The two sliders on the left set the base and the height. The triangle and the parallelogram on the right are both wired to the same two sliders, so moving either one reshapes both figures and updates both areas.
Compare the numbers
The board starts with a base of 6 and a height of 4. The parallelogram is 6 × 4 = 24. The triangle is 6 × 4 ÷ 2 = 12 — exactly half of 24.
Change the base or height — it stays half
Set the base to 9 and you get 36 for the parallelogram and 18 for the triangle. Base 5, height 3 gives 15 and 7.5.
When the parallelogram comes out odd, the triangle is still exactly half; it just ends in .5. However you set the base and height, the triangle stays at half the parallelogram.
Why half?
Draw a line across a parallelogram from one corner to the opposite corner. That line is called a diagonal. It splits the parallelogram into two triangles with exactly the same shape and size.
Each of those triangles has the same base and height as the parallelogram. Two identical halves make the whole, so one triangle is half the parallelogram.
It works the other way round too. Take a second copy of any triangle and turn it half a turn. Fit it against one of the sides that is not the base, and the two make a parallelogram with the same base and height.
And why is a parallelogram base × height? Cut the triangle off one end and move it to the other end, and it becomes a rectangle. The base and height do not change, so the area is the rectangle's: base × height.
Height is not the slanted side
Height is the distance measured from the base at a right angle. It is not the length of a slanted side. On the board, each figure draws its height as a dashed vertical line beside the shape.
A parallelogram's slanted side is longer than its height. Multiply by the slanted side instead and the area comes out too big.
Try it yourself
Set the base or the height to any number you like. Divide the parallelogram's area by 2. Does it match the triangle every time?