A square pyramid V–ABCD has base side a>0. Face VAD is equilateral and perpendicular to the base; M is the midpoint of AD. Find the pyramid volume.

01 / Standard#Your turn

Official paper · jm02-2025 · 1(b) · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Perpendicular planes make VM perpendicular to the base.

Hint 2

A pyramid uses one third of base area times height.

Worked solution

VM is perpendicular to the planes’ intersection AD within the face, so it is the pyramid height.
$h=VM=\frac{\sqrt3}2a,\quad A_{\rm base}=a^2$
Apply the volume formula.
$V=\frac13a^2\frac{\sqrt3}2a=\frac{\sqrt3}6a^3$

Volume √3a³/6.

Checks and common pitfalls: The volume coefficient is 1/3, not 1/2.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
VM is perpendicular to the planes’ intersection AD within the face, so it is the pyramid height.
$h=VM=\frac{\sqrt3}2a,\quad A_{\rm base}=a^2$
Apply the volume formula.
$V=\frac13a^2\frac{\sqrt3}2a=\frac{\sqrt3}6a^3$

Think first. Reveal a hint when the class is ready.

2025 JM02