Revise Geometry of 3D solids: volume and surface area for IB Mathematics: Analysis and Approaches (3.1)

A right prism has a constant cross-section — imagine slicing a loaf where every slice is identical. Its volume is the area of that cross-section (the base) multiplied by the length. A cylinder is simply a prism with a circular base, so its volume is the area of the circle times the height.

Volume of a cylinder: $V = \pi r^2 h$

Surface area of a closed cylinder: $A = 2\pi r h + 2\pi r^2$

Pyramids and cones taper to a point (the apex), and their volume is exactly one third of the prism or cylinder that would box them in. The crucial distinction here is between the perpendicular height $h$, measured straight down from the apex to the centre of the base, and the slant height $l$, measured down the sloping face. With the radius they form a right-angled triangle, so $l^2 = r^2 + h^2$, and you will constantly use Pythagoras to swap between them. Volume needs $h$; curved surface area needs $l$.

Volume of a cone: $V = \tfrac{1}{3}\pi r^2 h$

Surface area of a cone: $A = \pi r l + \pi r^2$ (curved surface + circular base)

A sphere is defined entirely by its radius, and so are its two formulas. Exam questions frequently glue solids together — a hemisphere on a cylinder to make a silo, a cone on a cylinder to make a pencil. For volume you simply add the parts. For surface area you must be selective: include only the surfaces on the outside, and drop every face hidden where two solids join.

Volume of a sphere: $V = \tfrac{4}{3}\pi r^3$

Surface area of a sphere: $A = 4\pi r^2$

The longest straight line inside a cuboid runs from one corner to the opposite corner, passing through the interior — the space diagonal. You find it with Pythagoras applied twice. First find the diagonal across the base rectangle; then treat that base diagonal and the vertical edge as the two shorter sides of a second right-angled triangle whose hypotenuse is the space diagonal. Combining the two steps gives a single neat formula.

Space diagonal of an $a \times b \times c$ cuboid: $d = \sqrt{a^2 + b^2 + c^2}$

Every 3D angle question reduces to the same move: find one right-angled triangle that contains the angle, redraw it flat, and use SOH-CAH-TOA. What changes is how you build that triangle. The angle between two lines is measured directly between them where they meet (or where they would meet if extended). The angle between a line and a plane is the angle between the line and its projection — its 'shadow' — on the plane; you build it by dropping a perpendicular from the far end of the line onto the plane. The angle between two planes (the dihedral angle) is found along the line where they meet: from a point on that line, draw a line in each plane perpendicular to the meeting line, and measure the angle between them.

In IB Mathematics the marks are analytic. A method mark (M) is earned for the correct approach — the right equation, the right triangle — even if the arithmetic later slips. An accuracy mark (A) is for a correct value and is DEPENDENT on the method being right (an A-mark cannot be earned without its M-mark). Follow-through (FT) means that once you have a wrong value, later marks are still awarded if your subsequent working is correct FOR THAT value. The engine also applies ISW ('ignore subsequent working') once a correct answer appears, accepts any equivalent exact form, and accepts a decimal that is correctly rounded to 3 significant figures. Study how each mark below is tied to a specific line.

The habit built here — reduce every solid to one right-angled triangle, redraw it flat, then use Pythagoras or a trig ratio — is exactly what the rest of the geometry and trigonometry unit demands. It feeds straight into the sine and cosine rules for non-right-angled triangles, into the area of a triangle with $\tfrac{1}{2}ab\sin C$, and into arc-length and sector problems, all of which reuse the same 'find the triangle, label the sides, choose the ratio' method. Master it now and the later trigonometry becomes variations on a routine you already own.