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Solid Shapes

Solid shapes are nothing but solids that consist of 3 dimensions, namely length, breadth, and height. Solid shapes are also known as 3D shapes. These solid shapes occupy space and are found in our day-to-day life. We touch, feel, and use them. In this fun lesson, you can check out some interactive examples to know more and try your hand at solving a few interesting practice questions at the end of the page.

1.	What are Solid Shapes?
2.	Solid Shapes and their Properties
3.	Faces, Edges, and Vertices of Solid Shapes
4.	FAQs on Solid Shapes

What are Solid Shapes?

In mathematics, we study shapes and their different types and try to apply them in real life. We will now learn about each solid shape in detail. Solid shapes are classified into several categories. Some of them have curved surfaces; some are in the shape of pyramids or prisms.

Solid Shapes Definition: Solid shapes are three-dimensional shapes that have length, breadth, and height as the three dimensions.

Three Dimensional Shape

Let us first learn about solid shapes with curved surfaces with examples.

Solid Shapes and their Properties

Solid shapes correspond to three-dimensional objects. Look around! Every other three-dimensional object, be it a laptop, cellphone, an ice-cream cone, balls, etc, are examples of solid shapes. These occupy some space, have length, width as well as height. Let's explore types of solid shapes-

Sphere
Cylinder
Cone
Pyramid
Prism

Sphere

A sphere is a solid shape, absolutely round in shape, defined in three-dimensional space. Every point on the surface is equidistant from the center.

Shape of a Sphere - example basketball

The table below shows the properties of a sphere:

Properties	Surface Area	Volume
It has no edges or vertices (corners). It has one surface. It is shaped like a ball and is perfectly symmetrical. All points on the surface are the same distance (r) from the center.	4πr²	(4/3)πr³

Cylinder

A cylinder is a solid shape defined on a three-dimensional plane. It holds two parallel bases, circular in shape, joined by a curved surface(like a tube), at a fixed distance.

Shape of a Cylinder - example bucket

The table below shows the properties of a cylinder:

Properties	Surface Area	Volume
It has a flat base and a flat top. It has one curved side. The bases are always congruent and parallel. It is a three-dimensional object with two identical ends that are either circular or oval.	2πr (r+h)	πr²h

Cone

A cone is a distinctive solid shape defined in a three-dimensional space. It has a flat surface and a curved surface, pointing towards the top. It is formed by a set of line segments connected from the circular base to a common point, known as the apex or vertex. Based on how the apex is aligned to the center of the base, a right cone or an oblique cone is formed.

Shape of a Cone - example ice cream cone

The table below shows the properties of a cone where r denotes radius, h represents height and s represents slant height of the cone:

Properties	Surface Area	Volume
It has a circular or oval base with an apex (vertex). It has one curved side. A cone is a rotated triangle.	π r(r + s)	1/3 πr²h

Pyramid

A pyramid is a solid shape or a polyhedron with a polygon base and all lateral faces are triangles. Pyramids are typically described by the shape of their bases. A pyramid with a:

Triangular base is called aTetrahedron.
A quadrilateral base is called a square pyramid.
Pentagon base is called a pentagonal pyramid.
Regular hexagon base is called a hexagonal pyramid.

Types of Pyramids

The table below shows the properties of a pyramid: (BA = base area, P = perimeter, A = altitude, and SH = slant height )

Properties	Surface Area	Volume
A pyramid is a polyhedron with a polygon base and an apex with straight lines. Based on their apex alignment with the center of the base, they can be classified into regular and oblique pyramids.	BA + 1/2 × P × (SH)	1/3 BA²

Prisms

A prism is a solid shape defined on a 3-dimensional plane with two identical shapes facing each other. The different types of prisms are triangular prisms, square prisms, pentagonal prisms, hexagonal prisms, etc. Prisms are also broadly classified into regular prisms and oblique prisms.

Types of Prisms

The table below shows the properties of a prism: (BA = base area, P = perimeter, H = height)

Properties	Surface Area	Volume
It has identical ends (polygonal) and flat faces. It has the same cross-section all along its length.	2 × (BA) + P × H	BA × H

Polyhedrons/Platonic Solids

Platonic solids have identical faces to regular polygons. There are five polyhedrons.

Tetrahedron with four equilateral-triangular faces
Octahedron with eight equilateral-triangular faces
Dodecahedron with twelve pentagon faces
Icosahedron with twenty equilateral-triangular faces
Hexahedron or cube with six square faces.

