Geometric Solid in Three Dimensional Geometry

We call geometry a branch of Math that focuses on the measurement and relationship of lines, angles, surfaces, solids, and points. An example of geometry is the calculation of a triangle's angles. Shapes that we study in geometry are 2-d and 3-d.

2-d shapes have two dimensions, such as x and the y-axis. Also, it has length and width. Examples are circle, square, rectangle, and so on. However, shapes like sphere, hemisphere, cylinder, cube, and cuboid have three dimensions, i.e., x, y, and z-axis. The walls of your room have a 3-d structure. Also, these shapes have length, height or depth, and width.

The walls of your room have a kind of 3-d space which you live in. And the geometry of this 3-d space is called the solid geometry or simply we can say it is a geometric solid.

What is Geometric Solid?

Before starting with what solid geometry is, let us go through some of the common 3-d shapes:

Image: Solid common 3-d shapes

We understood from the above text that these shapes have three dimensions having length, depth, and width. These shapes carry the following properties:

• Capacity or volume (think of how much water it could hold).

• Surface area (think of the amount of area covered by the surface of something).

• How many vertices (corner points), faces, and edges do they have.

Formulas of Solid Geometry

The following table gives the volume formulas and surface area formulas for the following 3-D solid shapes:

3-D shape	Image	Surface Area Formula	Volume Formula
Cube with side ‘s’		6s2	s3
Rectangular prism with length l, height h, and width w		2 (lw + wh + hl)	lwh
Prism with an area of base ‘B’, the perimeter of base B, ‘P’, and height ‘h’		2B + PH	Bh
Cylinder radius = r, height = h		2πr2 + 2πrh	πr2h
Sphere radius = r		4πr2	4/3 πr3
Cone radius = r, vertical height = h, slant height = h		πr2 + πrs	⅓ πr2h
Pyramid with an area of base B, the perimeter of base P, vertical height h, and slant height s		Regular pyramid B + 1 p/s	⅓ bh

Solved Examples of Geometric Solid

1. Find the volume and surface area of a cube whose side is 7 cm.

Solution:

Side, a = 7 cm.

The volume of a cube = a3 cubic units

V = 73

V = 7 x 7 x 7

V = 343 cm3

Therefore, the volume of a cube is 343 cubic centimeter.

The surface area of a cube = 6a2 square units

SA = 6 x (7)2 cm2

SA = 6 x (49)

SA = 294 cm2

Therefore, the surface area of a cube is 294 square centimeters.

2. Find the total surface area of a cuboid of dimensions 9 cm × 8 cm × 6 cm.

Solution:

Given dimensions of a cuboid: 9 cm x 8 cm x 6 cm

That means length = l = 9 cm, breadth = b = 8 cm, and height = h = 6 cm.

Total surface area of a cuboid = 2 (lb + bh +hl)

= 2 (9 x 8 + 8 x 6 + 6 x 9)

= 2 (72 + 48 + 54 )

= 348 cm2

Hence, the total surface area of the cuboid is 348 cm2.

From the above text on geometric solid, we understand that solid geometry is regarded with 3-D shapes. Examples of three-dimensional solid shapes are cubes, rectangular solids, prisms, cylinders, spheres, cones, and pyramids.