A regular triangular pyramid (a pyramid with an equilateral triangular base) is cut by two planes parallel to its base such that the altitude of the pyramid is trisected into three equal segments. The top, middle and bottom parts of the pyramid have volumes V1, V2 and V3 respectively. What is the ratio V1 : V2 : V3?

Introduction / Context:
This problem involves a regular triangular pyramid, also known as a pyramid with an equilateral triangular base and an apex directly above the centre. The altitude of the pyramid is divided into three equal segments by planes parallel to the base, creating three parts with different volumes: a small pyramid at the top and two frustum like sections below. Using similarity and the cubic nature of volume scaling, we can determine the volume ratios V1 : V2 : V3 without needing any actual measurements.

Given Data / Assumptions:

• The pyramid has an equilateral triangular base and a single apex vertically above the centre.
• The altitude (height) of the pyramid is H.
• Two planes parallel to the base cut the pyramid and trisect the altitude into three equal parts.
• The top part (nearest the apex) has volume V1.
• The middle frustum like part has volume V2.
• The bottom frustum like part (nearest the base) has volume V3.
• The cuts are parallel to the base, so the resulting cross sections are similar to the base.

Concept / Approach:
For any pyramid, cross sections parallel to the base at a fraction of the height form similar polygons. The volumes of similar pyramids scale with the cube of the linear scale factor. If we measure height from the apex downward, then the volume from the apex down to height h is proportional to (h / H)^3. We use this to find the volumes up to heights H / 3 and 2H / 3 from the apex, then subtract to obtain the volumes of the top, middle and bottom parts in terms of the total volume V.

Step-by-Step Solution:
Step 1: Let the total volume of the original pyramid be V and its full altitude be H. Step 2: The altitude is trisected, so planes are at heights H / 3 and 2H / 3 measured from the apex downwards. Step 3: The volume of a smaller pyramid cut off at height h from the apex is V(h) = V * (h / H)^3 due to similarity. Step 4: The top section is a small pyramid of height H / 3, so its volume V1 = V * (1 / 3)^3 = V * (1 / 27). Step 5: The volume from apex down to height 2H / 3 is V * (2 / 3)^3 = V * (8 / 27). Step 6: The middle section lies between H / 3 and 2H / 3, so its volume V2 = V * (8 / 27) − V * (1 / 27) = V * (7 / 27). Step 7: The bottom section is the remaining part between height 2H / 3 and H, so its volume V3 = V − V * (8 / 27) = V * (19 / 27). Step 8: Therefore the volumes in terms of V are V1 = V / 27, V2 = 7V / 27 and V3 = 19V / 27. Step 9: The ratio V1 : V2 : V3 is (1 / 27) : (7 / 27) : (19 / 27) = 1 : 7 : 19.

Verification / Alternative check:
We can check the correctness by verifying that the three parts sum to the total volume. Adding the fractions: (1 / 27) + (7 / 27) + (19 / 27) = 27 / 27 = 1, confirming that V1 + V2 + V3 = V. Additionally, the top part is much smaller compared to the bottom section, which is intuitive since the base is widest near the bottom, and volume increases sharply with height in a pyramid. This agrees with having a small coefficient 1, then 7 and then 19 for the three parts.

Why Other Options Are Wrong:
Ratios such as 1 : 8 : 27 or 1 : 8 : 19 emerge from using linear factors directly or squaring instead of cubing when scaling volumes. The ratio 2 : 9 : 27 or 1 : 6 : 20 does not match the correct fractional volumes derived from (h / H)^3 and fails the check that the parts must sum to the total volume V. Only 1 : 7 : 19 satisfies both the cubic scaling and the sum condition.

Common Pitfalls:
A typical mistake is to treat the volume as proportional to the height or area instead of the cube of the height for similar pyramids. Some learners also misinterpret “trisects the altitude” as meaning that volumes are equal, which is not true. Another pitfall is to forget to subtract the smaller partial volume from the larger one when computing the middle section. Carefully applying similarity and remembering that volume scales with the cube of linear scale resolves these issues.

Final Answer:
The required ratio of volumes is V1 : V2 : V3 = 1 : 7 : 19.