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# Solid|Definition & Meaning

## Definition

Any **three-dimensional object** is generally called a **solid.** These are shapes as we perceive them around us. Unlike 2D shapes, solids have the **third dimension** of depth in addition to length and width, which leads to the concept of **volume**. Pyramids, cylinders, cubes, spheres, etc., are all **examples** of solid 3D shapes, and their study is called **Solid Geometry.**

**Solids** form the basis of a field of **geometry** called Solid Geometry. It is **slightly different** from traditional geometry in the sense that the **third dimension** raises a degree of **complexity.** Some well-known solids include **pyramids, cubes,** rectangular **prisms, spheres, cones,** and **cylindrical** objects, etc. These objects are shown in the **figure** below:

**Figure 1: Examples of Solids**

## Explanation of Solids

As explained earlier, **solids** are a class of **three-dimensional** objects having **length, breadth,** and **height** (names of the three dimensions). If someone already has a background in **two-dimensional** shapes such as **circles, squares, triangles,** etc.

Such solid shapes may seem complex at first glance. However, with a little **intuition,** one can see the **relationship** between both domains and realize that **three-dimensional** geometry or solids can be explained directly from the **two-dimensional** counterpart.

To elaborate on this concept further, let us **consider** an **example.** The following **figure** shows a sphere (that is a solid).

**Figure 2: Relation between Solids and 2D Shapes**

Now if we slice the sphere at **three locations,** as shown in the **figure,** and then see the **cross-section** of the object. We will see that these cross sections are a **two-dimensional shape.** It turns out that it’s a **circle.** We may **generalize** from this example that if we **slice** any **solid** object along any **one** of the **three dimensions,** the resulting shape would be a two-dimensional shape.

The above concept is very **fundamentally** **vital** for deriving expressions and **formulae** of different **characteristics** of **solid** objects. Let us consider this same concept, but now, let us learn by way of **mathematical expressions.**

Consider the **equation** of a **unit circle** (radius = 1) at origin (center at origin) in 2D (xy plane):

x^{2} + y^{2} = 1

The **equation** of a **solid sphere** with a **unit radius** (radius = 1) located at the origin in 3d (xyz space) is shown below:

x^{2} + y^{2} + z^{2} = 1

**Comparing** the above two **equations,** can you guess the **relationship** between a circle and a sphere? Yes, it is very intuitive. **Two statements** can be **deducted** from these formulae.

**First,** the **2D shape** (circle) is a **subset** of the **3D solid** (sphere), which can be obtained by dropping any one of the dimensions. **Second,** the expressions for **2D shapes** can be **extended** to **3D solids** by adding another dimension. However, the latter mapping is a **one-to-many relationship** and would not have a **unique solution.**

**Figure 3: Equation of a Sphere and a Circle**

The **characteristics** of these solid objects, such as the number of **faces, vertices,** **edges,** etc., may be **frequently** found in the **literature.** So one may need to understand all these basic **terminologies** to grasp some higher-level concepts.

Examples and **visual depictions** of solids in the form of diagrams and figures can assist **beginners** in understanding such objects and developing the basic **intuition** that is existential for working with them.

Other than **basic solids** (spheres, cubes, cones, etc.), several other complicated forms are not that easy to remember and may only be **explained mathematically.** Such objects are usually illustrated with the help of **3D rendering** software such as **AutoCAD** and **SketchUp.**

## Applications of Solids

Although we are more concerned with **2D shapes** in our mathematics courses but the **real world** is **3D,** and to build anything useful with real-world **applications,** we have to understand **solid geometry.** Solids find applications in many fields of **engineering** and **art.**

**Architecture, Physics,** and **Graphics** are the most significant among these. Engineers may construct and test prototypes while architects use them to plan and visualize architectural ideas.

**Figure 4: Application of Solids**

## Some Common Characteristics of Solids

When studying **solid** shapes, it is essential to comprehend the **properties** of **solids** and how they may be **mathematically** stated. Solid shapes are, after all, three-dimensional objects with **length, width,** and **height.** They may be distinguished by their **characteristics,** such as the number of **faces, vertices,** and **edges,** and the knowledge of these terms is crucial for their **applications** in many areas.

