Area Calculator | 16 Popular Shapes! (2024)

There are many useful formulas to calculate the area of simple shapes. In the sections below, you'll find not only the well-known formulas for triangles, rectangles, and circles but also other shapes, such as parallelograms, kites, or annuli.

We hope that after this explanation, you won't have any problems defining what an area in math is!

Well, of course, it depends on the shape! Below you'll find formulas for all sixteen shapes featured in our area calculator. For the sake of clarity, we'll list the equations only - their images, explanations and derivations may be found in the separate paragraphs below (and also in tools dedicated to each specific shape).

Are you ready? Here are the most important and useful area formulas for sixteen geometric shapes:

• Square area formula: A = a²
• Rectangle area formula: A = a × b
• Triangle area formulas:
  • A = b × h / 2 or
  • A = 0.5 × a × b × sin(γ) or
  • A = 0.25 × √( (a + b + c) × (-a + b + c) × (a - b + c) × (a + b - c) ) or
  • A = a² × sin(β) × sin(γ) / (2 × sin(β + γ))
• Circle area formula: A = πr²
• Circle sector area formula: A = r² × angle / 2
• Ellipse area formula: A = a × b × π
• Trapezoid area formula: A = (a + b) × h / 2
• Parallelogram area formulas:
  • A = a × h or
  • A = a × b × sin(angle) or
  • A = e × f × sin(angle)
• Rhombus area formulas:
  • A = a × h or
  • A = (e × f) / 2 or
  • A = s² × sin(angle)
• Kite area formulas:
  • A = (e × f) / 2 or
  • A = a × b × sin(γ)
• Pentagon area formula: A = a² × √(25 + 10√5) / 4
• Hexagon area formula: A = 3/2 × √3 × a²
• Octagon area formula: A = 2 × (1 + √2) × a²
• Annulus area formula: A = π(R² - r²)
• Quadrilateral area formula: A = 1/2 × e × f × sin(angle)
• Regular polygon area formula: A = n × a² × cot(π/n) / 4

Want to change the area unit? Simply click on the unit name, and a drop-down list will appear.

Did you forget what's the square area formula? Then you're in the right place. The area of a square is the product of the length of its sides:

• Square Area = a × a = a², where a is a square side

That's the most basic and most often used formula, although others also exist. For example, there are square area formulas that use the diagonal, perimeter, circumradius or inradius.

The rectangle area formula is also a piece of cake - it's simply the multiplication of the rectangle sides:

• Rectangle Area = a × b

Calculation of rectangle area is extremely useful in everyday situations: from building construction (estimating the tiles, decking, siding needed or finding the roof area) to decorating your flat (how much paint or wallpaper do I need?) to calculating how many people your cake can feed.

1. Given base and height

• Triangle Area = b × h / 2

2. Given two sides and the angle between them (SAS)

• Triangle Area = 0.5 × a × b × sin(γ)

3. Given three sides (SSS) (This triangle area formula is called Heron's formula)

• Triangle Area = 0.25 × √( (a + b + c) × (-a + b + c) × (a - b + c) × (a + b - c) )

You can discover more in the Heron's formula calculator.

4. Given two angles and the side between them (ASA)

• Triangle Area = a² × sin(β) × sin(γ) / (2 × sin(β + γ))

There is a special type of triangle, the right triangle. In that case, the base and the height are the two sides that form the right angle. Then, the area of a right triangle may be expressed as:

Right Triangle Area = a × b / 2

The circle area formula is one of the most well-known formulas:

• Circle Area = πr², where r is the radius of the circle

In this calculator, we've implemented only that equation, but in our circle calculator you can calculate the area from two different formulas given:

1. Diameter

• Circle Area = πr² = π × (d / 2)²

2. Circumference

• Circle Area = c² / 4π

Also, the circle area formula is handy in everyday life – like the serious dilemma of which pizza size to choose.

