is marked: by Classfern's own checks, in your browser, straight away. Drawings are marked by what they show: any correct drawing counts (a shape started from any corner, any net that folds up), within the small allowance the question shows. Spot a mistake? Use the Report button.

Written questions – check with your teacher

Classfern does not mark these questions, and they are not part of your score above. They need a written answer or a drawing that a person has to read. Answer each one in the box or on paper, then ask your teacher or a parent to check it with the marking guide. What you type here stays on this device and is not sent anywhere.

Draw a circle of radius 6 cm. Using only a ruler and compasses, construct a square with all its corners on the circle. Leave in your construction lines.
3 marks
Do the drawing on paper with a ruler and compasses. Notes (optional):
Marking guide (for your teacher or a parent)

Give a mark for each point the answer shows. Tick as you go:
- Circle of radius 6 cm (to within 2 mm). (1 mark)
- Two perpendicular diameters constructed (a diameter and its perpendicular bisector). (1 mark)
- The four ends of the diameters joined to give a square. (1 mark)
One answer that would get full marks:
Draw a diameter. Construct its perpendicular bisector through the centre to get a second diameter at 90°. Join the four points where the diameters meet the circle.
Draw a straight line l and mark a point P about 3 cm from it. Using only a ruler and compasses, construct the perpendicular from P to the line l. Leave in your construction arcs.
3 marks
Do the drawing on paper with a ruler and compasses. Notes (optional):
Marking guide (for your teacher or a parent)

Give a mark for each point the answer shows. Tick as you go:
- An arc centred on P cutting l at two points. (1 mark)
- Equal arcs from those two points crossing on the other side of l (or on the same side, away from P). (1 mark)
- A line from P through the crossing point, meeting l at 90° (to within 2°). (1 mark)
One answer that would get full marks:
P is about 3 cm from l. With the compass point on P, draw an arc that cuts l at two points. From each of these points draw arcs of the same radius that cross on the far side of l. Join P to the crossing point: this line is perpendicular to l.
Draw an angle of 105° with a protractor. Then, using only a ruler and compasses, construct the bisector of the angle. Leave in your construction arcs.
3 marks
Do the drawing on paper with a ruler and compasses. Notes (optional):
Marking guide (for your teacher or a parent)

Give a mark for each point the answer shows. Tick as you go:
- Angle drawn as 105° (to within 2°). (1 mark)
- An arc centred on the vertex cuts both arms; equal arcs from those two points cross inside the angle. (1 mark)
- A line from the vertex through the crossing point, making two angles of 52.5° (to within 2°). (1 mark)
One answer that would get full marks:
Draw the 105° angle. Put the compass point on the vertex and draw an arc cutting both arms. From each of those two points draw arcs of the same radius so they cross inside the angle. Join the vertex to the crossing point: each half is 52.5°.

GCSE-style practice by Classfern. Not an official test and not affiliated with or endorsed by any exam board, Ofqual or the Department for Education.

Curriculum: the GCSE subject content covered here – G15 – measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings; G2 – use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); use these to construct given figures and solve loci problems; know that the perpendicular distance from a point to a line is the shortest distance to the line.

Contains public sector information licensed under the Open Government Licence v3.0. Curriculum statements from: Department for Education, Mathematics GCSE subject content and assessment objectives, © Crown copyright 2013.

Learn it first: Constructions and loci (Year 8)

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