Prism calculator
Volume and surface areas of a regular prism from base side and height.
Fill in the fields and the result will appear here automatically.
A right prism with a regular polygon base is determined by side count n, side length a and perpendicular height h. The base apothem runs from the centre to a side’s midpoint; base area is half the perimeter times that apothem. Rectangular lateral faces give total lateral area equal to base perimeter times height. At n = 4 the base is square; this model does not specify a general rectangle.
How it works
Formula and logic
Apothem = side ÷ (2 × tan(π ÷ n)). Base area = perimeter × apothem ÷ 2. Volume = base area × height, and the lateral surface is perimeter × height.
Example
A hexagonal prism with a 4 cm side and 10 cm height holds 415.69 cm³.
Fields and units
- Length unit — list option
- Sides of the base — 1
- Base side length — cm
- Prism height — cm
How to use
- — Choose the length unit for every input.
- — Enter how many sides the base polygon has.
- — Enter the length of one base side.
- — Enter the height of the prism.
- — Side count must be an integer from 3 to 100; side length and height are positive. This is the page’s limit, and a fractional count is not rounded to another polygon.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: sine and tangent in a right triangle OpenStax: solid volumes and surface areas
- Limitation
- The base is regular and lateral edges are perpendicular to it. Surface formulas do not describe an oblique prism or an unequal-sided base. The unit selector sets interpretation only; entered numbers are not converted.
FAQ
What makes a prism regular?
A regular polygon as the base and side faces perpendicular to it. Slanted prisms have the same volume but a larger lateral surface, which this calculation does not cover.
Why does the base area need an apothem?
Because a regular polygon splits into identical triangles from its centre, each with the side as base and the apothem as height. Summing them gives perimeter × apothem ÷ 2.
Is a cuboid a prism?
Yes, a prism with a four-sided base. Entering four sides gives exactly the square-based case, and the formulas reduce to the familiar ones.
What happens as the number of sides grows?
Compared with a circle of the same circumradius, the polygon’s area fraction is sin(2π/n)/(2π/n), tending to 1. At n = 100 the difference is about 0.0658%. Holding side length alone fixed changes the radius and overall size, a different comparison.