An icosahedron is a 3D shape with 20 flat faces. A regular icosahedron, one of the five Platonic solids, has 20 equilateral triangle faces, 30 edges and 12 vertices, with 5 triangles meeting at each vertex. For edge length a, its surface area is 5√3 a² ≈ 8.66a² and its volume is 5(3 + √5)/12 × a³ ≈ 2.18a³. A 20-sided die (d20) is an icosahedron.
Reviewed by the Helping with Math Team · Updated 10 October 2026
| Property | Value |
|---|---|
| Faces | 20 equilateral triangles |
| Edges | 30 |
| Vertices | 12 (5 faces meet at each) |
| Euler check (V – E + F) | 12 – 30 + 20 = 2 |
| Angle in each face | 60° |
| Surface area | 5√3 a² ≈ 8.66a² |
| Volume | 5(3 + √5)/12 × a³ ≈ 2.18a³ |
| Dual solid | Dodecahedron (12 faces, 30 edges, 20 vertices) |
Example: with a = 2 cm, surface area ≈ 8.66 × 4 = 34.64 cm² and volume ≈ 2.18 × 8 = 17.45 cm³.
What is an icosahedron?
The name “icosahedron,” which derives from the Greek terms “íkosi” (which means twenty) and “hédra” (which implies seat or face), describes a solid with twenty faces.
Icosahedron definition
A polyhedron is a three-dimensional form with flat sides and straight edges, and the icosahedron is one of them. But this particular polyhedron is not just any polyhedron but a special kind known as a Platonic Solid. The icosahedron is one of the five Platonic Solids known to exist.
So, what makes an icosahedron unique? It has 20 faces, all identical equilateral triangles, 12 vertices (corners), and 30 edges. Each vertex of an Icosahedron is where five triangles meet.
Icosahedron faces, edges, vertices and formulas
Regular Icosahedron: A regular icosahedron has 20 faces that are equilateral triangles. In a regular icosahedron, all 30 edges are of equal length and have 12 vertices.
Faces: The flat surfaces of the icosahedron are called faces. In an icosahedron, there are 20 faces, and each one is an equilateral triangle.
Vertices: The points where the faces of the icosahedron meet are called vertices. In an Icosahedron, there are 12 vertices.
Edges: The line segments where two faces meet are called edges. In an icosahedron, there are 30 edges.
The volume of Icosahedron: The formula to calculate the volume of a regular icosahedron is
Volume=$\frac{5(3+\sqrt{5})}{12}$a3, where a is the length of the edge.
Total surface area of Icosahedron: The formula to find the total surface area of a regular icosahedron is:
Total Surface Area=$5\sqrt{3}$ a2, where a is the length of the edge.
Properties, parts and nets of an icosahedron
A platonic solid called an icosahedron has twenty equilateral triangular faces. It belongs to the group of five regular polyhedra known as the platonic solids, which include the tetrahedron, cube, octahedron, icosahedron, and dodecahedron.
An easy way to picture an icosahedron is to think about a d20 die, the 20-sided die used in tabletop games like Dungeons and Dragons. It is shaped like an Icosahedron – each face numbered 1 to 20.
Parts of Icosahedron
The vertex is the point of intersection of five edges.
The face is the flat surface of an icosahedron.
The edge is the line where two adjacent faces meet.
Properties of Icosahedron
To help us identify this shape, the following are the properties of an icosahedron:
An icosahedron has 20 faces. Each face is an equilateral triangle, meaning all sides and angles are equal.
An edge is the section of a line where two faces intersect. An icosahedron has 30 edges. Two edges of the same face meet at 60°, because every face is an equilateral triangle.
The corners of an icosahedron, where the faces meet, are called vertices. There are 12 vertices in an icosahedron. There are five faces at each vertex.
Icosahedron has the most number of faces among all platonic solids.
For the same surface area, the icosahedron holds the most volume of the five Platonic solids. It is the closest to a sphere.
Nets of Icosahedron
An icosahedron can be made using a geometric net. A net is a two-dimensional shape in geometry that takes on a three-dimensional appearance when folded in a particular way.
Folding the following geometric nets shown below will result in the formation of an icosahedron.
Formulas Involving Icosahedron
Total Surface Area of an Icosahedron
The total surface area of a regular icosahedron is the surface area of all its faces. An equilateral triangle has an area of A=$\frac{\sqrt{3}}{4}$ a2 where a is the edge length. Since there are 20 equilateral triangles in an icosahedron, we have,
Total Surface Area=20×$\frac{\sqrt{3}}{4}$ a2
Therefore, the formula for calculating the total surface area of a regular icosahedron is:
Total Surface Area of an icosahedron = 5√3 a2, where a is the length of the edge.
The total surface area of an icosahedron is expressed in square units.
Volume of an Icosahedron
The volume of a regular icosahedron is the amount of space it takes up. The formula for calculating the volume of a regular icosahedron is given by:
VolumeIcosahedron=$\frac{5(3+\sqrt{5}) a^3}{12}$, where a is the length of the edge.
The volume of a regular icosahedron is expressed in cubic units.
Icosahedron surface area and volume examples
Example 1
Find the surface area of an icosahedron whose edge length is 4 inches.
