BTT Mathematics / Primary Mathematics Learning Hub / Convex Hulls
The convex hull of a finite set of points is the smallest convex shape containing every point. Imagine stretching an elastic loop around nails on a board: when released, the loop catches the outermost nails and forms the convex hull.
A set is convex when the straight segment joining any two points of the set stays inside the set. Hull vertices are extreme points: they cannot be written as points strictly inside a segment joining other points of the hull.
Points strictly inside the hull do not affect its boundary. A point lying on a straight hull edge may be on the boundary without being an extreme vertex.
This guide is Primary Mathematics enrichment. It extends Geometry, Angles, Symmetry and Coordinates, Pick’s Theorem, Lattice Points and Coordinate Area and Optimisation, Minimum Moves and Efficient Constructions.
Formal computational-geometry algorithms are optional extension. The practice uses small coordinate sets so the hull can be checked visually and numerically.
Convexity · Extreme points · Turn tests · Build a hull · Perimeter and area · Interior and boundary points · 24 questions · Worked answers
1. Convex shapes contain every chord between their points
A rectangle, triangle and regular hexagon are convex. A shape with an inward notch is not convex because two points can be chosen whose joining segment leaves the shape.
Worked example A: Rectangle point set
Points A=(0,0), B=(4,0), C=(4,3), D=(0,3) form a convex rectangle.
Worked example B: Add an interior point
Add P=(2,1). The convex hull remains A-B-C-D. P lies inside and does not change the enclosure.
Worked example C: Add an outside point
Add Q=(5,1). The hull must expand to include Q, so Q becomes a new extreme point.
Worked example D: Collinear boundary point
E=(2,0) lies on segment AB. It is on the hull boundary but is not a new corner; A and B remain the extreme endpoints of that edge.
2. Extreme points determine the hull polygon
If all points are strictly inside a triangle formed by three outer points, the hull is that triangle regardless of how many interior points are added.
Worked example E: Triangle with interior data
For A=(0,0), B=(6,0), C=(0,4), plus P=(1,1) and Q=(2,1), the hull vertices are A,B,C.
Worked example F: Rectangle with edge points
For rectangle corners plus E=(2,0) and F=(4,1), E and F lie on hull edges but do not create new extreme directions. The corner set remains sufficient.
Worked example G: Every point extreme
For points forming a convex pentagon with no three consecutive hull points collinear, every listed point is a hull vertex.
Worked example H: One point removed
Removing an interior point never changes the hull. Removing an extreme point may change the boundary because the next outermost points must be connected.
3. A signed turn test distinguishes left, right and collinear
For A=(x1,y1), B=(x2,y2), C=(x3,y3), compute
T=(x2−x1)(y3−y1)−(y2−y1)(x3−x1).
T>0 gives a counterclockwise turn, T<0 a clockwise turn and T=0 collinearity.
Worked example I
A=(0,0), B=(4,0), C=(4,3). T=4×3−0×4=12>0, so the turn is counterclockwise.
Worked example J
A=(0,0), B=(4,0), C=(4,−2). T=4×(−2)=−8, so the turn is clockwise.
Worked example K
A=(0,0), B=(2,1), C=(4,2). T=2×2−1×4=0, so the points are collinear.
Why turn tests matter
When walking around a convex hull in counterclockwise order, the boundary should keep making non-clockwise turns. A clockwise bend indicates an inward dent or incorrectly ordered point.
4. Build the hull by keeping only outward points
For a small set, plot the points, identify the leftmost and rightmost extremes, then trace the outer boundary while rejecting interior dents.
Worked example L: Five-point set
Points (0,0),(4,0),(4,3),(0,3),(2,1) have hull (0,0),(4,0),(4,3),(0,3).
Worked example M: Add a rightward extreme
Add (5,1). The hull becomes (0,0),(4,0),(5,1),(4,3),(0,3).
Worked example N: Add an edge midpoint
Add (2,0) to the rectangle set. The hull polygon does not need a new vertex because the point lies between existing hull vertices.
Worked example O: All collinear points
If every point lies on one line, the convex hull degenerates to the line segment joining the two extreme endpoints. There is no positive-area hull polygon.
5. Once the hull is known, perimeter and area use ordinary geometry
For the 4-by-3 rectangle hull, perimeter=2(4+3)=14 and area=12.
Worked example P: Right-triangle hull
Hull vertices (0,0),(6,0),(0,8) form a right triangle. Area=24. Side lengths are6,8,10, so perimeter=24.