Types of Polyhedrons

The table below shows the properties of platonic shapes: (EL = edge length)

Properties of Cube	Surface Area	Volume
It has 6 faces, each with 4 edges (and is a square). It has 12 edges. It has 8 vertices (corner points) where 3 edges meet.	6 × (EL)²	(EL)³

Faces, Edges, and Vertices of Solid Shapes

As mentioned before, solid shapes and objects are different from 2D shapes and objects because of the presence of the three dimensions - length, breadth, and height. As a result of these three dimensions, these objects have faces, edges, and vertices. Let's understand these three in detail.

Faces of Solid Shapes

A face refers to any single flat surface of a solid object.
Solid shapes can have more than one face.

Edges of Solid Shapes

An edge is a line segment on the boundary joining one vertex (corner point) to another.
They serve as the junction of two faces.

Vertices of Solid Shapes

A point where two or more lines meet is called a vertex.
It is a corner.
The point of intersection of edges denotes the vertices.

For example:

Cuboid showing faces, edges, and vertices

Solid Shapes	Faces	Edges	Vertices
Sphere	1	0	0
Cylinder	2	2	0
Cone	1	1	1
Cube	6	12	8
Rectangular Prism	6	12	8
Triangular Prism	5	9	6
Pentagonal Prism	7	15	10
Hexagonal Prism	8	18	12
Square Pyramid	5	8	5
Triangular Pyramid	4	6	6
Pentagonal Pyramid	6	10	6
Hexagonal Pyramid	7	12	7

Tips and Tricks

Rhyme to remember solid shapes:

"Solid shapes are fat, not flat.
Find a cone in a birthday hat!
You see a sphere in a basketball,
And a cuboid in a building so tall!
You see a cube in the dice you roll,
And a cylinder in a shiny flag pole!"

Moving your fingers along geometric shapes will help you understand the concept of faces, edges, and vertices.

Important Points

Solids or three-dimensional objects have 3 dimensions, namely length, breadth, and height.
Solid shapes have faces, edges, and vertices.
Learning about solid shapes will help us in our day-to-day life as most of our activities revolve around and depend on them.

Solid Shapes Examples

Example 1: A construction worker wants to build a solid sphere using cement. He wants to know the amount of cement required to construct a sphere of radius 10 inches. Find the volume of the sphere using the given radius.

Solution:

The radius of the sphere (r) = 10 inches. We know the formula for the volume of a sphere: v = 4/3 π r³. Substituting the value of the radius in the above formula, we get: v= 4/3 π r³= 4/3 π (10)³= 4188.8 inches³. Therefore, the volume of the cemented sphere is 4188.8 inches³
Example 2: Identify the regular polyhedron from the images shown below.

Solution:

Regular polyhedrons include:
Prisms
Pyramids
Platonic solids

The given examples of polyhedrons must come under these categories. Thus, the Egyptian pyramids and Rubik's cubes are polyhedrons.
Example 3: Find the area of the square-shaped floor room which is made up of 100 square tiles of side 15 inches.

Solution:

Area of one tile = 15 inch × 15 inch= 225 square inches. We know that there are 100 tiles on the floor of the room. Thus, the area occupied by 100 tiles is the floor area = 100 × 225 square inches = 22500 square inches. Therefore, the area of the floor is 22500 square inches.

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Practice Questions on Solid Shapes

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FAQs on Solid Shapes

What is a Solid Shape?

The objects that are three-dimensional with length, breadth, and height defined are known as solid shapes.

What can a Solid Shape Also be Called?

In geometry, a solid shape can also be called a three-dimensional shape.

How Many Solid Shapes are There?

The list of solid shapes includes cube, cuboid, sphere, cone, hemisphere, prism, cylinder, pyramid, etc.

What is the Volume of a Solid Shape?

The volume of solid shapes refers to the amount of cubic space filled within the shapes. To find the volume, we need the measurements of the three dimensions.

Is a Sphere Solid or Flat Shape?

A sphere is a solid shape with no edges or vertices (corners).

What are the Properties of Solid Shapes?

Solid shapes are three-dimensional objects. They have three dimensions - length, width, and height. Being three-dimensional, they take up space in the universe. Solid shapes are identified according to the features- edges, vertices, faces, etc.

What are the Flat and Solid Shapes?

Flat shapes are also known as plane shapes which are 2D shapes consisting of straight lines, curved lines, or both. Whereas solid shapes are 3D shapes consisting of length, breadth, and height. The difference between flat and solid shapes is the dimensions. For example, a square is a flat shape and its counter solid shape is a cube which is a 3D shape.

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