Use distances, angles and constraints to see why a route or a drawing behaves the way it does.
6 original activities1 interactive model3 quick questions
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Across the floor or along the aisles?
Choose the width and depth of a fictional empty rectangular floor. Compare the straight-line lower bound with travel along two perpendicular edges.
Solid: edge routeDashed: diagonal
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Pause at each question before opening the explanation. Examples are invented for learning unless stated otherwise.
IDEA 01
The shortest line may not be a usable route
Is a 10-metre diagonal always the route a warehouse picker can take?
TRY / THINK
Set the floor to 6 metres by 8 metres above. Compare the diagonal with the two-edge route. Imagine adding a shelf across the diagonal.
Explore the explanationWHY IT WORKS
For a right triangle, the square of the long side equals the sum of the squares of the other two sides. This gives geometric distance. A usable route must also satisfy access and safety constraints.
Worked example
Diagonal = √(6² + 8²) = 10 m. Along two edges = 6 + 8 = 14 m. Shelves may prevent taking the 10 m route.
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An unconstrained optimum is a useful comparison, not automatically an executable plan.
IDEA 02
Three corners share one total
If two angles of a flat triangle are 45° and 70°, what remains?
TRY / THINK
Draw your own triangle on paper, label two angles, and subtract them from 180°. Then test a different pair whose sum is less than 180°.
Explore the explanationWHY IT WORKS
In ordinary flat Euclidean geometry, the interior angles of a triangle total 180°. The rule needs a positive third angle. It does not apply unchanged to triangles drawn along curved surfaces such as a globe.
Worked example
180° − 45° − 70° = 65°. Angles of 100° and 100° cannot belong to an ordinary flat triangle.
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Check feasibility before optimizing. Some requested combinations cannot fit the rules.
IDEA 03
Find a centre using equal distances
Where are all points that are equally far from the ends of a line segment?
TRY / THINK
Draw two chords across a circle. For each chord, mark its midpoint and draw a line at a right angle through that midpoint.
Explore the explanationWHY IT WORKS
The perpendicular bisector of a chord passes through the circle centre because that centre is equally far from both endpoints. Two non-parallel bisectors meet at the centre. Hand drawing introduces measurement error.
Worked example
A horizontal chord centred on x = 0 has a vertical perpendicular bisector. A second tilted chord supplies another line and locates their intersection.
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Two independent constraints can reveal a location that either one alone leaves uncertain.
IDEA 04
Two distances, one fixed total
Does every oval have the same geometric definition?
TRY / THINK
Imagine two fixed points, A and B. Keep the distance from A to a moving point plus its distance to B fixed at 10 cm. Sketch possible positions.
Explore the explanationWHY IT WORKS
An ellipse is defined by that constant sum, with the sum greater than the distance between its two fixed points, called foci. A circle is the special case where the foci coincide. Not every hand-drawn oval is an ellipse.
Worked example
If the foci are 6 cm apart and the constant sum is 10 cm, the ellipse has semi-major axis 5 cm and semi-minor axis √(25 − 9) = 4 cm.
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Different shapes impose different distance constraints. A pleasing drawing is not enough to verify them.
IDEA 05
A close-looking path can still be longer
Will a staircase route eventually have the same length as a diagonal if its steps become tiny?
TRY / THINK
Draw a 6 m by 8 m rectangle and cross it using only rightward and upward moves. Split those moves into smaller steps without moving backward.
Explore the explanationWHY IT WORKS
Every such path still accumulates 6 m horizontally and 8 m vertically. Its length stays 14 m, even when it looks close to the 10 m diagonal. Visual closeness alone does not imply equal length.
Worked example
With 100 steps, each horizontal piece can be 0.06 m and each vertical piece 0.08 m: 100 × (0.06 + 0.08) = 14 m.
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A smoother-looking chart or finer time bucket does not guarantee that an underlying quantity has changed.
IDEA 06
Stretching a network does not add a road
If two route diagrams look different, can they describe the same connections?
TRY / THINK
Draw three locations with links A–B and B–C. Move the locations on paper without adding a link. Ask whether A now connects directly to C.
Explore the explanationWHY IT WORKS
A network separates connections from the exact shape of the drawing. Stretching or bending a line does not create a new connection. Distance and travel time still require extra information beyond connectivity.
Worked example
A–B–C requires two links to reach C from A. Moving C closer on the page does not create an A–C link.
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A supply-network picture should distinguish real relationships from decorative line placement.
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The explanations, examples, questions and diagrams on this page were written for Supply Chain Ideas. The underlying principles are established science and mathematics, not claims of invention. The following references support further study; their text and illustrations are not reproduced.
The ellipse ↗ — OpenStax, College Algebra 2e. Educational reference; accessed 3 November 2026.
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