Table of Contents

In geometry, $3D$ shapes (or three-dimensional shapes) are solid shapes or figures that have three dimensions – length, width(or breadth), and height. The most common of these shapes are cube, cuboid, cone, cylinder, and sphere. 3D shapes are defined by their respective properties such as edges, faces, vertices, curved surfaces, lateral surfaces, and volume.

We come across a number of objects of different shapes and sizes in our day-to-day life. There are matchboxes, ice cream cones, coke cans, footballs, and so on.

Let’s understand $3D$ shapes and their properties.

## What are 3D Shapes?

$3D$ shapes are solid shapes or objects that have three dimensions – length, width(or breadth), and height(or thickness), as opposed to two-dimensional objects which have only a length and a width(or breadth). They have depth and so they occupy some volume.

Some $3D$ shapes have their bases or cross-sections as $2D$ shapes. For example, a cube has all its faces in the shape of a square. Similarly, a cuboid has all its faces in the shape of a rectangle.

## Vertices, Edges, and Faces of 3D Shapes

Let’s understand the terms associated with 3D shapes such as faces, edges, and vertices.

**Vertices**

- A point where two or more lines meet is called a vertex.
- It is a corner.
- The point of intersection of edges denotes the vertices.

**Edges**

- An edge is a line segment on the boundary joining one vertex (corner point) to another.
- They serve as the junction of two faces.

**Faces**

- A face refers to any single flat or curved surface of a solid object.
- 3D shapes can have more than one face.

The following table shows the faces, edges, and vertices of a few $3$-dimensional shapes ($3D$ shapes).

## Euler’s Formula

Euler’s formula shows a relation between the number of vertices, edges, and faces in a solid shape. According to the formula, the number of vertices and faces together is exactly two more than the number of edges. We can write Euler’s formula as: $\text{Faces} + \text{Vertices} = \text{Edges} + 2$, i.e., $F + V = E + 2$ or, $F + V – E = 2$

where, $F =$ number of faces, $V =$ number of vertices and $E =$ number of edges

### Examples

**Ex 1:** Find the number of faces in a solid shape having $7$ vertices and $12$ edges.

$V = 7$ and $E = 12$

According to Euler’s formula $F + V – E = 2 => F + 7 – 12 = 2 => F – 5 = 2 => F = 7$.

Therefore, the number of faces in a solid shape having $7$ vertices and $12$ edges is $7$.

**Ex 2:** Find the number of vertices in a solid shape having $4$ faces and $6$ edges.

$F = 4$ and E = 6$

According to Euler’s formula $F + V – E = 2 => 4 + V – 6 = 2 => V – 2 = 2 => V = 4$.

Therefore, the number of faces in a solid shape having $4$ faces and $6$ edges is $4$.

**Ex 3:** Is it possible to have a solid shape with $5$ vertices, $3$ edges, and $2$ faces?

$V = 5$, $E = 3$ and $F = 2$

Since, $F + V – E = 2 + 5 – 3 = 4 \ne 2$, therefore, it is not possible to have a solid shape with $5$ vertices, $3$ edges, and $2$ faces.

## 3D Shape Names

These are some of the most common 3D shape names.

### Cube

A cube is a three-dimensional shape (3D shape) that has six square faces, eight vertices, and twelve edges. A cube occupies volume and has a surface area. The length, width, and height of a cube are the same.

### Cuboid

A cube is a three-dimensional shape (3D shape) that has six rectangular faces, eight vertices, and twelve edges. A cuboid occupies volume and has a surface area. The length, width, and height of a cuboid are different.

### Cylinder

A cylinder is a 3D shape that has two circular faces, one at the top and one at the bottom, and one curved surface. A cylinder has a height and a radius. The height of a cylinder is the perpendicular distance between the top and bottom faces.

Some important features of a cylinder are listed below.

- It has one curved face.
- The shape stays the same from the base to the top.
- It is a three-dimensional object with two identical ends that are either circular or oval.
- A cylinder in which both circular bases lie on the same line is called a right cylinder. A cylinder in which one base is placed away from another is called an oblique cylinder.

### Cone

A cone is another three-dimensional shape (3D shape) that has a flat base (which is of circular shape) and a pointed tip at the top. The pointed end at the top of the cone is called ‘Apex’. A cone also has a curved surface. Similar to a cylinder, a cone can also be classified as a right circular cone and an oblique cone.

- A cone has a circular or oval base with an apex (vertex).
- A cone is a rotated triangle.
- Based on how the apex is aligned to the center of the base, a right cone or an oblique cone is formed.
- A cone in which the apex (or the pointed tip) is perpendicular to the base is called a right circular cone. A cone in which the apex lies anywhere away from the center of the base is called an oblique cone.
- A cone has a height and a radius. Apart from the height, a cone has a slant height, which is the distance between the apex and any point on the circumference of the circular base of the cone.

