Circle: area = π × radius²
A circle's area comes from squaring its radius and multiplying by π (approximately 3.14159). For a radius of 7: π × 7² = π × 49 = 153.938 (rounded to 4 decimal places). For a larger radius of 10: π × 10² = π × 100 = 314.1593.
Square: area = side²
A square's area is simply its side length multiplied by itself. For a side of 12: 12 × 12 = 144 — no rounding needed, since 12² is a whole number. A square is really just a rectangle where both sides happen to be equal, so this is the same formula as a rectangle's area with length and width set to the same value.
Rectangle: area = length × width
A rectangle's area is its length multiplied by its width. For an 8-by-5 rectangle: 8 × 5 = 40. Note that a 10-by-4 rectangle also has area 40 (10 × 4 = 40) — different dimensions can produce the exact same area, which the companion article on perimeter versus area explores further.
Why the formulas look different but ask the same question
All three formulas answer "how much flat surface does this shape cover?" — they differ because each shape's boundary relates to its interior differently. A rectangle's area is a direct multiplication because its sides are already straight and perpendicular. A square's formula is a special case of a rectangle's. A circle has no straight sides at all, so its area formula instead comes from the geometric constant π, which captures the fixed relationship between any circle's radius and the space it encloses.