Worked example Q: Adding an interior point
Adding (1,1) to the triangle changes neither perimeter nor area because the hull vertices are unchanged.
Worked example R: Adding an exterior point
If a point lies beyond one hull edge, the new hull may have larger area and a different perimeter. The effect must be computed after the boundary is updated.
6. Boundary membership and extremeness are different questions
A point may be strictly inside the hull, lie on a hull edge, or be an extreme hull vertex.
Worked example S: Interior point
P=(2,1) inside the 4×3 rectangle is neither boundary nor extreme.
Worked example T: Edge point
E=(2,0) is boundary but not an extreme vertex because it lies between A=(0,0) and B=(4,0).
Optimization consequence
For many “outer boundary” problems, interior points can be ignored after the hull is identified. The hull compresses a large point set into the smaller set that actually controls enclosure.
7. Practice: 24 original questions
Questions 1–8: Convexity and extreme points
1. Is a rectangle convex?
2. Does adding an interior point change a convex hull?
3. In rectangle corners plus P=(2,1), is P extreme?
4. Is E=(2,0) on the boundary of the rectangle from (0,0) to (4,3)?
5. Is E=(2,0) an extreme vertex?
6. For triangle A=(0,0),B=(6,0),C=(0,4) plus P=(1,1), list hull vertices.
7. What happens to the hull if an interior point is removed?
8. What can happen if an extreme point is removed?
Questions 9–16: Turn tests and hull construction
9. Compute T for A=(0,0),B=(4,0),C=(4,3).
10. Classify that turn.
11. Compute T for A=(0,0),B=(4,0),C=(4,−2).
12. Classify that turn.
13. Compute T for A=(0,0),B=(2,1),C=(4,2).
14. What does T=0 mean?
15. Find the hull of (0,0),(4,0),(4,3),(0,3),(2,1).
16. If all points are collinear, what is the convex hull?
Questions 17–24: Perimeter, area and boundary status
17. Find the perimeter of the 4×3 rectangle hull.
18. Find its area.
19. Hull vertices (0,0),(6,0),(0,8): find area.
20. Find its perimeter.
21. Does adding (1,1) to that triangle change its hull area?
22. Classify P=(2,1) in the 4×3 rectangle: interior, boundary non-vertex or extreme vertex.
23. Classify E=(2,0) in the same rectangle.
24. Why can interior points often be ignored in an enclosure calculation after the hull is known?
8. Worked answers
Answers 1–8
1. Yes.
2. No.
3. No.
4. Yes.
5. No. It lies between two extreme endpoints on one straight hull edge.
6. A,B,C.
7. Nothing changes.
8. The hull boundary may change.
Answers 9–16
9. 12.
10. Counterclockwise.
11. −8.
12. Clockwise.
13. 0.
14. The three points are collinear.
15. (0,0),(4,0),(4,3),(0,3).
16. The line segment joining the two extreme endpoints.
Answers 17–24
17. 14 units.
18. 12 square units.
19. 24 square units.
20. 24 units. Side lengths6,8,10.
21. No.
22. Interior.
23. Boundary non-vertex.
24. The hull boundary and enclosure measurements are determined by the extreme outer points; strictly interior points do not alter that boundary.
9. Teaching and transfer
If a learner joins points in input order, ask whether the result contains an inward dent. The hull is an outer enclosure, not merely a polygon through every point.
When boundary points are called extreme automatically
Use a midpoint on a straight hull edge. It is on the boundary but can be removed without changing the hull corners.
When turn signs are memorised
Plot one positive and one negative example. The sign records direction of turn, not distance or area by itself.
When an interior point distracts the solution
Mark it only after the hull is drawn. Enclosure problems become simpler once the controlling extreme points are separated from the rest.
When the hull is measured before it is correct
Boundary identification comes first. Perimeter and area calculations are only meaningful after the outer polygon has been verified.
Connect to optimisation
The hull is a data-reduction step: many interior points can be discarded for outer-enclosure tasks. This is a geometric example of keeping only information that controls the objective.
Continue through this enrichment collection
For constraint grids, use Latin Squares, Sudoku Structure and Constraint Propagation. For one-sided surfaces, use Möbius Strips, One-Sided Surfaces and Topology. For codeword distance, use Hamming Distance, Error-Correcting Codes and Parity Checks.
Return to the BTT Primary Mathematics Learning Hub.
Original enrichment guide with 24 original practice questions and separate worked answers. Coordinate hull examples are finite and small enough for direct plotting and verification.