Three Dimensional Shapes
In mathematics, 3D shapes are nothing but solids that comprises 3 dimensions, namely - length, width, and height. The "D" here stands for "Dimensional." These 3D shapes preoccupy space and are applied in our day-to-day life. We can touch, use and feel them. 3D shapes are classified into various categories, of which some of them have curved surfaces; some are in the shape of prisms or pyramids. A 2-D solid shape that can be moulded to form a 3-D object is called a geometrical net.
3D Shapes and their Properties
Solid shapes that are inclusive of a curved or round edge are not polyhedrons. Polyhedrons can only consist of straight sides. Many of the geometrical objects around us will include a minimum of some curves. In Euclidean geometry, the most common curved solids are cones, cylinders, spheres and tori (plural form for torus). Following are the common 3-D shapes with curves:
1. Cone
A cone consists of a circular or oval base and an apex (or vertex). The side of the cone narrows down smoothly to the apex. A cone is the same as a pyramid but is marked different as a cone has a single circular base and a curved side.
2. Cylinder
A cylinder consists of a similar cross-section from one end to the other. Cylinders possess two identical ends of either an oval or a circle. Although identical, cylinders are not prisms.
3. Sphere
Shaped like a globe or a ball, a sphere is a round object. Every point on the surface of a sphere is equidistant to the centre of the sphere.
4. Tori or Torus
Shaped like a ring, bangle, a tire or a doughnut, a regular ring torus is created by rotating a smaller circle around a bigger circle. There is even a more complex form of tori.
More Examples of three Dimensional Shapes
3-D shapes with curved surfaces with examples are as follows:
1. Pyramid
A pyramid is a polyhedron with a polygon base and an apex with straight lines. Depending upon its apex alignment with the centre of the base, they can be categorized into regular and oblique pyramids. Moreover, a pyramid with a triangular base is known as a Tetrahedron, the quadrilateral base is known as the square pyramid, pentagon base is known as a pentagonal pyramid and a regular hexagon is called a hexagonal pyramid
2. Prisms
Prisms are solid shapes with flat parallelogram sides and similar polygon ends. It consists of a similar cross-section all along its length.
Prisms are also widely classified into regular prisms and oblique prisms. Various kinds of prisms are as follows-
triangular prisms
square prisms
pentagonal prisms
Hexagonal prisms, etc.
Next, let's learn about 3-D shapes with regular polyhedrons (Platonic Solids).
3. Polyhedrons / Platonic solids
There are 5 polyhedrons.
4- equilateral-triangular faces are called Tetrahedron
8- equilateral-triangular faces are Octahedron
12- Pentagon faces are called Dodecahedron
20- equilateral-triangular faces are called Icosahedron
6- square face is a Cube
They have similar faces of regular polygons.
Solved Examples
Example1:
Evaluate the surface area of a cuboid having length 5 inches, breadth 7 inches, and height 12 inches.
Solution:
Given that,
Length of the cuboid = 5 inches
Breadth of the cuboid = 7 inches
Height of the cuboid = 12 inches
Now, applying the formula for Surface area of the cuboid i.e
2 * (lb + bh+ lh)
=2(5×7+7×12+5×12)
=2(35+ 84 + 60)
= 2(109)
= 218 square inches
Example2:
A carpenter wants to construct a 3D sphere using cement. He seeks to know the amount of cement needed to build the sphere of radius 20 inches. Determine the volume of the sphere using the given value of radius.
Solution:
Given that,
The radius of the sphere (r) = 20 inches
Apply the formula for the volume of a sphere: 4/3πr³
The volume of the cement sphere v= 4/3πr³
Substituting the value of the radius in the formula, we obtain:
4/3 * 3.14 * 20³ (We are taking the value of pie π = 3.14)
= 33493.33
FAQs on Three Dimensional Shapes
1. What are three-dimensional shapes?
Three-dimensional shapes, often called 3D shapes or solids, are objects that have three dimensions: length, width, and height. Unlike flat two-dimensional shapes, 3D shapes occupy space and have volume. You can think of them as objects you can hold in your hand, like a box or a ball.
2. What are some common examples of 3D shapes in everyday life?
You can find examples of three-dimensional shapes all around you. Some common ones include:
- Cube: A dice or a sugar cube.
- Cuboid: A book, a brick, or a shoebox.
- Sphere: A football, a marble, or an orange.
- Cylinder: A soda can, a battery, or a pipe.
- Cone: An ice cream cone or a birthday party hat.
3. What are the faces, edges, and vertices of a 3D shape?
The three key properties of many 3D shapes, especially polyhedrons, are its faces, edges, and vertices.
- Faces are the flat surfaces of the shape.
- Edges are the straight lines where two faces meet.
- Vertices (singular: vertex) are the corners where two or more edges meet.
For example, a cube has 6 faces, 12 edges, and 8 vertices.
4. How can you tell the difference between a two-dimensional (2D) and a three-dimensional (3D) shape?
The primary difference lies in the number of dimensions they possess. A 2D shape, like a square or a circle, is completely flat and only has length and width. You can draw it on a piece of paper. A 3D shape, however, has an additional dimension of height or depth, which gives it volume and makes it a solid object that takes up space.
5. What is a polyhedron, and can you give an example of a 3D shape that is not a polyhedron?
A polyhedron is a specific type of 3D shape made up entirely of flat surfaces (polygonal faces), straight edges, and sharp corners (vertices). Common examples include cubes, prisms, and pyramids. Shapes with any curved surfaces are not polyhedrons. For instance, a sphere, a cylinder, or a cone are not polyhedrons because their surfaces are curved, not flat polygons.
6. Is there a mathematical relationship between the number of faces, vertices, and edges in a polyhedron?
Yes, for any convex polyhedron, there is a consistent relationship known as Euler's Formula. The formula states that the number of Faces (F) plus the number of Vertices (V) minus the number of Edges (E) always equals 2. The formula is written as: F + V - E = 2. This is a fundamental property for verifying the structure of polyhedrons. For a cube, this would be 6 faces + 8 vertices - 12 edges = 2.