Solution
Let us substitute the given edge length, a=4 inches, into the formula.
Total Surface Area=5√3 a2 (Formula)
Total Surface Area=5√3 (4)2 (Substitution)
Total Surface Area=80√3 (Simplify)
Total Surface Area≈138.56
Hence, the surface area of an icosahedron with an edge length of 4 inches is 138.56 in2.
Example 2
Calculate the volume of an icosahedron whose edge length measures 8 centimeters.
Solution
Using the given edge length, a=8 cm, we have,
Volume=$\frac{5(3+\sqrt{5})}{12}$a3 (Formula)
Volume=$\frac{5(3+\sqrt{5})}{12}$ (8)3 (Substitution)
Volume≈1117.03
Therefore, the volume of an icosahedron whose edge length measures 8 cm is 1117.03 cm3.
Example 3
If the total surface area of a regular icosahedron is 405√3 square units, find its volume.
Solution
Since we know that the total surface area is 405√3 square units, let us first calculate the edge length of the icosahedron. Using the formula for getting the total surface area we have,
5√3 a2=405√3
$\frac{5\sqrt{3}a^2}{5\sqrt{3}}$=$\frac{405\sqrt{3}}{5\sqrt{3}}$ (Dividing both sides by 5$\sqrt{3}$)
a2=81 (Simplify)
√(a2)=√81
a=9 units (Edge length)
Now, using the formula for getting the volume of a regular icosahedron, we shall have,
Volume=$\frac{5(3+\sqrt{5})}{12}$a3 (Formula)
Volume=$\frac{5(3+\sqrt{5})}{12}$ (9)3 (Substitution)
Volume≈1590.46
Therefore, the volume of a regular icosahedron is 1590.46 cubic units if TSA is 405√3 square units.
Icosahedrons in real life
Problem 1
The edge of an icosahedron-shaped banner is 4.5 centimeters. Determine its surface area.
Solution
Surface Area=5√3 a2 (Formula)
Surface Area=5√3 (4.5)2 (Substitution)
Surface Area=101.25√3 (Simplify)
Surface Area≈175.37
Hence, the surface area of the icosahedron-shaped banner is 175.37 cm2.
Problem 2
Find the volume of an icosahedron-shaped water bomb whose edge measures five units.
Solution
Volume=$\frac{5(3+\sqrt{5})}{12}$a3 (Formula)
Volume=$\frac{5(3+\sqrt{5})}{12}$ (5)3 (Substitution)
Volume≈272.71
Therefore, the volume of the icosahedron-shaped water bomb is 272.71 cubic units.
The icosahedron is not just a shape in a math book. You can find it in nature and in everyday objects. So, the next time you’re rolling a d20 or hitting a golf ball, remember the icosahedron!
Icosahedron practice questions
1. How many faces does an Icosahedron have?
2. What shape are the faces of an Icosahedron?
3. How many vertices does an Icosahedron have?
4. How many edges does an Icosahedron have?
5. Calculate the surface area and volume of a regular icosahedron whose edge length measures 10 cm.
Answers:
1. 20 faces
2. Equilateral triangle
3. 12 vertices
4. 30 edges
5. Surface Area=866.03 cm2; Volume≈2181.69 cm3
More practice: drawing and describing 2d and 3d figures 7th grade worksheets.
For teachers: grade level and standards
Grade level
Kids between the ages of 8 and 15 can understand the icosahedron. It is typically introduced to kids in middle school arithmetic as they learn about geometry and three-dimensional shapes.
Math domain
The icosahedron falls within the scope of Geometry, a field of mathematics that deals with size, shape, relative placement of objects, and spatial characteristics.
Common Core standards
The study of the icosahedron is aligned with the following Common Core standards:
CCSS.MATH.CONTENT.5.G.B.3 – Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
CCSS.MATH.CONTENT.5.G.B.4 – Classify two-dimensional figures in a hierarchy based on properties.
CCSS.MATH.CONTENT.6.G.A.4 – Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.
Icosahedron FAQs
How many faces, edges and vertices does an icosahedron have?
A regular icosahedron has 20 faces, 30 edges and 12 vertices. Euler’s formula checks out: 12 – 30 + 20 = 2.
What shape are the faces of an icosahedron?
Each face of a regular icosahedron is an equilateral triangle. All 20 faces are the same size, and 5 of them meet at every vertex.
What is the formula for the surface area of an icosahedron?
Surface area = 5√3 × a², which is about 8.66 × a², where a is the edge length. It comes from 20 triangles, each with area (√3/4)a². An edge of 4 inches gives about 138.56 in².
What is the formula for the volume of an icosahedron?
Volume = 5(3 + √5)/12 × a³, which is about 2.18 × a³. An edge of 8 cm gives about 1,117.03 cm³.
What is the difference between an icosahedron and a dodecahedron?
An icosahedron has 20 triangle faces, 30 edges and 12 vertices. A dodecahedron has 12 pentagon faces, 30 edges and 20 vertices. They are duals: the face and vertex counts swap.
Where do you see icosahedrons in real life?
Twenty-sided dice (d20), the shells of many viruses such as the common cold virus, some geodesic domes and lamps, and the dimple layout of some golf balls.
