Dodecahedron - Definition, Properties and Examples Last Updated : 23 Jul, 2025 Comments Improve Suggest changes Like Article Like Report A dodecahedron is a 3D geometric shape that belongs to the category of polyhedra, specifically known as a Platonic solid.It has twelve flat faces, with each face being a regular pentagon. Its unique structure gives it a total of 20 vertices and 30 edges, making it both symmetrical and aesthetically pleasing.The dodecahedron can be visualized as a solid made up of 12 pentagonal faces, arranged in such a way that three faces meet at each vertex. This structure gives the dodecahedron its distinctive and aesthetically pleasing shape. The term "dodecahedron" comes from the Greek words "dodeca," meaning twelve, and "hedron," meaning face. Thus, a dodecahedron translates to a twelve-faced figure. In mathematical terms, the dodecahedron is one of the five Platonic solids, which are convex polyhedra with identical faces composed of congruent regular polygons. The other Platonic solids are the tetrahedron, cube (or hexahedron), octahedron, and icosahedron.Regular DodecahedronA regular dodecahedron is a specific type of dodecahedron with all faces being regular pentagons, meaning all sides and angles are equal. This geometric shape has the following properties:Properties of a Regular DodecahedronFaces: 12 regular pentagonal faces.Vertices: 20 vertices.Edges: 30 edges.Dihedral Angle: The angle between any two faces is approximately 116.57 degrees.Structure of a DodecahedronThe structure of a dodecahedron, specifically a regular dodecahedron, consists of several key elements.Faces, Edges, and VerticesFaces: The dodecahedron has 12 flat faces. Each face is a regular pentagon, meaning all five sides and angles are equal.Vertices: There are 20 vertices in a dodecahedron. At each vertex, three pentagonal faces meet.Edges: The dodecahedron has 30 edges. Each edge is shared by two pentagonal faces.Symmetry and GeometryIcosahedral Symmetry: The regular dodecahedron has icosahedral symmetry, meaning it has rotational symmetries similar to those of an icosahedron.Rotational Symmetry: It has 60 rotational symmetries.Reflection Symmetry: The shape also has reflective symmetries.Dihedral Angle: The angle between any two faces is approximately 116.57 degrees.Face Angles: Each internal angle of the pentagonal faces is 108 degrees.Dodecahedron NetA net for a dodecahedron consists of 12 regular pentagons arranged in such a way that they can be folded along the edges to form the 3D shape. Here’s a simplified version of the net:Net of a DodecahedronEuler's FormulaFor any convex polyhedron, Euler's formula states that: V−E+F = 2For the dodecahedron: 20 − 30 + 12 = 2Mathematical Properties of a DodecahedronSome of the common mathematical properties of the dodecahedron are listed below:VolumeThe volume V of a regular dodecahedron with edge length a is given by: V = \frac{15 + 7\sqrt{5}}{4} a^3 \approx 7.663 \, a^3Surface AreaThe surface area A of a regular dodecahedron with edge length a is given by: A = 3\sqrt{25 + 10\sqrt{5}} \, a^2 \approx 20.6457 \, a^2Radius of Circumscribed SphereThe radius R of the circumscribed sphere (sphere that passes through all vertices) for a regular dodecahedron with edge length a is given by: R = \frac{a}{4} \sqrt{3} \left(1 + \sqrt{5}\right) \approx 1.401 \,aRadius of Inscribed SphereThe radius r of the inscribed sphere (sphere tangent to all faces) for a regular dodecahedron with edge length a is given by: r = \frac{a}{2} \sqrt \frac{25 + 11\sqrt 5}{10} \approx 1.113aGolden Ratio in DodecahedronThe vertices of a regular dodecahedron centered at the origin can be represented in Cartesian coordinates.These coordinates involve permutations and combinations of: (±1, ±1, ±1)(±ϕ, ±1/ϕ, 0)(±1/ϕ, 0, ±ϕ)(0, ±ϕ, ±1/ϕ)Where ϕ = (1 + √5)/2 (the golden ratio).Applications of DodecahedronSome of the common applications of the dodecahedron are:Study of Symmetry: The dodecahedron is often used in the study of polyhedral symmetry, group theory, and geometric transformations due to its high degree of symmetry.Topology: It serves as an example in the study of topological spaces and polyhedral theory.Molecular Structures: Certain molecules and crystal structures exhibit dodecahedral symmetry, such as boron-based compounds and complex organic molecules.Fullerenes: Some forms of fullerenes (molecular carbon structures) resemble the shape of a dodecahedron.Puzzles: It is used in 3D puzzles and brain teasers to challenge spatial reasoning and problem-solving skills.Dice: The dodecahedron is used as a 12-sided die in role-playing games and other gaming applications.Read More,Triangular PrismHexagonal PrismOctagonal PrismPentagonal PyramidRectangular PyramidRegular Tetrahedron Comment More infoAdvertise with us S somesh_barthwal Follow Improve Article Tags : Mathematics School Learning Geometry Similar Reads Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields 5 min read Basic ArithmeticWhat are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or 15+ min read Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in 9 min read Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E 7 min read What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t 10 min read ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t 9 min read PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a 5 min read AlgebraVariable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch 5 min read Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one 9 min read CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is 8 min read Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari 14 min read Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−) 3 min read GeometryLines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang 9 min read Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two 2 min read Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of 10 min read Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the 10 min read Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction 14 min read Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su 6 min read Trigonometry & Vector AlgebraTrigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give 4 min read Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might 9 min read Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t 10 min read Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in 6 min read Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function 11 min read Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv 9 min read CalculusIntroduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia 6 min read Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp 12 min read Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li 13 min read DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee 2 min read Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f 11 min read IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y 3 min read Probability and StatisticsBasic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors 7 min read Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba 13 min read Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows 13 min read Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat 5 min read What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi 7 min read Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o 11 min read Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple, 3 min read PracticeNCERT Solutions for Class 8 to 12The NCERT Solutions are designed to help the students build a strong foundation and gain a better understanding of each and every question they attempt. This article provides updated NCERT Solutions for Classes 8 to 12 in all subjects for the new academic session 2023-24. The solutions are carefully 7 min read RD Sharma Class 8 Solutions for Maths: Chapter Wise PDFRD Sharma Class 8 Math is one of the best Mathematics book. It has thousands of questions on each topics organized for students to practice. RD Sharma Class 8 Solutions covers different types of questions with varying difficulty levels. The solutions provided by GeeksforGeeks help to practice the qu 5 min read RD Sharma Class 9 SolutionsRD Sharma Solutions for class 9 provides vast knowledge about the concepts through the chapter-wise solutions. These solutions help to solve problems of higher difficulty and to ensure students have a good practice of all types of questions that can be framed in the examination. Referring to the sol 10 min read RD Sharma Class 10 SolutionsRD Sharma Class 10 Solutions offer excellent reference material for students, enabling them to develop a firm understanding of the concepts covered. in each chapter of the textbook. As Class 10 mathematics is categorized into various crucial topics such as Algebra, Geometry, and Trigonometry, which 9 min read RD Sharma Class 11 Solutions for MathsRD Sharma Solutions for Class 11 covers different types of questions with varying difficulty levels. Practising these questions with solutions may ensure that students can do a good practice of all types of questions that can be framed in the examination. This ensures that they excel in their final 13 min read RD Sharma Class 12 Solutions for MathsRD Sharma Solutions for class 12 provide solutions to a wide range of questions with a varying difficulty level. With the help of numerous sums and examples, it helps the student to understand and clear the chapter thoroughly. Solving the given questions inside each chapter of RD Sharma will allow t 13 min read Like