3D Solids

• Apply V = lwh for rectangular prisms
• Apply V = πr²h for cylinders and recognize when h is not the vertical dimension
• Apply V = (1/3)πr²h for cones and V = (4/3)πr³ for spheres
• Compute surface area of cylinders (2πr² + 2πrh) and spheres (4πr²)
• Break composite solids (cylinder-plus-cone, box-minus-hemisphere) into parts

3D solid problems test two things: which formula to use and how to read the geometry of the figure. Memorize these: rectangular prism (box) volume = length × width × height. Cylinder volume = πr²h, where r is the radius of the circular base and h is the perpendicular distance between the two bases. Cone volume = (1/3)πr²h. Exactly one-third the cylinder with the same base and height. Sphere volume = (4/3)πr³. Pyramid volume = (1/3) × (area of base) × height. Surface area is the outside covering. Think about painting the shape. Cylinder surface area = 2πr² (two circular ends) + 2πrh (rectangular wrap-around side). Sphere surface area = 4πr². For a rectangular prism, surface area = 2(lw + lh + wh). Six rectangular faces, three pairs. Composite solids are the act's favorite trap: an ice-cream-cone shape (cone plus hemisphere), a silo (cylinder plus hemisphere on top), a tank with a cylindrical hole drilled through it. Break the composite into its basic pieces, compute each volume or surface area, and add (if the pieces are combined) or subtract (if one is carved out). For surface area in composites, be careful. The surfaces where two pieces meet are internal and do not count toward the outside. For example, a cylinder with a cone on top does not include the top circle of the cylinder or the bottom circle of the cone in its surface area, because those surfaces are hidden inside. Displacement: an object sunk in a cylindrical tank raises the water by a cylinder of water with the tank's radius, so gravel that raises the water 3 inches in a tank of radius 10 inches has volume π(10²)(3) = 300π cubic inches. Cross sections: a slice through a pyramid's apex is a triangle whose height is the pyramid's height, so the slice along the base's diagonal (the longest base line) beats a slice through the midpoints of two sides. Units are squared for surface area, cubed for volume. If the answer choices mix in² and in³, you can often eliminate half immediately.

• Cone and pyramid both have the (1/3) factor. Sphere has (4/3). Write these three fractions on scratch paper at the start.
• In a cylinder problem, the 'height' is the distance between the two circular bases. Which may not be vertical in the diagram.
• For composite solids, draw the split line and compute each piece separately. Do not try to combine formulas.
• In composite surface area, subtract any circular or flat face where two pieces meet. Those are internal.
• Volume uses cubed units, surface area uses squared. Check the unit on the answer choices as a sanity check.