Mensuration

Formula sheet

Everything for this chapter, in one place.

Read the weight before you read the page. This chapter is measured at 0.067 questions per paper — nine questions since 2017, and exactly one in the current format across five years. It is the lightest chapter in the syllabus. Learn it last.

Volumes


Cube, side $a$	$a^3$
Cuboid	$lbh$
Cylinder	$\pi r^2 h$
Cone	$\dfrac{1}{3}\pi r^2 h$
Sphere	$\dfrac{4}{3}\pi r^3$
Hemisphere	$\dfrac{2}{3}\pi r^3$
Prism	base area $\times$ height
Pyramid	$\dfrac{1}{3} \times$ base area $\times$ height

Surface areas


Cube	$6a^2$
Cuboid	$2(lb+bh+hl)$
Cylinder, curved	$2\pi rh$ ; total $2\pi r(r+h)$
Cone, curved	$\pi r l$ with $l = \sqrt{r^2+h^2}$ ; total $\pi r(r+l)$
Sphere	$4\pi r^2$
Prism	base perimeter $\times$ height, plus the two ends

Diagonals


Cube	$a\sqrt3$
Cuboid, space diagonal	$\sqrt{l^2+b^2+h^2}$
Box inscribed in a sphere	the space diagonal is the diameter

Scaling — the idea that does the work


Lengths scale by $k$	areas scale by $k^2$ , volumes by $k^3$
A cone cut parallel to its base at fraction $k$ of the height from the apex	the small cone has $k^3$ of the volume
A frustum	the whole cone minus the small one

Melting and recasting


Volume	is conserved
Surface area	is not — cutting a solid up always increases it
Equal-volume pieces from several blocks	the common volume is the gcd of the block volumes

Mensuration

Formula sheet

Why this chapter matters