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.
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.Â
3.Â SphereÂ Â Â
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.Â Â
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.
3-D shapes with curved surfaces with examples are as follows:
1.Â 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.
Â 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(5Ã—7+7Ã—12+5Ã—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: 4/3Ï€rÂ³ Â
The volume of the cement sphere v= 4/3Ï€rÂ³ Â
Substituting the value of the radius in the formula, we obtain:Â Â
4/3 * 3.14 * 20Â³ (We are taking the value of pie Ï€ = 3.14)
= 33493.33
1. How Do We Identify 3-Dimensional Objects and Shapes?
Answer: Remember that 3-dimensional objects and shapes are distinguished from 2 Dimensional objects and shapes due to the existence of the 3 dimensions - length, width, and height. As an outcome of these 3 dimensions, these objects comprise edges, faces, and vertices. For example, 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.
2. What Do We Understand by Edges of 3D Shapes?
Answer: An edge is a line segment on the boundary connecting one corner point (mathematically called a vertex) to another. An edge of 3-D objects serves as the juncture of two faces.
3. What Do We Understand by Faces of 3D Shapes?
Answer: A face is any single flat surface of a solid object. Remember that a 3 Dimensional shape can have more than one face.
4. What Do We Understand by Faces of 3D Shapes?
Answer: A point of corner where two or more lines meet is known as a vertex. Vertices are also denoted as a point of intersection of edges.
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