Area Calculator

Usually, Area is the Size of a Surface. In other words, you can describe it as space occupied by any flat shape. In fact, you can think of the area as a space needed to cover a surface. There are different formulas to calculate the area and we have listed all of them in the further modules.

Procedure to Calculate the Area

The Area Calculation entirely depends on the shape and we have covered many shapes. Check out the Formulas for each shape along with detailed explanations and derivations that are necessary. Have a glance at the most important and useful formulas for calculating the shapes.

The formula for Square Area A = a²

Rectangle Area A = a*b

Circle Area Formula A = πr²

Cirlce Sector Area Formula A = r² * angle / 2

Formulas for Triangle Area

• A = b * h / 2
• A = 0.5 * a * b * sin(γ)
• A = 0.25 * √( (a + b + c) * (-a + b + c) * (a – b + c) * (a + b – c) )
• A = a² * sin(β) * sin(γ) / (2 * sin(β + γ))

Trapezoid Area Formula A = (a + b) * h / 2

Ellipse Area Formula A = a * b * π

Parallelogram Area Formulas

• A = a * h or
• A = a * b * sin(angle) or
• A = e * f * sin(angle)

Formulas for Kite Area

• A = (e * f) / 2 or
• A = a * b * sin(γ)

Pentagon Area Formula

• A = a² * √(25 + 10√5) / 4

Area of Rhombus Formula

• A = a * h or
• A = (e * f) / 2 or
• A = s² * sin(angle)

Formula for Hexagon Area A = 3/2 * √3 * a²

Octagon Area Formula A = 2 * (1 + √2) * a²

Area of Annulus Formula A = π(R² – r²)

Area of Regular Polygon Formula A = n * a² * cot(π/n) / 4

Quadrilateral Area A = e * f * sin(angle)

If you have irregular shapes try dividing them in your mind as regular shapes and then your calculation work becomes much simple.

Area of Square Formula

If you wish to find the Area of Square then this is the place. Usually, Square Area is nothing but the Product of the Length of its sides.

Let us consider the side length be “a” therefore Area of Square = a*a = a²

This is the most basic and often used formula although there are many formulas out there. There is area of square formulas dealing with diagonal, perimeter, circumradius, or inradius.

Area of a Rectangle Formula

The Rectangle area is nothing but the multiplication of the rectangle sides.

Rectangle Area = a*b

You might need to calculate rectangular area in your day to day life as in estimating tiles for roof area or decorating your flat, how many people can your cake sheet feed, etc.

Triangle Area Formula

There are various formulas to determine the triangle area based on the criteria given and the laws and theorems involved. Have a glance at the different formulas for triangle area

• Given base and height Triangle Area = b * h / 2
• Given two sides and the angle between them (SAS) Triangle Area = 0.5 * a * b * sin(γ)
• Given three Sides(SSS) Heron’s Formula Triangle Area = 0.25 * √( (a + b + c) * (-a + b + c) * (a – b + c) * (a + b – c) )
• Given two angles and the side between them (ASA) Area of Triangle = a² * sin(β) * sin(γ) / (2 * sin(β + γ))

You will have a special case i.e. when the Triangle is Right Triangle. In this case, base and height will form the right triangle and the Area of the Right Triangle is given as under

Area of the Right Triangle = a * b / 2

Area of Circle Formula

Circle Area Formula is the most known formulas and you can calculate it when different parameters are given. We have listed all the formulas to calculate the Area of the Circle below.

Area of Circle = πr² where r, is the radius.

If you know the diameter Circle Area = πr² = π * (d / 2)²

When you have Circumference, Circle Area = c² / 4π

Sector Area Formula

You can find the Sector Area by taking a proportion of the circle. The area of the sector is proportional to its angle. Thus, we can write the formula

α / 360° = Sector Area / Circle Area

As per Angle Conversion 360° = 2π

α / 2π = Sector Area / πr²

Rewriting the above formula we get the Sector Area as such

Sector Area = r² * α / 2

Ellipse Area Formula

While finding the Ellipse Area you need to recall the area of a circle formula πr². When it comes to ellipse there will not be a single value for radius and has two different values a and b. Ellipse Area Formula is replacing r² in circle area formula with the product of semi-major and semi-minor axes, a*b

Ellipse Area = π * a * b

Area of a Trapezoid Formula

You can find the Trapezoid Area according to the formula

Trapezoid area = (a + b) * h / 2, where a, b are lengths of parallel sides, and h is the height.

