Measure the length of the four sides of the quadrangle using the ruler. Let the measurements be represented by P, Q, R and S. Starting with P, label the sides in a clockwise direction, ending with S being to the left of P.
Draw a diagonal line that joins the corner between P and Q to the corner between R and S. Measure the length of this diagonal line, D. This diagonal divides the quadrangle into two triangles: Triangle 1 with sides P, D and S, and Triangle 2 with sides Q, R and D.
Find Angle d in Triangle 1 using the Cosine Rule. Angle d is the angle between sides P and S found directly opposite to side D. The Cosine Rule applied to Triangle 1 is described by the formula: 2 x P x S x cos(d) = P^2 + S^2 -- D^2. It follows that d = arccos{(P^2 + S^2 -- D^2)/( 2 x P x S)}. Arccos, or arccosine, is the inverse of the cosine function. For example, if P = 3, S = 4, and D = 5, then angle d = 90 degrees = arccos{(3^2 + 4^2 -- 5^2)/( 2 x 3 x 4)} = arccos{(P^2 + S^2 -- D^2)/( 2 x P x S)}.
Note: "^" represents a superscript 2.
Find Angle d in Triangle 2 using the Cosine Rule. Angle d is the angle between Sides Q and R found directly opposite to Side D. The Cosine Rule applied to Triangle 2 is described by the formula: 2 x Q x R x cos(d) = Q^2 + R^2 -- D^2. It follows that d = arccos{(Q^2 + R^2 -- D^2)/( 2 x Q x R)}. For example, if Q = 4, R = 3, and D = 5, then Angle d = 90 degrees = arccos{(4^2 + 3^2 -- 5^2)/( 2 x 4 x 3)} = arccos{(Q^2 + R^2 -- D^2)/( 2 x Q x R)}.
Find Angle s in Triangle 1 using the Cosine Rule. Angle s is the angle between Sides P and D found directly opposite to Side S. The Cosine Rule applied to Triangle 1 is described by the formula: 2 x P x D x cos(s) = P^2 + D^2 -- S^2. It follows that s = arccos{(P^2 + D^2 -- S^2)/( 2 x P x D)}. For example, if P = 3, S = 4, and D = 5, then Angle s = 53.13 degrees = arccos{(3^2 + 5^2 -- 4^2)/( 2 x 3 x 5)} = arccos{(P^2 + D^2 -- S^2)/( 2 x P x D)}.
Find Angle r in Triangle 2 using the Cosine Rule. Angle r is the angle between sides Q and D found directly opposite to side R. The Cosine Rule applied to Triangle 2 is described by the formula: 2 x Q x D x cos(r) = Q^2 + D^2 -- R^2. It follows that r = arccos{(Q^2 + D^2 -- R^2)/( 2 x Q x D)}. For example, if R = 3, Q = 4, and D = 5, then angle r = 36.87 degrees = arccos{(4^2 + 5^2 -- 3^2)/( 2 x 4 x 5)} = arccos{(Q^2 + D^2 -- R^2)/( 2 x Q x D)}.
Add Angle s from Triangle 1 to Angle r from Triangle 2 to obtain the total angle formed between Sides P and Q in Quadrangle PQRS. For example, if Angle s from Triangle 1 is 53.13 degrees and Angle r from Triangle 2 is 36.87 degrees, then angle formed between Sides P and Q in Quadrangle PQRS is: 90 degrees = 53.13 + 36.87.
Find the fourth angle in the quadrangle. The fourth angle is the angle between Sides S and R of quadrangle PQRS. The interior angles of a quadrangle add up to 360 degrees, so the angle between Sides S and R is the difference between 360 and the sum of the other 3 angles of the quadrangle. For example, if Quadrangle PQRS has sides P = R = 3 and S = Q = 4, then the angle between S and R is 90 degrees = 360 -- (90 + 90 + 90) = 360 -- [(angle between S and P) + (angle between Q and R) + (angle between P and Q)].