Chapter 3.3 — Exercise 3.2 — SSSSD Construction

The General Method

For a quadrilateral ABCD with all four sides and diagonal AC given:

• Draw one side of the triangle that the diagonal completes — say AB — as a plain line segment.
• Swing an arc from each end of that side, using the diagonal's length from one end and the adjacent side's length from the other, to locate the third point of that first triangle.
• Join both ends of the base to that third point, completing the first triangle using the diagonal as one of its sides.
• Swing two more arcs from the two ends of the diagonal, using the two remaining side lengths as radii, to locate the fourth vertex.
• Join the fourth vertex to both ends of the diagonal, completing the quadrilateral.

Worked Example

Construct quadrilateral ABCD with AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and diagonal AC = 7 cm.

1. Draw line segment AB of length 4.5 cm.
2. With B as centre, draw an arc of radius 5.5 cm.
3. With A as centre, draw an arc of radius 7 cm, cutting the previous arc at C.
4. Join A–C and B–C to complete triangle ABC.
5. With A as centre, draw an arc of radius 6 cm.
6. With C as centre, draw an arc of radius 4 cm, cutting the previous arc at D.
7. Join A–D and C–D to complete quadrilateral ABCD.

What the Diagonal Is Really Standing In For

It helps to think of the diagonal in this method as a temporary stand-in for information you don't have yet — namely, the angle between the two sides it connects. Without a diagonal, four side lengths alone can't fix a quadrilateral's shape at all, because a four-sided figure can flex like a hinge even with every side length locked (imagine pushing on one corner of a rectangle to lean it into a slanted parallelogram shape — all four sides stay the same length throughout). Adding the diagonal locks that hinge in place by fixing the distance between two opposite corners directly, which is exactly enough extra information to remove the flex and pin down one specific shape, rather than a whole family of shapes that all happen to share the same four side lengths — picture that same rectangle leaning further and further until it's nearly flat, all without a single side length changing — the diagonal is precisely the measurement that stops that leaning at one exact point.

Why Building the Triangle First Works

This method only works because a triangle's shape is completely fixed once its three sides are known — there's no ambiguity left once AB, BC, and diagonal AC are all set, which is exactly why the first three steps produce one definite triangle rather than a range of possible ones. The second half of the construction repeats the identical logic on triangle ACD, using AC (now already drawn) as a known side. Every SSSSD construction is really two SSS triangle constructions sharing one common side — the diagonal.

Checking the Diagonal Doesn't Break the Triangle

Not every combination of three lengths can actually form a triangle — the sum of any two sides must exceed the third, a fact worth checking mentally before starting to construct. In the worked example, triangle ABC uses AB = 4.5 cm, BC = 5.5 cm and diagonal AC = 7 cm: 4.5 + 5.5 = 10, comfortably more than 7, so the arcs are guaranteed to intersect. If a diagonal were instead longer than the sum of the two sides meeting it, the two arcs would never cross at all, and no quadrilateral could be constructed from those particular measurements — a useful sanity check before picking up the compass.

Applying It to a Parallelogram and a Rhombus

• Parallelogram ABCD (AB = 6 cm, AD = 4.5 cm, BD = 7.5 cm): opposite sides equal gives CD = 6 cm and BC = 4.5 cm before construction starts, and the given diagonal BD anchors the first triangle exactly as AC did above.
• Rhombus NICE (NI = 4 cm, IE = 5.6 cm): all sides of a rhombus are equal, so NI = 4 cm alone fixes IC = CE = NE = 4 cm too, leaving IE as the one diagonal needed to construct the first triangle.

As in Exercise 3.1, the compass-and-arc steps themselves never change between the general quadrilateral and these special cases — only the reasoning that fills in the unstated side lengths beforehand does.

What the Diagonal Reveals About the Quadrilateral's Shape

Beyond making the construction possible, the diagonal you're given carries information about the quadrilateral's overall shape that the four sides alone don't. A short diagonal relative to the sides pulls the two triangles into a shape closer to a "kite" pinched at that diagonal, while a longer diagonal (closer to the sum of its two adjacent sides) stretches the quadrilateral out and flattens the angle at the vertices it connects. Noticing this before construction — comparing the diagonal's length to the two sides meeting at each of its endpoints — is a useful way to predict roughly what the finished shape will look like before a single arc is drawn.

Verifying the Second Triangle Independently

Once ABCD is complete, it's worth checking triangle ACD on its own terms, the same way you'd check any SSS construction: does AD really measure 6 cm, and CD really 4 cm, when checked against the finished drawing, with AC (7 cm) shared from the first triangle? Because the second triangle depends entirely on the first being accurate — it uses A and C exactly as the first triangle placed them — any small error in triangle ABC quietly carries forward into ACD without an obvious sign that anything went wrong until the final measurements are checked.

Where This Construction Typically Breaks Down

• Choosing the wrong first triangle. The diagonal you're given splits the quadrilateral into two specific triangles — build the one that uses two of your other four known sides first, not an arbitrary pairing.
• Losing track of which side belongs to which triangle once both triangles share the diagonal as a common side — label each side clearly against the vertex names before starting to draw.
• Assuming any three sides can replace the diagonal. The diagonal is a specific measurement (a distance between two opposite vertices), not interchangeable with any of the four sides.

One Diagonal Becomes Two

Splitting a quadrilateral into two triangles that share a common line is the same idea taken further in Exercise 3.3, where two diagonals are given instead of one. For the SSSSA method this exercise builds on, revisit Exercise 3.1.