ALL EXPERIMENTS

📐 Area Explorer

Drag the sliders to see how rectangles, triangles, parallelograms, rhombi and trapeziums build up their area formulas — and why the perimeter never tells the whole story.

Rectangles & Squares

Area is the number of unit squares that pack a region. For a rectangle, that number is simply length × width. Drag the sliders and watch the unit-square grid.

Area = length × width
Rangoli example from the chapter: a 7 cm × 4 cm rectangle needs 28 unit squares of powder, while an 8 cm × 3 cm rectangle needs only 24 — even though both have similar-looking sizes!

Why Can't Perimeter Measure Area?

Two rectangles can share the same perimeter yet have very different areas. Adjust each rectangle independently and compare.

Area of a Triangle

A triangle enclosed in a rectangle always fills exactly half of it. Drag the vertex along the top and change the base/height to see the ½ × base × height rule hold every time — even for obtuse triangles.

Area = ½ × base × height
Notice: the apex can slide anywhere on the dashed line (a line parallel to the base) — the area never changes, because base and height stay the same!

Parallelogram → Rectangle Dissection

Cut a triangular piece off one end of a parallelogram and slide it to the other end — it becomes a rectangle of the very same area. Press "Dissect" to watch it happen.

Area = base × height

Rhombus & its Diagonals

A rhombus's diagonals are perpendicular bisectors of each other. That lets us split it into 4 right triangles that can be rearranged into a rectangle of sides d₁ and d₂⁄2.

Area = ½ × d₁ × d₂

Area of a Trapezium

Split the trapezium into a rectangle and two triangles, or picture two copies joined to form a parallelogram. Either way, the area comes out to half the height times the sum of the parallel sides.

Area = ½ × height × (a + b)

Area of Any Polygon

Any polygon — quadrilateral, pentagon, hexagon — can be divided into triangles from one chosen vertex. Once we know each triangle's area, we can add them up to get the whole polygon's area. Choose the number of sides and see the triangulation.

A polygon with n sides splits into (n − 2) triangles when triangulated from one vertex.

Units & Conversion

1 in = 2.54 cm, so a 1 in × 1 in square equals a 2.54 cm × 2.54 cm square — that's 2.54² = 6.4516 cm² per square inch. Try converting either way below.

1 ft = 12 in  •  1 acre = 43,560 ft²  •  1 km² = 1,000,000 m²