The Dodecahedron Surface Area Calculator calculates the total surface area of a regular dodecahedron — a solid with 12 identical regular pentagon faces — from its edge length.
The formula is A = 3√(25 + 10√5) × a², about 20.6457 × a². The calculator also gives the volume, and a second mode works backward from a known surface area to the edge length.
Dodecahedron Surface Area Calculator
Surface area and volume of a regular dodecahedron (12 pentagonal faces) from edge length.
Formula
A = 3√(25 + 10√5) × a² ≈ 20.6457a²
V = ((15 + 7√5) ÷ 4) × a³ ≈ 7.6631a³
Edge from area: a = √(A ÷ (3√(25 + 10√5)))
Where:
- A = total surface area
- V = volume
- a = edge length (the side of each pentagon)
How the Calculation Works
Step 1: Find the area of one regular pentagon face: ¼√(5(5 + 2√5)) × a² ≈ 1.7205a².
Step 2: Multiply by 12, because the dodecahedron has 12 identical faces. This simplifies to 3√(25 + 10√5) × a².
Step 3: For the volume, the calculator applies ((15 + 7√5) ÷ 4) × a³.
Variables Explained
| Variable | Meaning | Unit |
|---|---|---|
| a | Edge length | mm, cm, in |
| A | Surface area | mm², cm², in² |
| V | Volume | mm³, cm³, in³ |
Example Calculation
Given: edge length = 3 cm
Formula: A = 20.6457 × a²
Calculation: A = 20.6457 × 9
Result: A ≈ 185.81 cm²; volume ≈ 7.6631 × 27 ≈ 206.90 cm³
Working backward
Given: surface area = 500 cm²
Result: a = √(500 ÷ 20.6457) ≈ 4.92 cm
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 cm² | 100 mm² |
| 1 in² | 6.4516 cm² |
| 1 m² | 10,000 cm² |
Shape Properties
A regular dodecahedron is one of the five Platonic solids. It has 12 pentagonal faces, 30 edges and 20 vertices, with three faces meeting at each vertex. The 12-sided dice used in tabletop games (d12) are dodecahedrons.
Dimensions Explained
Measure one edge of a pentagon face. The distance across a face or between opposite faces is not the edge length and will give the wrong result if entered here.
Frequently Asked Questions
What is the formula for the surface area of a dodecahedron?
A = 3√(25 + 10√5) × a², approximately 20.6457 × edge².
How many faces does a dodecahedron have?
12 regular pentagonal faces, with 30 edges and 20 vertices.
How do you find the volume of a dodecahedron?
V = ((15 + 7√5) ÷ 4) × a³, about 7.6631 × edge³.
How do you find the edge length from the surface area?
a = √(A ÷ 20.6457). Use the “From surface area” mode.
What is the surface area of one face of a dodecahedron?
About 1.7205 × a² — one twelfth of the total.