Following are some of the important and **commonly used terminologies:**

**Volume:**Volume is the**amount of space**a solid shape takes up. It is very commonly used, and**standard formulae**exist for standard solid models.**Surface Area:**Surface area is the**total area**of all of a solid shape’s**faces.****Cross-Sectional Area:**The**area of any slice**or cross-section of a solid object is called the cross-sectional area.**Edges:**Line segments that connect two of its**faces**or**vertices**are called**edges**of a solid body.**Vertices:**The point of**intersections**on a solid body where two or more**edges**meet are called**vertices.****Faces:**The**flat**surfaces or regions comprising points with the same**surface normal**are called the**faces**of the solid object.**Radius:**For**spherical**or**circular areas**of a solid body, the radius is defined as the distance between the**circumferential points**and the**center**of the circle or sphere.

There are also other, more advanced terms, but these are enough at this level.

## Some Elementary Solids

Let us consider some of the **elementary solids** in **detail.**

### Solid Cube

A **cube** has **six square faces, eight vertices,** and **twelve edges.** They are one of the easiest solid objects to work with, and because of their three-dimensional construction, they are frequently used as **building blocks** and **toys.**

### Rectangular Box

Rectangular **prisms** can have any configuration of **rectangles** on their **faces,** much like cubes. Similar to each other, these objects contain **eight vertices** and **twelve edges.** Rectangular prisms may be used to replicate genuine objects like **furniture** and **buildings** as well as **boxes** and **containers.**

### Solid Sphere

**Spheres** are round objects with **no vertices or edges.** Instead, their **center** and **radius** operate as distinguishing **features.** Because they have a smooth, continuous surface, spheres are commonly used to symbolize objects like **planets** and **basketballs.**

### Solid Cones

A cone’s **base** is **circular** and has a **single vertex.** The cone’s axis, which has **one flat face** and **one curved face,** runs from the vertex to the center of the base. Cones are used to mimic a range of objects, such as **ice cream cones** and **party hats.**

### Cylinders

Cones and cylinders are related; however, cones only have one curved face, whereas **cylinders** have **two parallel faces.** The route that joins the centers of the two circular faces is known as the **cylinder’s axis.** Cylinders are used to create models for objects like **cans** and **tubes.**

## Daily Life Examples of Solids

As referenced earlier, the following are some **examples** of real-world solids:

- Solid Cube:
**Lego building blocks**and**toys** - Rectangular Box: Packaging
**boxes**and**containers** - Solid Sphere:
**planets**and**basketballs** - Solid Cones:
**Ice cream**cones and**party hats** - Cylinders:
**Cans**and**tubes**

Of course, there are many others.

## Numerical Problems of Solids

(a) Calculate the **volume** of a **cylinder** with a height of **6 cm** and a radius** of 3 cm.**

(b) Calculate the **volume** of a **sphere** with a radius** of 3 cm.**

(c) Calculate the **volume** of a **cone** with a height of **6 cm** and a radius** of 3 cm.**

### Solution

#### Part (a): Volume of a Cylinder

The formula for a **cylinder’s volume** is as follows:

V = πr^{2}h

where h is the **height** and r is the **radius.** **Substituting** given values:

V = 3.14 x (3 cm)^{2} (6 cm)

V = 3.14 x 9 x 6 cubic centimeters

V = 169.56 cubic centimeters

The **cylinder’s volume** is **169.56 cubic centimeters.**

#### Part (b): Volume of a Sphere

The formula for a **sphere’s volume** is as follows:

V = (4/3)πr^{3}

where r is the **radius. Substituting** given values:

V = (4/3) (3.14) (3 cm)^{3}

V = 113.1 cubic centimeters

The **sphere’s volume** is **113.1 cubic centimeters.**

#### Part (c): Volume of a Cone

The formula for a **cone’s volume** is as follows:

V = πr^{2}h/3

where h is the **height** and r is the **radius.** **Substituting** given values:

V = 3.14 x (3 cm)^{2} (6 cm) / 3

V = 3.14 x 9 x 6 cubic centimeters

V = 56.55 cubic centimeters

The **cone’s volume** is **56.55 cubic centimeters.**

*All images were created with GeoGebra.*