Area & Volume

Area = length × width. Perimeter = 2(length + width).

Area = ½ × base × perpendicular height.

Area = πr². Circumference = 2πr (or πd). If the radius is a multiple of 7, use π = 22/7.

A sector is a slice. Area = (θ/360)πr²; arc length = (θ/360)×2πr.

Split into shapes you know and add; or subtract the inside from the outside (a path, a hole).

Volume = l×w×h. Surface area = 2(lw + lh + wh). An open-top box leaves out one face.

Volume = πr²h. Curved area = 2πrh. Closed: total = 2πrh + 2πr².

Volume = ⅓πr²h. Curved area = πrl, where l is the slant height.

Sphere: V = (4/3)πr³, SA = 4πr². Hemisphere: V = (2/3)πr³.

Add the two volumes (cylinder + cone, cylinder + hemisphere, cone + hemisphere).

When a solid is melted and re-made, the volume is unchanged. Put V₁ = V₂ and solve.

Space left over = container volume − object volume (sphere in a box, cylinder in a box).

When an object sinks, the water rises by exactly the object’s volume: πr²(rise) = V.