What is a prism in geometry?

A prism is a three-dimensional solid with two identical, parallel ends (bases) and flat rectangular sides. The bases can be triangles, rectangles, hexagons, or any other polygon, which gives the prism its specific name.

How is the total surface area of a prism calculated?

The total surface area is the sum of the lateral sides (Lateral Area = Base Perimeter × Prism Height) plus the area of the two parallel ends: Total Surface Area = Lateral Area + (2 × Base Area).

What is the difference between a prism and a pyramid?

A prism has two identical parallel bases connected by rectangular sides, keeping its thickness constant throughout. A pyramid has only one base that slants upward to meet at a single point (apex), meaning its volume is exactly one-third of a prism with the same dimensions.

Geometry Tool

Prism Calculator

Calculate the volume, lateral surface area, base area, base perimeter, and total surface area of rectangular, triangular, and regular polygonal prisms.

Interactive Prism Calculator Workspace

Live Solver

Base Length (l)

Base Width (w)

Prism Height (H)

Prism Outputs

Volume (V)240.0000

Total Area248.0000

Lateral Area200.0000

Base Area24.0000

Step-by-Step Explanation

1. Base Area (B): Calculated l × w = 24.0000

2. Base Perimeter (Pb): Calculated = 20.0000

3. Volume (V): Base Area × Prism Height = 24.0000 × 10 = 240.0000

4. Total Surface Area: (Perimeter × Prism Height) + 2 × Base Area = (20.0000 × 10) + 2 × 24.0000 = 248.0000

Mathematical Formula & Variables

Equation ModelVolume = Base Area × Height, Total Surface Area = (Perimeter × Height) + (2 × Base Area)

Finds the volume and surface areas of any solid prism by calculating the base area and perimeter first, then multiplying by height.

Variable Definitions

Symbol	Description
Base Area (B)	The flat surface area of the two matching polygon ends.
Base Perimeter (Pb)	The outer boundary line distance of one base polygon.
H (Prism Height)	The perpendicular depth or distance separating the two parallel bases.
V (Volume)	The 3D space enclosed within the prism: V = B × H.

How to Use the Prism Calculator

Select the Prism Base Shape from the options (Rectangular Prism, Triangular Prism, or Hexagonal Prism).
Input the dimensions corresponding to your base shape (e.g., base lengths, base width, triangle base & legs).
Input the vertical Prism Height (distance between the two parallel bases).
Check the updated volume, lateral surface area, and total surface area output boards.

Practical Example Calculation

Scenario Context: A researcher is evaluating a triangular glass prism. The base is a right triangle with legs of 3 cm and 4 cm, and the prism height is 10 cm.

Step 1: Identify base shape parameters: Right triangle with legs a = 3 cm, b = 4 cm. Prism height H = 10 cm.

Step 2: Calculate Base Hypotenuse (c): c = √(3² + 4²) = √25 = 5 cm.

Step 3: Calculate Base Perimeter (Pb): Pb = a + b + c = 3 + 4 + 5 = 12 cm.

Step 4: Calculate Base Area (B): B = 0.5 × Base × Height = 0.5 × 3 × 4 = 6 square centimeters.

Step 5: Calculate Volume (V): V = B × H = 6 × 10 = 60 cubic centimeters.

Step 6: Calculate Lateral Area (LA): LA = Pb × H = 12 × 10 = 120 square centimeters.

Step 7: Calculate Total Surface Area (TA): TA = LA + (2 × B) = 120 + (2 × 6) = 132 square centimeters.

The glass prism has a volume of 60.00 cm³, a lateral area of 120.00 cm², and a total surface area of 132.00 cm².

Frequently Asked Questions

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