1. Drag the shape
A parallelogram has two side lengths, a and b, and an angle θ between them. It also has two diagonals, c and d. Bend the shape and watch the diagonals change. Then look at the last two lines of the table.
Drag corner D to change b and θ. Drag corner B to change a. Keyboard: Tab to a corner, then use the arrow keys.
| Sides | a = , b = , θ = |
|---|---|
| Diagonals (measured) | c = , d = |
| Squares | c² = , d² = |
| c² + d² | |
| 2(a² + b²) |
Nothing is faked. The page places the four corners, then measures the diagonals with the distance formula. It never uses the identity to fill in the table. The two totals agree because geometry makes them agree.
2. Why the totals always match
The proof needs only the Law of Cosines and one fact about cosine. Go through the five steps. The shaded triangle is the one each step uses, and the numbers come from your shape in step 1.
3. Two special shapes
A rectangle and a rhombus are both parallelograms, so the law holds for them too. Each one adds a bonus fact.
4. Why it works
The correction terms cancel. The Law of Cosines is Pythagoras plus a correction, −2ab cos θ. In a parallelogram the two diagonals face angles that add to 180°, so one diagonal gets the correction subtracted and the other gets the same amount added. Push the shape flatter and one diagonal shrinks exactly as much (in squares) as the other grows.
With arrows. Call the two sides from corner A the arrows u and v. The diagonals are u + v and v − u. Squaring their lengths gives |u|² + |v|² ± 2 u·v. Add the two and the ± terms cancel again.
In words. In any parallelogram, the squares on the two diagonals add up to the squares on all four sides: c² + d² = a² + b² + a² + b².
A cousin. Each diagonal cuts the other in half. Halve d and the same equation becomes Apollonius's theorem about the median of a triangle: a² + b² = 2(d/2)² + c²/2.
Where it fits. This page goes with Trigonometry Lesson 7.4, The Law of Cosines. Steps 2 and 3 of the proof use the identity cos(180° − θ) = −cos θ, from the unit circle in Unit 3.
5. Check yourself
Try each one on paper first. The short answers are hidden until you reveal them. Full worked solutions are in the answer key PDF for the parent or teacher.
1. A parallelogram has sides 5 and 8 with a 60° angle between them. Find both diagonals.
2. A parallelogram has sides 6 and 9, and one diagonal is 7. Find the other diagonal without finding any angle. Then find the angle opposite the 7.
3. A rhombus has sides of 13 cm, and one diagonal is 10 cm. How long is the other diagonal?
4. A carpenter measures a rectangular frame with sides of 9 ft and 12 ft. Use the law to find each diagonal, and say which earlier theorem you just used.
5. Someone claims a parallelogram has sides 4 and 7 and diagonals 5 and 10. Can that be right?