Three Dimensional Shapes

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