## Introduction

The amount of space in a two-dimensional place, which is acquired by a shape is called the area of that particular shape. The area measurement unit is dimension square. Suppose we are measuring something in meters and if we calculate the area, then area measurement units will be m^{2}.

There are many useful formulas to calculate the space acquired by easy shapes and simple shapes. In the sections below you will find not only the well-known formulas for triangles, rectangle, and circles but also other shapes, like parallelogram, kites or annulus.

There are many area calculator apps available on the internet. Some of the area calculation tools are available for free online. Apart from this, there are many area measurement calculators also available on the internet.

## Area Measurement

As we discussed, the area is the amount of space acquired by shape, so it depends on the shape and size. We'll discuss different types of shapes and formulas for their formulas.

### 1. Area of Square

Square is a quadrilateral in which all sides are equal length and all angles are equal to 90°. Let the length of sides of the square is a unit then it’s the area will be a x a = a^{2}.

### 2. Area of Rectangle

A rectangle is a quadrilateral in which opposite sides are of equal length and all the angles are equal to 90°. Let a be it’s length and b be its breadth then we can calculate its area by the formula. a x b = ab.

### 3. Area of Triangle

There are several formulas for the area of the triangle. We use them depending on the given information in the question.

When Base and Height are Given

When base and height are given and we need to find the area of a triangle then use the formula ½ x b x h. where b is the base of the triangle and h is given height.

When Two Sides and the Angle Between Them is Given

When two sides and angles between them are given and we need to find the area of a triangle then use the formula ½ x a x b x sin ℽ. where a and b are lengths of the given sides and ℽ is the angle between them.

When all Three Sides are Given

When all three sides are given of a triangle and we need to find the area of it then we can use the formula \[\sqrt{s(s-a)(s-b)(s-c)}\] where s is \[\frac{a+b+c}{2}\] . Here, a, b and c are the given sides of the triangle. This formula is also known as Heron's formula.

When Two Angles and a Side is Given

When two angles and a side is given and we need to find the area of a triangle then use the formula \[\frac{a^{2}\times sin\beta \times sin\gamma }{2sin(\beta +\gamma )}\] . where a is the length of given the side and β , 𝛄 are given two angles.

Area of a Right-Angled Triangle

There is a special type of triangle whose area can be formulated. A right-angled triangle is a triangle whose one angle is right-angled. If the base and height of the triangle are given to us as a and be then, the area will be ½ x base x height.

### 4. Area of Circle

The area of the circle can be calculated by the formula πr^{2} , where r is the radius of the circle. Let us suppose we have given diameter in the question then we’ll use the formula π(d/2)^{2} because we know that d = r/2.

### 5. Sector Area Formula

Area of sector formula can easily be calculated by the formula \[\frac{\alpha}{2\pi }=\frac{SectorArea}{CircleArea}\] Here, 𝞪 is the angle of the arc. On further solving we get,

Sector area = 𝞪/2π x πr^{2 } because we know the area of a circle is πr^{2}.

Finally, we’ll get the area as \[\frac{\alpha \times r^{2}}{2}\].

### 6. Ellipse Area Formula

Recall the area formula of the circle. It is nothing but for πr^{2}. Where r is the radius, the distance from the centre to each point of the circumference. In the case of an ellipse, this distance is different (as shown in the picture). Hence instead of r, we need to counter these parameters. So, the formula of the area of the ellipse we get as π x a x b. Where a is semi-minor axes and b is semi-major axes.

### 7. Trapezoid Area Formula

The area of the trapezoid can be calculated by the formula ½ x (a + b) x h. Here, a and b are the length of parallel sides and h is the distance between them. There is another formula of area of trapezoid but it can be computed by simple manipulation of the above-mentioned formula. We know that arithmetic means of a and b is given as (a + b)/2 . Now let’s represent this by c then the area formula will become c x h.

### 8. Area of Parallelogram Formula

There are several formulas for the area of the parallelogram. We use them depending on the given information in the question.

When Base and Height are Given

When base and height are given and we need to find the area of a parallelogram then use the formula b x h.where b is the base of the parallelogram and h is given height.

When Sides and Angle Between Them are Given

When sides and angle between them are given and we need to find the area of the parallelogram then use the formula a x b x sin𝞪. Where a and b are given sides and 𝞪 is the angle between them.

When Diagonals and Angle Between Them are Given

When diagonals and angles between them are given and we need to find the area of parallelogram then use the formula e x f x sinθ where e and f are the length of diagonals and θ is the angle between them.

### 9. Area of Rhombus Formula

Similar to the parallelogram, we have several formulas to find the area of the rhombus. We use them depending on the given data.

When Side and Height are Given

When a side and height of the rhombus are given to us and we need to find the area of it then use the formula a x h. Where a is the length of the side and h is the given height of a rhombus.

When the Length of Diagonals are Given

When the length of diagonals are given to us and we need to find the area of the rhombus then use the formula ½ x e x f. Where e and f are the lengths of diagonals.

When a Side and an Angle is Given

When side and angle are given and we need to find the area of rhombus then us the formula a^{2} x sin𝞪. Where a is the length of sides and 𝞪 is the given angle.

### 10. Regular Polygon Area Formula

A regular polygon is a two-dimensional shape whose all sides and angles are equal. It can be further divided depending on the number of sides. For example, If the number of sides is 5 then it’s a pentagon. It's area can be calculated by the formula ¼ x n x a^{2} x cot(π/n) Where n is the number of sides and a is the length of a side.

### Did You Know?

In higher mathematics to find areas of non-regular or simple shapes, we use Integration. The integration uses the core concept of the Riemann sum. In simple words, Riemann sum is a sum only and a new way to represent sum. What Integration does is, takes small areas and adds them up to get the total area on the plane. For finding such a small area, we use the area of the rectangle formula.

1. The Area of a Rectangle is 48 m^{2}. If its Breadth is 6 m then Find its Length.

Ans: We know that the area of a rectangle is its length multiplied by its breadth.

So, length x breadth = 48

Now breadth is given to us as 6 m. On putting this value

length x 6 = 48

On solving

length = 48/6

length = 8m

Hence the length of the rectangle is 8m.

2. Finding the Area of a Rectangle Whose Length is 12 m and Breadth 9 m.

Ans: We know that the area of a rectangle is its length multiplied by its breadth.

The length of a rectangle is given as 12m and breadth is 9m. On putting these values into the formula, we get

Area of rectangle = 12 x 9 = 108 m^{2}.