Geometry extends to 3D: surface area measures the outer wrapping; volume measures the space inside. Each shape has distinct formulas.
Rectangle: l×w · Triangle: ½bh · Parallelogram: bh · Trapezoid: ½(b₁+b₂)h · Circle: πr² · Regular polygon: ½×perimeter×apothem
SA is the total area of all faces. Rectangular prism: 2(lw+lh+wh). Cylinder: 2πr²+2πrh. Sphere: 4πr². Cone: πr²+πrl (l = slant height).
Prism/cylinder: V = Base Area × height. Pyramid/cone: V = ⅓×Base Area×height. Sphere: V = (4/3)πr³. Doubling a dimension: V scales proportionally.
Two solids with equal heights and equal cross-sectional areas at every level have equal volumes. This justifies why oblique and right prisms with same base and height have equal volume.
Q1: A cylinder has radius 3 and height 10. Its volume is:
Q2: A cone and a cylinder have the same base and height. The cone's volume is:
Q3: A sphere has radius 6. Its surface area is: