Geometry & Measures
Grade 4-6Constructions & Loci
Constructions and loci combine practical compass-and-straight-edge skills with the ability to describe, in precise mathematical language, exactly where a moving point can lie. This lesson covers the three standard constructions, describing single and combined loci fully, and finding the area or perimeter of common locus shapes such as a goat grazing on a rope.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The three standard constructions
The perpendicular bisector of a line segment is constructed by drawing an arc from each end with the same radius (more than half the length of the segment), so the arcs cross above and below the line, then joining these two crossing points. The angle bisector of an angle is constructed by drawing an arc from the vertex across both arms, then drawing equal arcs from those two points so they intersect, and joining the vertex to that intersection. A perpendicular to a line from a given point uses similar arcs from that point to find two equal points on the line, then two more arcs from those points to find where to draw the perpendicular.
All of these constructions must be shown using compass arcs left visible on the diagram, since marks are awarded for the correct method as well as the correct final line.
Simple loci
A locus is the set of all points that satisfy a given rule. A point a fixed distance from a single point traces a circle. A point a fixed distance from a straight line traces two parallel lines (one on each side). A point equidistant from two fixed points traces the perpendicular bisector of the line joining them. A point equidistant from two lines traces the angle bisector between them.
A point that is always exactly 5 cm from a fixed point A traces a circle of radius 5 cm, centred on A. A point that is always the same distance from two fixed points A and B traces the perpendicular bisector of AB.
Combined loci and regions
Many questions combine two conditions, so the point must satisfy both loci at once, meaning it lies in the region where the two individual loci overlap, or lies at a small number of intersection points.
If point P lies closer to A than to B, and also within 4 cm of C, then P must lie on the A-side of the perpendicular bisector of AB, and inside a circle of radius 4 cm centred on C, so P lies in the overlap of these two regions.
Area and perimeter of locus regions
Real-world locus problems, such as an animal tied to a rope, often ask for the area or perimeter of the region reached. A wall or fence can block part of a full circle, leaving only a fraction of it, found using the angle available in the same way as a sector.
A goat is tied to a corner of a barn with an 8 m rope, and the barn wall blocks a 90° corner, leaving 270° available. The area grazed is (270 ÷ 360) × π × 8² = 0.75 × 3.14 × 64 = 150.72 m².
Worked Examples
Three exam-style questions, fully solved.
A point moves so that it is always exactly 5 cm from a fixed point A. Describe fully the locus of the point.
Easy- 1.Identify the rule: a fixed distance from a single fixed point
Answer: A circle of radius 5 cm, centred on A
A goat is tied to a corner of a rectangular barn, using a rope of length 8 m. The rope cannot cross the barn walls, and both walls at that corner are longer than 8 m. Find the area the goat can graze. Use π = 3.14.
Medium- 1.Recognise that the barn wall blocks a 90° corner, leaving a 270° region available
- 2.Find the fraction of the full circle available: 270 ÷ 360
- 3.Find the area: 0.75 × π × 8²
Answer: 150.72 m²
A point moves so that it is always exactly 4 cm from a straight line segment XY, where XY has length 10 cm (unlike an infinite line, the locus must curve around each end of the segment). Describe fully the shape of the locus, and find its total perimeter. Use π = 3.14.
Hard- 1.Describe the shape: two straight lines parallel to XY, 4 cm from it, joined by a semicircular arc of radius 4 cm at each end
- 2.Find the length of the two straight parts: 2 × 10 = 20 cm
- 3.Find the length of the two semicircular arcs, which together form a full circle of radius 4 cm: 3.14 × (2 × 4) = 25.12 cm
- 4.Add the straight and curved parts together: 20 + 25.12
Answer: 45.12 cm
Avoid These
The most common mistakes students make.
Erasing or not showing the compass arcs used in a construction, which loses method marks even if the final line is drawn in the correct place.
Confusing the perpendicular bisector of a line segment (equidistant from two points) with the angle bisector of an angle (equidistant from two lines).
Forgetting that a locus a fixed distance from a line segment (rather than an infinite line) must curve around each end, rather than staying as two infinite parallel lines.
When a wall or fence blocks part of a circle, using the full 360° instead of subtracting the blocked angle first.
Describing only one of the two conditions in a combined locus question, rather than identifying the overlap region that satisfies both.
FAQ
Questions parents and students ask.
Before this topic, make sure you know
What to learn next
Want a plan built around your child specifically?
Get our free 8-video course, or book a free Roadmap Call for a personalised plan.