3D Shapes - Definition, Properties, Formulas, Types Of ... - Cuemath

3D Shapes

3D shapes are solids that consist of 3 dimensions - length, breadth (width), and height. 3D in the word 3D shapes means three-dimensional. Every 3D geometric shape occupies some space based on its dimensions and we can see many 3D shapes all around us in our day-to-day life. Some examples of 3D shapes are cube, cuboid, cone, and cylinder.

1.	What are 3D Shapes?
2.	Types of 3D Shapes
3.	Properties of 3D Shapes
4.	3D Shapes Formulas
5.	3D Shapes - Faces, Edges, Vertices
6.	3D Shapes Nets
7.	FAQs on 3D Shapes

What are 3D Shapes?

3D shapes are solid shapes or objects that have three dimensions (which are length, width, and height), as opposed to two-dimensional objects which have only a length and a width. Other important terms associated with 3D geometric shapes are faces, edges, and vertices. 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. Let us now learn about each 3-dimensional shape (3D shape) in detail. 3D shapes are classified into several categories. Some of them have curved surfaces; some are in the shape of pyramids or prisms.

Real-Life Examples of 3D Geometric Shapes

In mathematics, we study 3-dimensional objects in the concept of solids and try to apply them in real life. Some real-life examples of 3D shapes are shown below which are a soccer ball, a cube, a bucket, and a book.

Types of 3D Shapes

There are many 3 dimensional shapes (3D shapes) that have different bases, volumes, and surface areas. Let us discuss each one of them.

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.

Cube and Cuboid

Cube and cuboid are three-dimensional shapes (3D shapes) that have the same number of faces, vertices, and edges. The main difference between a cube and a cuboid is that in a cube, all its six faces are squares and in a cuboid, all its six faces are rectangles. A cube and a cuboid occupy different volumes and have different surface areas. The length, width, and height of a cube are the same, while for a cuboid, length, height, and width 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 the 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.

Torus

A torus is a 3D shape. It is formed by revolving a smaller circle of radius (r) around a larger circle with a bigger radius (R) in a three-dimensional space.

• A torus is a regular ring, shaped like a tire or doughnut.
• It has no edges or vertices.

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.

Prisms

Prisms are solids 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.

Now, let us learn about 3-dimensional shapes that are platonic solids.

Polyhedrons

A polyhedron is a 3D shape that has polygonal faces like (triangle, square, 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.

Properties of 3D Shapes

Every 3D shape has some properties which help us to identify them easily. Let us discuss each of them briefly.

3D Shapes	Properties
Sphere (With radius - r)	It has no edges or vertices (corners). It has one curved surface. It is perfectly symmetrical. All points on the surface of a sphere are at the same distance (r) from the center.
Cylinder	It has a flat base and a flat top. The bases are always congruent and parallel. It has one curved side.
Cone	It has a flat base. It has one curved side and one-pointed vertex at the top or bottom known as the apex.
Cube	It has six faces in the shape of a square. The sides are of equal lengths. 12 diagonals can be drawn on a cube.
Cuboid	It has six faces which are rectangular in shape. All the sides of a cuboid are not equal in length. 12 diagonals can be drawn on a cuboid.
Prism	It has identical ends (polygonal) and flat faces. It has the same cross-section all along its length.
Pyramid	A Pyramid is a polyhedron with a polygon base and an apex with straight lines. Based on their apex alignment with the center of the base, they can be classified into regular and oblique pyramids.

3D Shapes Formulas

As discussed, all 3 Dimensional shapes have a surface area and volume. The surface area is the area covered by the 3D shape at the bottom, top, and all the faces including the curved surfaces, if any. Volume is defined as the amount of space occupied by a 3D shape. Every three-dimensional shape (3D shape) has different surface areas and volumes. The following table shows different 3D shapes and their formulas.

3D Shape	Formulas
Sphere	Diameter = 2 × r; (where 'r' is the radius) Surface Area = 4πr2 Volume = (4/3)πr3
Cylinder	Total Surface Area = 2πr(h+r); (where 'r' is the radius and 'h' is the height of the cylinder) Volume = πr2h
Cone	Curved Surface Area = πrl; (where 'l' is the slant height and l = √(h2 + r2)) Total Surface Area = πr(l + r) Volume = (1/3)πr2h
Cube	Lateral Surface Area = 4a2; (where 'a' is the side length of the cube) Total Surface Area = 6a2 Volume = a3
Cuboid	Lateral Surface Area = 2h(l + w); (where 'h' is the height, 'l' is the length and 'w' is the width) Total Surface Area = 2 (lw + wh + lh) Volume = (l × w × h)
Prism	Surface Area = [(2 × Base Area) + (Perimeter × Height)] Volume = (Base Area × Height)
Pyramid	Surface Area = Base Area + (1/2 × Perimeter × Slant Height) Volume = [(1/3) × Base Area × Altitude]

3D Shapes - Faces Edges Vertices

As mentioned earlier, 3D shapes and objects are different from 2D shapes and objects because of the presence of the three dimensions - length, width (breadth), and height. As a result of these three dimensions, these objects have faces, edges, and vertices. Observe the figure given below to identify the face, vertex, and edge of a 3D shape.

Faces

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

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.

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.

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

3D shapes	Faces	Edges	Vertices
Sphere	1	0	0
Cylinder	3	2	0
Cone	2	1	1
Cube	6	12	8
Rectangular Prism	6	12	8
Triangular Prism	5	9	6
Pentagonal Prism	7	15	10
Hexagonal Prism	8	18	12
Square Pyramid	5	8	5
Triangular Pyramid	4	6	4
Pentagonal Pyramid	6	10	6
Hexagonal Pyramid	7	12	7

3D Shapes Nets

We can understand the three-dimensional shapes and their properties by using nets. A 2-dimensional shape that can be folded to form a 3-dimensional object is known as a geometrical net. A solid may have different nets. In simple words, the net is an unfolded form of a 3D figure. Observe the following figure to see the nets that are folded to make a 3D shape.

☛ Related Topics:

• Closed Shapes
• Plane Shapes
• Visualizing Solid Shapes
• Oval Shape

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