Using the mean you can get the Trapezoid Area and the formula for that is

Trapezoid Area = m * h, where m is the arithmetic mean of length of two parallel sides.

Formula for Area of a Parallelogram

You can calculate the area of a parallelogram if you are given base and height, diagonals of the parallelogram, and the angle between them, sides, and angles. Check out the different formulas to determine the Parellogram Area depending on your need.

• Base and height Parallelogram Area = a * h
• Sides and an angle between them Area of Parallelogram = a * b * sin(α)
• Diagonals and angle between them Parallelogram Area = e * f * sin(θ)

Area of Rhombus Formula

We have includes the Rhombus Area Formulas depending on the inputs you have. Use them and find the Area of Rhombus easily. They are as follows

• If you have side and height Rhombus Area = a * h
• When Diagonals are given Area of Rhombus = (e * f) / 2
• Side and any angle, e.g., α Rhombus Area = a² * sin(α)

The Formula for Kite Area

In order to determine the Area of Kite, you need to use two equations depending on what is known for you.

Area of Kite Formula when you know the diagonals Kite Area = (e * f) / 2

Given two non-congruent side lengths and the angle between those two sides Area of Kite = a * b * sin(α)

Pentagon Area Formula

You can calculate the Area of the Pentagon using the formula

Pentagon Area = a² * √(25 + 10√5) / 4 where a is the side of the regular pentagon

Have a glance at the tool dedicated to Pentagon if you know different parameters like Side, Diagonal, Height and Parameter, circumcircle, inradius.

Area of Hexagon Formula

The Basic Formula for Hexagon is Hexagon Area = 3/2 * √3 * a² where a is the regular hexagon side.

Think of Regular Hexagon as the collection of 6 congruent Equilateral Triangles. In order to determine the Hexagon Area just find the area of one triangle and multiply with six.

Equilateral Triangle Area = (a² * √3) / 4

Hexagon Area = 6 * Equilateral Triangle Area

= 6 *(a² * √3) / 4

= 3/2 * √3 * a²

Area of an Octagon Formula

In order to evaluate the Octagon Area, you need to know the octagon side length and the formula

Octagon Area = 2 * (1 + √2) * a²

The other formula to determine the Octagon Area is given as such

Octagon Area = perimeter * apothem / 2

Perimeter in the Case of Octagon is 8*a

Apothem is nothing but the distance between the center of the polygon and the midpoint of the side. In fact, it is also the height of the triangle obtained by taking a line from the vertices of the octagon to its center. The Triangle is Isosceles Triangle and you can calculate the height of it using the Pythagoras’ theorem.

h = (1 + √2) * a / 4

At last, we get the equation as

Octagon Area = Perimeter * Apothem / 2 = (8 * a * (1 + √2) * a / 4) / 2 = 2 * (1 + √2) * a²

Area of an Annulus Formula

An annulus is a ring-shaped object and it’s a region bounded by two concentric circles having different radii. It’s easy to determine the Annuls Area if you know the Circle Area. The Area of the Annular Ring is the difference between the areas of larger circle radius R and Smaller one having radius r.

Annulus Area = πR² – πr² = π(R² – r²)

Area of a Quadrilateral Formula

Quadrilateral Area if two diagonals and angle between them is given by

Quadrilateral Area = e * f * sin(α) where e, f are two diagonals.

You can use any two angles to calculate their sine. We know that two adjacent angles are supplementary and thus we can state as such

sin(angle) = sin(180° – angle).

Area of Regular Polygon Formula

Regular Polygon Area is given as follows

Regular Polygon Area = n * a² * cot(π/n) / 4 where n is the number of sides and a is the length of the side.

There are other equations to find the Regular Polygon Area if you know the circumradius or perimeter. When you are dealing with an irregular polygon do remember to split the shape into simpler figures. Find the Area of the Shapes individually and then sum up at the end.

---