History of Area

The first recorded use of areas was in ancient Babylon, where they used it to measure the amount of land that was owned by different populations for taxation purposes. Later in 287 BC, the great mathematician Archimedes from Greece discovered the area and the perimeter of the circle and the relationship between spheres. Archimedes, no doubt wasn't the first to realize the fact. However, he was, as far as we know, the first to prove it formally. He also gave the earliest proofs for the volume of the sphere and surface area.

Definition of Area

In geometry, the area can be defined as the space occupied by the surface of an object or any flat shape. The area of an object is the number of unit squares that cover the surface of a closed object. The area is measured in square units such as square feet, square centimetres, square inches, etc.

The origin of the word is from ‘area’ in Latin, which translates to a vacant piece of level ground. This further led to a derivation of the area as a particular amount of space contained within a set of boundaries.

To determine the size of the carpet to be bought, we often find the area of the room floor.

To cover the floor with tiles, to cover the wall with paint or wallpaper or building a swimming pool are other examples, where the area is computed.

In reality, not every plane surface can be clearly classified as a rectangle, square or a triangle. For finding the area of a composite figure that contains more than one shape, we will find the sum of the area of all the shapes forming the composite figure.

The area of the outside surface of a three-dimensional shape or a solid is called Surface Area of that surface. For example, a rectangular prism has 6 rectangular bases and lateral faces. Thus, the total surface area is equal to the sum of the areas of all 6 rectangles.

Area Formulas

In general, we can say that the area of shapes can be defined as the quantity of paint required to cover the surface with a single coat. These are the following ways to calculate the area based on the number of sides that exist in the shape, as illustrated below in fig.

Area Formula for Different Shapes





π × r2

r = radius of the circle


½ × b × h

b = base

h = height



a = length of side


l × w

l = length

w = width


b × h

b = base

h = vertical height


½(a + b) × h

a & b are length of parallel sides

h = height



a = ½ minor axis

b = ½ major axis

What are 3D Shapes?

The three-dimensional shapes, also known as solid shapes, are those which have three dimensions such as length, breadth and thickness. The two different measures which are used to define the three-dimensional shapes are surface area and volume. In general, the 3D shapes are obtained from the rotation of two-dimensional shapes. Thus, the surface area of any two-dimensional shapes should be a 2D shape. If we want to calculate the surface area of any solid shape, we can easily calculate it from the area of 2D shapes.

Area of 3D Shapes Formula

According to the International System of Units (SI), the standard unit of area is the square meter (written as m2) and it is also the area of a square whose sides are one meter long. For example, a definite shape with an area of three square meters would have the same area as three such squares. The surface area of any solid object is a measure of the total area which the surface of the object occupies.

For 3D/ solid shapes like cuboid, cube, cylinder, sphere and cone, the area is updated to the concept of the surface area of the shapes. The formulas for 3D shapes are given in the table below:

Formulas for 3D Shapes


Surface Area




a = length of the edge

Rectangular prism

2(wl + hl + hw)

l = length

w = width

h = height


2πr(r + h)

r = radius of the circular base

h = height of the cylinder


πr(r + l)

r = radius of the circular base

l = slant height



r = radius of the sphere



r = radius of the hemisphere

FAQs (Frequently Asked Questions)

Question 1. How do you Find the Area?

Answer: The most commonly used area calculations are for rectangles and squares. For finding the area of a rectangle, multiply its height by its width. For a square, you only need to find the length of one of the sides (as each side is of the same length) and then multiply this by itself to find the area.

Question 2. Which is the Area of the Figure?

Answer: The area is measured in "square" units. The area of a given surface is the number of squares required to envelop it completely, like tiles on a floor. Area of a square = side times side. Since every side of a square is the same, the solution can simply be the length of one side squared.

Question 3. What is the Area Formula?

Answer: Given a rectangle with length l and width w, the formula for the area goes as A = lw (rectangle). That is, the area of the rectangle is the width multiplied by the length. As a unique case, as l = w, in the case of a square, the area of a square with side length s is given by the formula: A = s2 (square).

Question 4. What are Area and Volume?

Answer: The area is the quantity of space occupied by a two-dimensional flat object in a plane. Volume is defined as the space occupied by the 3D object. It is always measured in square units. It is always measured in cubic units. It is measured in 2 dimensions.