- What is area in math? Area definition
- How to calculate area?
- Square area formula
- Rectangle area formula
- Triangle area formula
- Circle area formula
- Sector area formula
- Ellipse area formula
- Trapezoid area formula
- Area of a parallelogram formula
- Area of a rhombus formula
- Area of a kite formula
- Pentagon area formula
- Area of a hexagon formula
- Area of an octagon formula
- Area of an annulus formula
- Area of a quadrilateral formula
- Regular polygon area formula
If you're wondering how to calculate the area of any basic shape, you're in the right place - this area calculator will answer all your questions. Use our intuitive tool to choose from sixteen different shapes, and calculate their area in the blink of an eye. Whether you're looking for an area definition or, for example, the area of a rhombus formula, we've got you covered. Keep scrolling to read more or just play with our tool - you won't be disappointed!
What is area in math? Area definition
Simply speaking, area is the size of a surface. In other words, it may be defined as the space occupied by a flat shape. To understand the concept, it's usually helpful to think about the area as the amount of paint necessary to cover the surface. Look at the picture below – all the figures have the same area, 12 square units:
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!
How to calculate area?
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 / 2or
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 × hor
A = a × b × sin(angle)or
A = e × f × sin(angle)
- Rhombus area formulas:
A = a × hor
A = (e × f) / 2or
A = s² × sin(angle)
- Kite area formulas:
A = (e × f) / 2or
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.
Square area formula
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
ais 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.
Rectangle area formula
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.
Triangle area formula
There are many different formulas for triangle area, depending on what is given and which laws or theorems are used. In this area calculator, we've implemented four of them:
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
Circle area formula
The circle area formula is one of the most well-known formulas:
Circle Area = πr², where
ris 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:
Circle Area = πr² = π × (d / 2)²
Circle Area = c² / 4π
Also, the circle area formula is handy in everyday life – like the serious dilemma of which pizza size to choose.
Sector area formula
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
Sector Area = r² × α / 2
Ellipse area formula
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:
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
Trapezoid area formula
The area of a trapezoid may be found according to the following formula:
Trapezoid area = (a + b) × h / 2, where
bare the lengths of the parallel sides and
his 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
Area of a parallelogram formula
Whether you want to calculate the area given base and height, sides and angle, or diagonals of a parallelogram and the angle between them, you are in the right place. In our tool, you'll find three formulas for the area of a parallelogram:
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)
Area of a rhombus formula
We've implemented three useful formulas for the calculation of the area of a rhombus. You can find the area if you know the:
1. Side and height
Rhombus Area = a × h
Rhombus Area = (e × f) / 2
3. Side and any angle, e.g., α
Rhombus Area = a² × sin(α)
Area of a kite formula
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)
Pentagon area formula
The area of a pentagon can be calculated from the formula:
Pentagon Area = a² × √(25 + 10√5) / 4, where
ais 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.
Area of a hexagon formula
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
Area of an octagon formula
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
Area of an annulus formula
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)
Area of a quadrilateral formula
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
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.
Regular polygon area formula
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.
What quadrilateral has the largest area?
For a given perimeter, the quadrilateral with the maximum area will always be a square.
What shape has the largest area given perimeter?
For a given perimeter, the closed figure with the maximum area is a circle.