## What are Solid Shapes in Geometry?

Look at these two shapes.

While the rectangle only has a length and breadth, the cuboid also has the dimension of height, making it a solid shape. The rectangle is a plane figure, whereas the cuboid looks more like a real solid figure.

You will find multiple objects that fit the solid shape criteria if you look around. Living in a three-dimensional world, we will likely come across solid shapes more often than two-dimensional shapes. There are endless examples of solid shapes surrounding us, from a matchbox to a birthday cap.

There are four features of solid shapes that make them different from plane shapes, which are discussed below.

## Elements of Solid Shapes

- Faces: A face is a single flat surface of a solid shape, and there can be more than one face of a shape.

- Vertices: A vertex is a point where two or more lines meet, forming a corner. It is also the point of intersection of edges.

- Edges: An edge is a line segment at the boundary of a solid shape that joins one vertex to another.

- Volume: Every solid shape occupies some volume, which is not the case in a two-dimensional object.

## Types of Solid Shapes

### Sphere

A sphere is round in shape, like the moon or a ball. It is perfectly symmetrical, and it has no edges or vertices. So, it has only one surface. Every point on a sphere is located such that it is at an equal distance from a central point on the sphere.

The following are formulas related to the sphere shape:

Total Surface Area = 4πr², where r is the radius of the sphere.

Volume = $\frac{4}{3}$πr³, where r is the radius of the sphere.

### Cylinder

Consider a can of your favorite fruit juice. It has a flat base and top, both the same size. Going from the base to the top, the shape of the cylinder remains the same. Also, it has one curved side that connects the base to the top.

A cylinder is a solid shape with a curved surface joining its top and bottom circular bases. Think of it as a can of toffees.

The following are formulas related to the cylinder shape:

Curved Surface Area = 2πrh, where r is the radius of the base and h is the height.

Total Surface Area = 2πr(h + r), where r is the radius of the base and h is the height.

Volume = $\pi$$r^2$$h$, where r is the radius of the base and h is the height.

### Cuboid

Any box-shaped object resembles the shape of a cuboid. It has a total of six faces, and all the angles of the cuboid stand at a right angle. An example of a cuboid shape is a matchbox or a smartphone. It is a solid rectangular shape with six faces, each of which is a rectangle. It has a total of eight vertices and twelve edges. Often, it is also referred to as a rectangular prism.

The following are formulas related to the cuboid shape:

Curved Surface Area = 2h (l + b), where l = length, b = width, h = height of the cuboid.

Total Surface Area = 2 (lb + bh + hl) ,where l = length, b = width, h = height of the cuboid.

Volume = lbh, where l = length, b = width, h = height of the cuboid.

### Cube

Consider a cube of ice in the tray of your fridge. All six faces of the cube are the same, which also makes it a square prism. Even a Rubik’s Cube or a playing dice are examples of a cube. This is what makes it a solid 3D object. A cube is a symmetrical three-dimensional shape contained within six equal squares. It may be solid or hollow.

The following are formulas related to the cube shape:

Curved Surface Area = $4a^2$, where a = edge length of the cube.

Total Surface Area = $6a^2$, where a = edge length of the cube.

Volume = $a^3$, where a = edge length of the cube.

### Cone

A cone is a distinctive three-dimensional geometric figure with a flat and curved surface pointed toward the top. A cone has 3 dimensions—its radius, height, and slant height.

A birthday cap and a funnel are some examples of the cone shape.

The following are formulas related to the cone shape:

Curved Surface Area = πrl, where r is the radius of the base and l is the slant height.

Total Surface Area = πr (l + r), where r is the radius of the base and l is the slant height.

Volume = $\frac{1}{3}$ $\pi$$r^2$$h$, where r is the radius of the base and h is the height.

### Pyramid

The mention of a pyramid brings to mind images of the great structures built in Egypt. The pyramid’s base is flat with straight edges, and the remaining faces of the pyramid are triangles. There are no curves in a pyramid.

