MathLabs
TheoremProved

Volume ratio for triangular pyramids sharing an apex

Statement

Given a triangular pyramid S.ABCS.ABC and points A′∈SAA' \in SA, B′∈SBB' \in SB, C′∈SCC' \in SC on its three lateral edges, the pyramid S.A′B′C′S.A'B'C' satisfies VS.A′B′C′VS.ABC=SA′SA⋅SB′SB⋅SC′SC\dfrac{V_{S.A'B'C'}}{V_{S.ABC}} = \dfrac{SA'}{SA}\cdot\dfrac{SB'}{SB}\cdot\dfrac{SC'}{SC}.

Why is it true?

Sliding each of the three points independently along its own edge from the apex stretches the pyramid independently in three different directions, so the volume should scale by the product of the three independent stretch factors, just as scaling the three sides of a box independently multiplies its volume by the product of the three scale factors.

Proof sketch

Place the apex SS at the origin and let u⃗=SA→\vec{u} = \overrightarrow{SA}, v⃗=SB→\vec{v} = \overrightarrow{SB}, w⃗=SC→\vec{w} = \overrightarrow{SC}. The volume of a tetrahedron spanned by three edge vectors from a common vertex is given by the scalar triple product VS.ABC=16∣u⃗⋅(v⃗×w⃗)∣V_{S.ABC} = \frac{1}{6}\left|\vec{u}\cdot(\vec{v}\times\vec{w})\right|.

Since A′∈SAA' \in SA, B′∈SBB' \in SB, C′∈SCC' \in SC, we can write SA′→=k1u⃗\overrightarrow{SA'} = k_1\vec{u}, SB′→=k2v⃗\overrightarrow{SB'} = k_2\vec{v}, SC′→=k3w⃗\overrightarrow{SC'} = k_3\vec{w} where k1=SA′SAk_1 = \frac{SA'}{SA}, k2=SB′SBk_2 = \frac{SB'}{SB}, k3=SC′SCk_3 = \frac{SC'}{SC}. Then VS.A′B′C′=16∣(k1u⃗)⋅((k2v⃗)×(k3w⃗))∣V_{S.A'B'C'} = \frac{1}{6}\left|(k_1\vec{u})\cdot\big((k_2\vec{v})\times(k_3\vec{w})\big)\right|.

The scalar triple product is trilinear (linear in each of its three vector arguments), so (k1u⃗)⋅((k2v⃗)×(k3w⃗))=k1k2k3(u⃗⋅(v⃗×w⃗))(k_1\vec{u})\cdot\big((k_2\vec{v})\times(k_3\vec{w})\big) = k_1 k_2 k_3 \left(\vec{u}\cdot(\vec{v}\times\vec{w})\right). Taking absolute values and dividing by 6 on both sides gives VS.A′B′C′=k1k2k3⋅VS.ABCV_{S.A'B'C'} = k_1k_2k_3 \cdot V_{S.ABC}, which rearranges exactly to VS.A′B′C′VS.ABC=SA′SA⋅SB′SB⋅SC′SC\dfrac{V_{S.A'B'C'}}{V_{S.ABC}} = \dfrac{SA'}{SA}\cdot\dfrac{SB'}{SB}\cdot\dfrac{SC'}{SC}.

Topics that use this theorem

Polyhedra and their volumes

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

  1. Euclid; trans. T. L. Heath (1908). Euclid's Elements, Book XII (method of exhaustion; pyramid and prism volumes)
  2. Weisstein, Eric W. (2024). Platonic solid