⬡ Polygon Prism Calculator
Volume, side length, and height for regular polygon-based prisms
Enter Polygon Prism Dimensions
Hexagon • n=6 sides
📊 Results Hexagonal Prism
Number of Sides
6
Side Length
5 cm
Height
10 cm
Apothem
4.33 cm
Base Area
64.95 cm²
Volume
649.5 cm³
Perimeter
30 cm
Surface Area
429.9 cm²
📝 Step-by-Step Solution
Given: n = 6 sides, s = 5 cm, h = 10 cm
Apothem: a = s / (2×tan(π/n)) = 5 / (2×tan(π/6)) = 4.33 cm
Base area: A = (n × s × a) / 2 = (6 × 5 × 4.33) / 2 = 64.95 cm²
Volume: V = A × h = 64.95 × 10 = 649.5 cm³
Surface area: SA = 2×A + n×s×h = 129.9 + 300 = 429.9 cm²
📐 Regular Polygon Prism Formulas
Apothem: a = s / (2 × tan(π/n))
Base Area: A = (n × s × a) / 2 = (n × s²) / (4 × tan(π/n))
Volume: V = A × h
Perimeter: P = n × s
Surface Area: SA = 2A + n×s×h
Understanding Regular Polygon Prisms
⬡ Regular Polygons
Equal sides and angles. n = number of sides. Examples: triangle (3), square (4), pentagon (5), hexagon (6), octagon (8).
📏 The Apothem
Center to midpoint of side. Key measurement for area. a = s / (2×tan(π/n)). Larger n → apothem approaches circumradius.
📦 Prism Volume
V = Base Area × Height. Same formula for any prism. The base can be any polygon shape.
♾️ Approaching Circle
More sides = more circular. As n→∞, polygon becomes circle. Area approaches πr².
Frequently Asked Questions
What is the volume formula for a polygon prism?
V = Base Area × Height. For regular polygon: V = (n × s² × h) / (4
× tan(π/n)), where n = sides, s = side length, h = height.
What is an apothem?
Distance from center to midpoint of any side. For regular polygon:
a = s / (2×tan(π/n)). Perpendicular to the side.
How do I find polygon base area?
A = (n × s × a) / 2. Or equivalently: A = (n × s²) / (4 ×
tan(π/n)). Splits into n triangles.
What is a regular polygon?
All sides equal length, all angles equal. Examples: equilateral
triangle, square, regular pentagon, regular hexagon.
How do I calculate hexagonal prism volume?
V = (3√3/2) × s² × h. Or use general formula with n=6. Hexagon
area = (3√3/2)s².
What is the pentagon volume formula?
V = (5 × s² × h) / (4 × tan(36°)). tan(36°) ≈ 0.7265. Or use: V ≈
1.72 × s² × h.
How does octagon prism volume work?
V = 2(1+√2) × s² × h. Eight-sided regular polygon. Common in
architecture (octagonal columns).
What is circumradius vs apothem?
Circumradius = center to vertex. Apothem = center to side
midpoint. R = a / cos(π/n).
How accurate as circle approximation?
12-gon within 3%, 20-gon within 1%. More sides = closer to πr².
Useful for approximating circular objects.
Can I use this for irregular polygons?
No, regular polygons only. Irregular polygons require different
methods (decomposition, coordinate geometry).
What is the interior angle formula?
θ = (n-2) × 180° / n. Triangle: 60°, Square: 90°, Pentagon: 108°,
Hexagon: 120°, Octagon: 135°.
How do I find side length from volume?
s = √(4V × tan(π/n) / (n × h)). Rearrange the volume formula. Our
calculator solves this automatically.
What is surface area formula?
SA = 2×(base area) + (perimeter × height). Two polygon bases plus
n rectangular sides.
Why are hexagonal prisms common?
Efficient packing. Hexagons tessellate perfectly. Used in
honeycomb, nuts, pencils, and architecture.
What is the area of a triangle prism?
V = (√3/4) × s² × h for equilateral triangle. Our calculator
handles triangular prisms with n=3.
How does this relate to cylinders?
Cylinder is limit case. As n→∞, polygon prism approaches cylinder.
Polygon area → πr².
What units should I use?
Any consistent units. All dimensions in same unit. Volume output
can be converted to liters, gallons, etc.