It is a polyhedron with a polygon base and all lateral faces in a triangle shape. Depending on the alignment with the center of the base, a pyramid can be further classified as a regular or an oblique pyramid.

### Prism

A prism is a solid object with identical ends, flat faces, and no curves.

A prism has two identical shapes facing each other and can be of different types, such as triangular prisms, hexagonal prisms, pentagonal prisms, etc. A unique feature of a prism is that it has the same cross-section throughout its length.

## Solved Examples

**Example 1: If we want to build a solid sphere by filling it with cement, how much cement will be required to construct one sphere of radius 10 cm?**

**Solution**: We know that the volume of a sphere is given by

Volume = $\frac{4}{3}$πr³, where r is the radius of the sphere.

Here r = 10 cm,

Therefore, Volume of given sphere = $\frac{4}{3}$ ✕ 3.14 ✕ 10 ✕ 10 ✕ 10 = 4186.6 cubic centimeters

**Example 2: Calculate the volume of a cylinder with a radius of 3 cm and a height of 9 cm.**

**Solution**: Using the formula to calculate the volume of a cylinder, we get

Volume = $\pi$$r^2$$h$, where r is the radius of the base and h is the height.

Here r = 3 cm and h = 9 cm

Therefore, volume of the given cylinder is,

V = 3.14 ✕ 3 ✕ 3 ✕ 9 = 254.34 cm³.

**Example 3: What will be the surface area of a cuboid whose dimensions are as follows:**

**Length = 8 cm**

**Breadth = 5 cm**

**Height = 7 cm**

**Solution**: To calculate the total surface area of a cuboid, we can use the formula

Total surface area = 2 ✕ (lb+bh+lh), where l = length, b = width, h = height of the cuboid.

Here l = 8 cm, b = 5 cm, h = 7 cm

Therefore, total surface area of the given cuboid is

Total Surface Area = 2 ✕ ( 8 ✕ 5 + 5 ✕ 7 + 8 ✕ 7)

= 2 ✕ (40 + 35 + 56)

= 2 ✕ 131

= 262 square centimeters.

## Practice Problems

## Solid Shapes - Definition, Types, Properties, Examples

### What will be the volume of a sphere with a radius of 7 cm?

Volume = $\frac{4}{3}$πr³, where r is the radius of the sphere.

Here r = 7 cm,

Therefore, Volume of given sphere = $\frac{4}{3}$ ✕ 3.14 ✕ 7 ✕ 7 ✕ 7 = 1436.02 cubic centimeters

### Find the volume of a cube that has an edge of length 5 cm.

We know that volume of a cube is given by

Volume = $a^3$, where a = edge length of the cube.

For the given cube, a = 5 cm

Therefore, Volume of given cube = 5 ✕ 5 ✕ 5 = 125 cm³

### Which of the following is not a solid shape?

A circle is a two-dimensional object since it is a closed and flat figure. We can call it a plane figure with no sides or edges.

### Calculate the volume of a right cylinder that has a radius of 5 cm and a height of 10 cm.

Using the formula to calculate the volume of a cylinder

We get Volume = $\pi$$r^2$$h$, where r is the radius of the base and h is the height.

Here r = 5 cm and h = 10 cm

Therefore, volume of the given cylinder is,

V = 3.14 ✕ 5 ✕ 5 ✕ 10 = 785 cm³

## Frequently Asked Questions

**What are platonic solids?**

Platonic solid shapes have identical faces and are also known as polyhedrons, which can be of five types, namely, tetrahedron, octahedron, dodecahedron, icosahedron, and hexahedron.

**How many edges does a cylinder have?**

A cylinder does not have any edges, but it has two faces.

**Does every solid shape have at least one face, edge, and vertex?**

Solid shapes don’t need all faces, edges, and vertices. For example, a sphere has one rolling surface but no edges or vertices.