The sector area formula may be found by taking a proportion of a circle. The area of the sector is proportional to its angle, so knowing the circle area formula, we can write that:

α / 360° = Sector Area / Circle Area

Angle conversion tells us that 360° = 2π

α / 2π = Sector Area / πr²

so:

• Sector Area = r² × α / 2

To find an ellipse area formula, first recall the formula for the area of a circle: πr². For an ellipse, you don't have a single value for radius but two different values: a and b. The only difference between the circle and ellipse area formula is the substitution of r² by the product of the semi-major and semi-minor axes, a × b:

• Ellipsis Area = π × a × b

The area of a trapezoid may be found according to the following formula:

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

Also, the trapezoid area formula may be expressed as:

Trapezoid area = m × h, where m is the arithmetic mean of the lengths of the two parallel sides

1. Base and height

• Parallelogram Area = b × h

2. Sides and an angle between them

• Parallelogram Area = a × b × sin(α)

3. Diagonals and an angle between them

• Parallelogram Area = e × f × sin(θ)

1. Side and height

• Rhombus Area = a × h

2. Diagonals

• Rhombus Area = (e × f) / 2

3. Side and any angle, e.g., α

• Rhombus Area = a² × sin(α)

To calculate the area of a kite, two equations may be used, depending on what is known:

1. Area of a kite formula, given kite diagonals

• Kite Area = (e × f) / 2

2. Area of a kite formula, given two non-congruent side lengths and the angle between those two sides

• Kite Area = a × b × sin(α)

The area of a pentagon can be calculated from the formula:

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

Check out our dedicated pentagon calculator, where other essential properties of a regular pentagon are provided: side, diagonal, height and perimeter, as well as the circumcircle and incircle radius.

The basic formula for the area of a hexagon is:

• Hexagon Area = 3/2 × √3 × a², where a is the regular hexagon side

So, where does the formula come from? You can think of a regular hexagon as the collection of six congruent equilateral triangles. To find the hexagon area, all we need to do is to find the area of one triangle and multiply it by six. The formula for a regular triangle area is equal to the squared side times the square root of 3 divided by 4:

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

Hexagon Area = 6 × Equilateral Triangle Area = 6 × (a² × √3) / 4 = 3/2 × √3 × a²

To find the octagon area, all you need to do is know the side length and the formula below:

• Octagon Area = 2 × (1 + √2) × a²

The octagon area may also be calculated from:

Octagon Area = perimeter × apothem / 2

A perimeter in octagon case is simply 8 × a. And what is an apothem? An apothem is a distance from the center of the polygon to the mid-point of a side. At the same time, it's the height of a triangle made by taking a line from the vertices of the octagon to its center. That triangle - one of eight congruent ones - is an isosceles triangle, so its height may be calculated using, e.g., Pythagoras' theorem, from the formula:

h = (1 + √2) × a / 4

So finally, we obtain the first equation:

Octagon Area = perimeter * apothem / 2 = (8 × a × (1 + √2) × a / 4) / 2 = 2 × (1 + √2) × a²

An annulus is a ring-shaped object – it's a region bounded by two concentric circles of different radii. Finding the area of an annulus formula is an easy task if you remember the circle area formula. Just have a look: an annulus area is a difference in the areas of the larger circle of radius R and the smaller one of radius r:

• Annulus Area = πR² - πr² = π(R² - r²)

The quadrilateral formula this area calculator implements uses two given diagonals and the angle between them.

• Quadrilateral Area = 1/2 × e × f × sin(α), where e and f are diagonals.

We can use any of two angles as we calculate their sine. Knowing that two adjacent angles are supplementary, we can state that sin(angle) = sin(180° - angle).

If you're searching for other formulas for the area of a quadrilateral, check out our dedicated quadrilateral calculator, where you'll find Bretschneider's formula (given four sides and two opposite angles) and a formula that uses bimedians and the angle between them.

The formula for regular polygon area looks as follows:

• Regular Polygon Area = n × a² × cot(π/n) / 4

where n is the number of sides, and a is the side length.

Other equations exist, and they use, e.g., parameters such as the circumradius or perimeter. You can find those formulas in a dedicated paragraph of our regular polygon area calculator.

If you're dealing with an irregular polygon, remember that you can always divide the shape into simpler figures, e.g., triangles. Just calculate the area of each of them and, at the end, sum them up. Decomposition of a polygon into a set of triangles is called polygon triangulation.

For a given perimeter, the quadrilateral with the maximum area will always be a square.

For a given perimeter, the closed figure with the maximum area is a circle.

To find the area under a curve over an interval, you have to compute the definite integral of the function describing this curve between the two points that correspond to the endpoints of the interval in question.