Dragging the Base and Height sliders deforms the triangle and the parallelogram at once and moves both areas. They are drawn on the same base and height, so the triangle is visibly always exactly half.
How to read this board
A figure card assigns its edges in connection order. For a triangle or a parallelogram the first is the base and the second the height; a trapezoid takes top, bottom and height; a circle takes a radius. That is how two sliders can drive both shapes.
The card also prints the formula with the numbers in it
(4×3÷2). Seeing the letters replaced by actual values means you can follow the
shape even when the formula itself has gone.
Where it trips you up
The places that are easy to get wrong, next to how to check each one on the board.
| Where it trips you up | Why that happens | How the board shows it |
|---|---|---|
| Height is not the slanted side | The slanted edge is the one that stands out, so it gets read as the height. The height is the perpendicular distance to the base. | The figure card draws the height as a dashed guide, separate from the slanted edge, so which line is which is visible. |
| Why the triangle is halved | Memorised as a formula, the reason does not survive. Two copies of the triangle make the parallelogram. | The triangle and the parallelogram take the same sliders. Move base or height and the two areas keep the doubling-and-halving relation. |
| The trapezoid formula will not stick | (top + bottom) × height ÷ 2 has a lot of parts. | Switch the figure card to a trapezoid and its connections become top, bottom and height. The formula prints with the numbers in it. |
| Circumference and area get mixed up | 2πr and πr² swap places in memory. | The circle card prints the perimeter alongside the area. Change the radius and one grows in proportion while the other grows with the square. |
| Doubling the base doubles the area; doubling both quadruples it | The words "I doubled it" are the same in both cases, so the results feel like they should be too. | Double one slider, then both. The rates of growth are visibly different. |
| Solids continue the same arithmetic | They look like a separate topic, but a prism or cylinder is base area × height — the area calculation becomes a part of it. | The board carries a cylinder card. Choose a prism and the triangle card's area feeds it directly as the base. |
The formulas for the triangle, the parallelogram and the trapezoid look unrelated, but moving and combining the figures connects them: cut the end off a parallelogram and slide it and you have a rectangle; the triangle is half of that; the trapezoid is a rectangle built on the average of its two parallel sides. They look unrelated because the formula does not carry the relation.
Try this
- Set the height to 1 — the triangle's area becomes half the base. The ÷2 is the whole difference.
- Drag the base out and back — both areas move in step and the ratio between them never changes.
- Grow the radius — the perimeter climbs steadily while the area pulls away from it — proportional against squared, side by side.