### Sphere

A sphere is round in shape. It is a 3D geometric shape that has all the points on its surface that are equidistant from its center. Our planet Earth resembles a sphere, but it is not a sphere. The shape of our planet is a spheroid. A spheroid resembles a sphere but the radius of a spheroid from the center to the surface is not the same at all points.

Some important characteristics of a sphere are as follows.

- It is shaped like a ball and is perfectly symmetrical.
- It has a radius, diameter, circumference, volume, and surface area.
- Every point on the sphere is at an equal distance from the center.
- It has one face, no edges, and no vertices.
- It is not a polyhedron since it does not have flat faces.

### Pyramid

A pyramid is a polyhedron with a polygon base and an apex with straight edges and flat faces. Based on their apex alignment with the center of the base, they can be classified into regular and oblique pyramids.

- A pyramid with a triangular base is called a Tetrahedron.
- A pyramid with a quadrilateral base is called a square pyramid.
- A pyramid with the base of a pentagon is called a pentagonal pyramid.
- A pyramid with the base of a regular hexagon is called a hexagonal pyramid.

### Prism

Prism is a solid shape with identical polygon ends and flat parallelogram sides. Some of the characteristics of a prism are:

- It has the same cross-section all along its length.
- The different types of prisms are – triangular prisms, square prisms, pentagonal prisms, hexagonal prisms, and so on.
- Prisms are also broadly classified into regular prisms and oblique prisms.

### Polyhedron

A polyhedron is a 3D shape that has polygonal faces like (a triangle, square, or hexagon) with straight edges and vertices. It is also called a platonic solid. There are five regular polyhedrons. A regular polyhedron means that all the faces are the same. For example, a cube has all its faces in the shape of a square.

Some more examples of regular polyhedrons are given below:

- A Tetrahedron has four equilateral-triangular faces.
- An Octahedron has eight equilateral-triangular faces.
- A Dodecahedron has twelve regular pentagon faces.
- An Icosahedron has twenty equilateral-triangular faces.
- A Cube has six square faces.

## Conclusion

3D shapes are solid shapes or figures that have three dimensions – length, width, and height. The main attributes associated with 3D shapes are vertices, edges, and faces. These shapes are characterized by surface areas and volumes.

## Practice Problems

- It has one curved face
- Cube
- Cuboid
- Cylinder
- Pyramid

- ______ and ________ have same number of faces.
- Cube, Sphere
- Cube, Cuboid
- Sphere, Cone
- Cone, Cube

- A cone is a rotated __________.
- triangle
- circle
- square
- rectangle

- Solid shape with $2$ plane surfaces and $1$ curved surface
- Cone
- Cube
- Sphere
- Cylinder

- A polyhedron has ___________ base.
- square
- rectangle
- triangle
- Any of the above

- Which of the following is true?
- All the edges in a cube are equal
- All the edges in a cuboid are equal

## Recommended Reading

- Area of Rectangle – Definition, Formula & Examples
- Area of Square – Definition, Formula & Examples
- Area of a Triangle – Formulas, Methods & Examples
- Area of a Circle – Formula, Derivation & Examples
- Area of Rhombus – Formulas, Methods & Examples
- Area of A Kite – Formulas, Methods & Examples
- Perimeter of a Polygon(With Formula & Examples)
- Perimeter of Trapezium – Definition, Formula & Examples
- Perimeter of Kite – Definition, Formula & Examples
- Perimeter of Rhombus – Definition, Formula & Examples
- Circumference (Perimeter) of a Circle – Definition, Formula & Examples
- Perimeter of Square – Definition, Formula & Examples
- Perimeter of Rectangle – Definition, Formula & Examples
- Perimeter of Triangle – Definition, Formula & Examples
- What Are 2D Shapes – Names, Definitions & Properties

## FAQs

### What are 3D shapes with examples?

In geometry, three-dimensional shapes or 3D shapes are solids that have three dimensions such as length, width, and height. Whereas 2d shapes have only two dimensions, i.e. length and width. Examples of three-dimensional objects can be seen in our daily life such as cone-shaped ice cream, cubical boxes, a ball, etc.

### What are basic 3D shapes called?

These are the names of the 3D shapes: Cube, Cuboid, Sphere, Hemisphere, Cone, Tetrahedron or Triangular-based pyramid, Cylinder, Triangular prism, Hexagonal prism, and Pentagonal prism.

### What is the most common 3D shape?

One of the most basic and familiar polyhedrons is the cube. A cube is a regular polyhedron, having six square faces, 12 edges, and eight vertices.