Polygon Aquarium Volume Calculators

Polygon aquariums use multi-sided geometric cross-sections — hexagons, octagons, and pentagons — to create visually distinctive tanks that approximate a cylinder while using flat glass panels for distortion-free viewing. The number of sides directly determines the cross-sectional area and therefore the water volume, so each polygon shape requires its own formula based on side count and side length. Use the calculators below to find the exact capacity for your polygon tank.

Hexagonal

Hexagonal aquariums provide six flat viewing panels that give a near-360-degree experience while using flat glass that avoids optical distortion.

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Flat Back Hexagon

Flat back hexagon aquariums are modified hex tanks with one flat rear panel designed to sit flush against a wall.

Calculate Volume

Octagon

Octagonal aquariums offer eight viewing panels that closely approximate a cylinder while using flat glass for distortion-free clarity.

Calculate Volume

Pentagon

Pentagon aquariums are five-sided tanks that offer an unusual geometric footprint, making them a conversation piece in any room.

Calculate Volume

How Polygon Tank Shapes Are Measured

Regular polygon tanks (hexagons, octagons, pentagons) require only two measurements: side length and height. The side length is the inside distance along any one of the identical flat glass panels, measured from one edge to the adjacent edge. Because all sides of a regular polygon are equal, a single panel measurement defines the entire cross-section. Always confirm that all panels are indeed the same length before relying on one measurement.

The apothem (the perpendicular distance from the center to the midpoint of a side) is a key geometric value used internally by the formula, but you do not need to measure it directly. Our calculators derive the apothem from the side length using trigonometric relationships specific to each polygon type. If a manufacturer specifies the tank diameter or apothem instead of the side length, you can convert: for a regular hexagon, the side length equals the circumscribed circle radius.

Flat back hexagon tanks break the regular polygon pattern. Instead of six equal sides, they have a wide front panel, two angled wing panels, and a flat rear panel. These tanks require a front panel width measurement and a depth measurement (front to back), making them more similar to measuring a custom-shaped tank than a true polygon.

Volume Formula Differences

Each regular polygon has a unique area formula based on the number of sides. The hexagon uses V = (3√3 ÷ 2) × s² × H, the octagon uses V = 2(1 + √2) × s² × H, and the pentagon uses V = (√(25 + 10√5) ÷ 4) × s² × H. The constant multiplier increases with the number of sides as the polygon approaches a circle, which is why an octagon of the same side length encloses more area than a hexagon.

The flat back hexagon uses a composite formula that combines the front panel width and depth to compute the area of its irregular cross-section. It cannot be reduced to the standard hexagon formula because the rear panel changes the geometry. For the most accurate result, always match the specific polygon shape to its dedicated calculator rather than approximating with a similar shape.

Common Mistakes for Polygon Tanks

  • Measuring the diameter instead of side length: The diameter (corner to opposite corner) is not the same as the side length. A regular hexagon with a 12-inch diameter has 6-inch sides, not 12-inch sides. Entering the diameter as the side length will dramatically overstate volume.
  • Using the regular hexagon formula on a flat back hex: A flat back hexagon is not a regular six-sided polygon. Its rear panel is wider and flat for wall placement. Using the regular hexagon formula will give an incorrect cross-sectional area and a wrong volume result.
  • Assuming all panels are equal on an irregular polygon: If alternating panels are different lengths, the tank is not a regular polygon and the standard formulas will not apply. Measure multiple panels to confirm regularity before using the polygon calculator.
  • Confusing polygon area with cylinder area: A hexagon inscribed in a circle captures about 83% of the circle area, not 100%. Using a cylinder formula (with the circumscribed circle radius) on a hexagon tank will overstate volume by approximately 20%.

Related Calculator Categories

Standard Aquarium Calculators Curved Aquarium Calculators Corner Aquarium Calculators Custom Aquarium Calculators

Frequently Asked Questions

What is a polygon aquarium?
A polygon aquarium has a cross-section with three or more straight sides forming a regular geometric shape — most commonly a hexagon (6 sides), octagon (8 sides), or pentagon (5 sides). Each shape uses a specific formula based on the number of sides and the side length.
How do I measure the side length of a hexagonal tank?
Measure the inside length of any one of the six identical flat glass panels from one edge to the adjacent edge. All sides of a regular hexagon must be equal. If you find unequal sides, your tank may be a modified hexagon (such as a flat back hex) and needs a different formula.
What is the difference between apothem and side length?
The side length is the distance along one flat edge of the polygon. The apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. Our calculators use side length as the input because it is easier to measure directly on an aquarium. The apothem is calculated internally from the side length.
Does a flat back hexagon use the same formula as a regular hexagon?
No. A flat back hexagon has one flat rear panel designed for wall placement, making it a five-sided shape rather than a true six-sided regular hexagon. It uses a different formula that combines the front panel width and depth measurement. Using the regular hexagon formula on a flat back hex will give an incorrect result.
How much more water does an octagon hold compared to a hexagon?
For tanks inscribed in the same circle (same outer diameter), a regular octagon has approximately 8% more cross-sectional area than a regular hexagon. This means an octagon tank of the same diameter and height holds about 8% more water, which can matter for stocking and filtration calculations.
Are pentagon aquariums common?
Pentagon aquariums are relatively uncommon compared to hexagonal and octagonal tanks. They are mostly available as custom-built or specialty models. The regular pentagon formula requires all five sides to be equal. Irregular five-sided tanks would need to be decomposed into simpler shapes for accurate volume calculation.
Can I approximate a polygon tank as a cylinder?
As the number of sides increases, a polygon more closely approximates a circle. An octagon captures about 90% of the area of its circumscribed circle, while a hexagon captures about 83%. For rough estimates a cylinder formula may suffice, but for accurate stocking and dosing calculations, always use the polygon-specific formula.