Let the pyramid $P$ have volume $V_P$ and height $H$. Let the frustum $F$ that is a sliced part of this pyramid have height $h$, so $0 \leq h \leq H$. You want to know the volume of the frustum. Call it $V_F$.

By calculus, $V_P = (1/3)AH$, where $A$ is the area of the base of the pyramid. The part of the pyramid above the frustum is a scaled version of the original pyramid. It has height $H-h$, so the smaller pyramid is a scaled-down version of the original pyramid by the factor $(H-h)/H = 1 - h/H$. In particular, the base of the smaller pyramid has area $(1-h/H)^2A$, so the volume of the smaller pyramid is $(1/3)(1-h/H)^2A(H-h)$. Therefore $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2A(H-h). $$ Replacing $A$ by $3V_P/H$, $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2\frac{3V_P}{H}(H-h) = \left(1 - \left(1 - \frac{h}{H}\right)^3\right)V_P. $$ So there is your frustum volume formula in terms of $V_P$, $H$, and $h$. As a reality check, when $h = H$ we get $V_F = V_P$ (and the frustum $F$ in this case is the full pyramid $P$) and when $h = 0$ we get $0$ (and the frustum $F$ in this case is the base of the pyramid, so it is planar).

PS: Until less than two weeks ago, I always thought the term in English was frustrum. To learn so late in life that it is actually frustum was quite... frustating

Let the pyramid $P$ have volume $V_P$ and height $H$. Let the frustum $F$ that is a sliced part of this pyramid have height $h$, so $0 \leq h \leq H$. You want to know the volume of the frustum. Call it $V_F$.

By calculus, $V_P = (1/3)AH$, where $A$ is the area of the base of the pyramid. The part of the pyramid above the frustum is a scaled version of the original pyramid. It has height $H-h$, so the smaller pyramid is a scaled-down version of the original pyramid by the factor $(H-h)/H = 1 - h/H$. In particular, the base of the smaller pyramid has area $(1-h/H)^2A$, so the volume of the smaller pyramid is $(1/3)(1-h/H)^2A(H-h)$. Therefore $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2A(H-h). $$ Replacing $A$ by $3V_P/H$, $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2\frac{3V_P}{H}(H-h) = \left(1 - \left(1 - \frac{h}{H}\right)^3\right)V_P. $$ So there is your frustum volume formula in terms of $V_P$, $H$, and $h$. As a reality check, when $h = H$ we get $V_F = V_P$ (and the frustum $F$ in this case is the full pyramid $P$) and when $h = 0$ we get $0$ (and the frustum $F$ in this case is the base of the pyramid, so it is planar).

PS: Until less than two weeks ago, I always thought the term in English was frustrum. To learn so late in life that it is actually frustum was quite… frustating.

Let the pyramid $P$ have volume $V_P$ and height $H$. Let the frustum $F$ that is a sliced part of this pyramid have height $h$, so $0 < h < H$. You want to know the volume of the frustum. Call it $V_F$

By calculus, $V_P = (1/3)AH$, where $A$ is the area of the base of the pyramid. The part of the pyramid above the frustum is a scaled version of the original pyramid. It has height $H-h$, so the smaller pyramid is a scaled-down version of the original pyramid by the factor $(H-h)/H = 1 - h/H$. In particular, the base of the smaller pyramid has area $(1-h/H)^2A$, so the volume of the smaller pyramid is $(1/3)(1-h/H)^2A(H-h)$. Therefore $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2A(H-h). $$ Replacing $A$ by $3V_P/H$, $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2\frac{3V_P}{H}(H-h) = \left(1 - \left(1 - \frac{h}{H}\right)^3\right)V_P. $$ So there is your frustum formula in terms of $V_P$, $H$, and $h$. As a reality check, when $h = H$ we get $V_P$ (indeed, the frustum $F$ is the full pyramid $P$) and when $h = 0$ we get $0$ (and in this case the frustum is just the base of the pyramid, so it is planar).

Let the pyramid $P$ have volume $V_P$ and height $H$. Let the frustum $F$ that is a sliced part of this pyramid have height $h$, so $0 \leq h \leq H$. You want to know the volume of the frustum. Call it $V_F$.

By calculus, $V_P = (1/3)AH$, where $A$ is the area of the base of the pyramid. The part of the pyramid above the frustum is a scaled version of the original pyramid. It has height $H-h$, so the smaller pyramid is a scaled-down version of the original pyramid by the factor $(H-h)/H = 1 - h/H$. In particular, the base of the smaller pyramid has area $(1-h/H)^2A$, so the volume of the smaller pyramid is $(1/3)(1-h/H)^2A(H-h)$. Therefore $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2A(H-h). $$ Replacing $A$ by $3V_P/H$, $$ V_F = V_P - \frac{1}{3}\left(1 - \frac{h}{H}\right)^2\frac{3V_P}{H}(H-h) = \left(1 - \left(1 - \frac{h}{H}\right)^3\right)V_P. $$ So there is your frustum volume formula in terms of $V_P$, $H$, and $h$. As a reality check, when $h = H$ we get $V_F = V_P$ (and the frustum $F$ in this case is the full pyramid $P$) and when $h = 0$ we get $0$ (and the frustum $F$ in this case is the base of the pyramid, so it is planar).

PS: Until less than two weeks ago, I always thought the term in English was frustrum. To learn so late in life that it is actually frustum was quite... frustating