What is Area?
The area is a measure of the amount of space inside a two-dimensional shape or figure. It quantifies the size of a surface and is expressed in square units, such as square meters (m²), square centimeters (cm²), or square inches (in²). Area applies to flat shapes like squares, rectangles, triangles, circles, and more. It does not apply to three-dimensional objects. Each shape has a specific formula for calculating its area. Let’s understand how each formula is derived with step-by-step explanations.
Area Formulas for Common 2D Shapes
Area of a Square
A square has four equal sides, so if the length of one side is 𝑎, then all sides are also units long. To calculate how much space the square covers, let’s find how many unit squares (1 unit x 1 unit) fit inside the larger square.
If the square is units long on each side, it can fit unit squares along the length and another unit squares along the width. Thus, the total number of unit squares inside the larger square is 𝑎, or 𝑎².
The formula for calculating the area of a square with a side length of a is:
Area of a Square = a x a = a²
Example:
Calculate the area of a square with side length 6 cm.
Side length = a = 6 cm
Area = a² = 6×6 = 36 square centimeteres = 36 cm²
Area of a Rectangle
Area is a measure of the space contained within a shape. For a rectangle, it represents the total number of square units that can fit inside it. To understand the area of a rectangle, imagine covering the rectangle with 1 cm x 1 cm squares (unit squares).
If the rectangle is cm long and cm wide, you can fit unit squares along the length and unit squares along the width. The total number of unit squares that can fit in the rectangle gives the area.
The formula for calculating the area of a rectangle is:
Area of a Rectangle = Length x Breadth (or) Length x Width = 𝓁𝒷 (or) 𝓁𝓌
Length is the longer side of the rectangle. Breadth (Width) is the shorter side of the rectangle.
Example:
If a rectangle is 4 cm long and 3 cm wide, you can visualize filling the rectangle with 1 cm squares. You would fit:
- 4 squares along the length.
- 3 squares along the width.
Thus, the area would be square centimeters.
Area of a Parallelogram
A parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length.
Imagine a parallelogram slanted to one side. You can “cut off” a triangle from one end of the parallelogram and move it to the other side. When you do this, the parallelogram becomes a rectangle without changing its total area.
The length of this rectangle is the base of the parallelogram and its breadth is the height of the parallelogram. Hence, the area of this rectangle is , which is the same formula used for the parallelogram.
The formula for calculating the area of a parallelogram is:
Area of a Parallelogram = Base x Height = bh
b is the base of the parallelogram and h is the perpendicular distance from the base to its opposite side.
Example:
If a parallelogram has a base of 8 cm and a height of 5 cm, its area is:
square centimeters.
Area of a Triangle
A triangle is a polygon with three sides and three angles. The area of a triangle refers to the amount of space enclosed by the three sides. Every triangle has a base and a height, which are key to calculating its area. The area formula for a triangle is derived by comparing it to a rectangle:
- If you place two identical triangles side by side, they can form a rectangle or parallelogram. The area of this rectangle or parallelogram would be .
- Since the triangle is half of this rectangle or parallelogram, its area must be half the area of the rectangle. This gives us the formula:
Area of a Triangle = $\frac{1}{2}$ x Base x Height = $\frac{1}{2}$bh
Base is any one side of the triangle (you can choose which side to use as the base).
Height is the perpendicular distance from the base to the opposite vertex. This is sometimes inside the triangle (for right triangles or acute triangles), but it can also be outside (in the case of obtuse triangles).
Example:
A triangle has a base of 6 cm and a height of 4 cm. Find its area.
Area = $\frac{1}{2}$ x Base x Height = $\frac{1}{2}$x6x4 = 12 cm²
Area of a Rhombus
A rhombus is a quadrilateral where all sides have equal length, the opposite angles are equal, and the diagonals bisect each other at right angles.
- The diagonal d1 of a rhombus divides it into two equal triangles, each with the base d1 and height $\frac{d2}{2}$, as the diagonals bisect each other at right angles.
- So, the area of each triangle is $\frac{1}{2}$ x Base x Height = $\frac{1}{2}$ x d1 x $\frac{d2}{2}$
- Adding these areas together gives us the area of the rhombus. $\frac{1}{2}$ x d1 x $\frac{d2}{2}$ + $\frac{1}{2}$ x d1 x $\frac{d2}{2}$
The formula for calculating the area of a rhombus is:
Area of a Rhombus = $\frac{1}{2}$ x d1 x d2
d1 and d2 are the lengths of the diagonals.
Example:
If a rhombus has diagonals measuring 10 cm and 15 cm, its area is
$\frac{1}{2}$x10x15 = 75 square centimeters.
Area of a Trapezium (Trapezoid)
A trapezium is a quadrilateral with at least one pair of parallel sides. Let’s name the parallel sides as a and b and the perpendicular distance between them as h (height of the trapezium).
- The diagonal of the trapezium splits it into two triangles, each with one of the parallel sides (a,b) as the base and h as the height.
- The area of the first triangle = $\frac{1}{2}$xaxh
- The area of the second triangle = $\frac{1}{2}$xbxh
- Adding them gives the area of the trapezium. $\frac{1}{2}$xaxh+$\frac{1}{2}$xbxh = $\frac{1}{2}$x(a+b)xh
The formula for calculating the area of a rectangle is:
Area of a Trapezium or Trapezoid = $\frac{1}{2}$h(a+b)
Example:
If the parallel sides of a trapezium are 8 cm and 5 cm, its height is 4 cm, find its area.
A = $\frac{1}{2}$xhx(a+b) = $\frac{1}{2}$x4x(8+5) = 26 cm²
Area of a Circle
A circle is a round shape where all points are equidistant from the center. The formula for the area of a circle can be derived using calculus, but here’s a simplified explanation:
- Imagine dividing the circle into many thin wedges (like pizza slices). As you increase the number of wedges, their total area can be approximated using triangles, where the base of each triangle is part of the circle’s circumference and the height is the radius.
- When you add up the areas of these wedges, you arrive at the area of the entire circle, which simplifies to .
The formula for calculating the area of a circle is:
Area of a Circle = 𝜋r²
(pi) is a mathematical constant approximately equal to 3.14159 or $\frac{22}{7}$.
is the radius of the circle, which is the distance from the center of the circle to any point on its circumference.
Example:
Calculate the area of a circle with a radius of 14 cm.
A = 𝜋r² = $\frac{22}{7}$x14x14 = 616 cm²
Area Formula Chart
The below table lists the formulas to find the area of common 2D shapes.
Understanding the “why” behind each area formula is essential for grasping the concept of area itself and applying it effectively in real-world situations. In everyday life, we often deal with physical spaces, such as flooring, gardening, or painting walls. Knowing why we use specific formulas helps us apply them correctly to solve practical problems, such as how much paint is needed for a wall or how many tiles are required for a floor. Understanding the reasoning behind these formulas not only aids in solving mathematical problems but also empowers students to apply their knowledge in practical and meaningful ways. This foundation is essential for developing a deeper appreciation of mathematics and its relevance in everyday life.