Use perpendicular height h for volume. For a right circular cone, slant height l=sqrt(r2+h2). Surface area counts only exposed surfaces. **Solid** | **Volume** | **Surface area** Prism | Bh | 2B+Ph for a right prism Cylinder | pi r2h | 2pi r2+2pi rh Right pyramid | 13 Bh | Base plus areas of triangular faces Right circular cone | 13pi r2h | pi r2+pi rl Sphere | 43pi r3 | 4pi r2 Hemisphere | 23pi r3 | Curved: 2pi r2; closed: 3pi r2 Here B is base area and P is base perimeter. Split a composite solid into familiar pieces. Add or subtract volumes; do not count joined or hidden faces in the exposed area. Example A hemisphere of radius 3 is joined to a cylinder of radius 3 and height 5, with its lower circular base exposed. V=pi(3)2(5)+ 23pi(3)3=63pi A=2pi(3)(5)+2pi(3)2+pi(3)2=57pi Area uses square units; volume uses cubic units. Convert all lengths to one unit before substitution.

Use perpendicular height \(h\) for volume. For a right circular cone, slant height \(l=\sqrt{r^2+h^2}\). Surface area counts only exposed surfaces.

Curved: \(2\pi r^2\); closed: \(3\pi r^2\)

Here \(B\) is base area and \(P\) is base perimeter. Split a composite solid into familiar pieces. Add or subtract volumes; do not count joined or hidden faces in the exposed area.

A hemisphere of radius \(3\) is joined to a cylinder of radius \(3\) and height \(5\), with its lower circular base exposed.

\(A=2\pi(3)(5)+2\pi(3)^2+\pi(3)^2=57\pi\)

Area uses square units; volume uses cubic units. Convert all lengths to one unit before substitution.

Mensuration (Pyramid, Cones and Spheres)

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Solid	Volume	Surface area
Prism	\(Bh\)	\(2B+Ph\) for a right prism
Cylinder	\(\pi r^2h\)	\(2\pi r^2+2\pi rh\)
Right pyramid	\(\dfrac13 Bh\)	Base plus areas of triangular faces
Right circular cone	\(\dfrac13\pi r^2h\)	\(\pi r^2+\pi rl\)
Sphere	\(\dfrac43\pi r^3\)	\(4\pi r^2\)
Hemisphere	\(\dfrac23\pi r^3\)	Curved: \(2\pi r^2\); closed: \(3\pi r^2\)

Surface Areas and Volumes

Surface Areas and Volumes