how do you find the area of a shape

If you're wondering how to calculate the area of any basic shape, yous're in the correct identify - this area calculator will answer all your questions. Use our intuitive tool to cull from sixteen different shapes, and calculate their expanse in the blink of an centre. Whether you're looking for an expanse definition or, for example, the area of a rhombus formula, we've got you lot covered. Keep scrolling to read more or just play with our tool - you won't be disappointed!

What is area in math? Surface area definition

Simply speaking, area is the size of a surface. In other words, information technology may exist defined as the space occupied past a flat shape. To understand the concept, it'due south usually helpful to think about the area as the amount of paint necessary to cover the surface. Look at the motion picture below - all the figures have the aforementioned expanse, 12 square units:

There are many useful formulas to summate the area of elementary shapes. In the sections beneath you lot'll detect not just the well-known formulas for triangles, rectangles, and circles, but also other shapes, such every bit parallelograms, kites or annulus.

We promise that after this explanation y'all won't have whatsoever issues defining what an expanse in math is!

How to calculate area?

Well, of course, it depends on the shape! Below you lot'll discover formulas for all xvi 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 besides in tools dedicated to each specific shape).

Are you lot gear up? Here are the virtually important and useful expanse formulas for sixteen geometric shapes:

• Foursquare area formula: A = a²
• Rectangle area formula: A = a * b
• Triangle surface 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 surface area formula: A = πr²
• Circumvolve sector area formula: A = r² * angle / 2
• Ellipse area formula: A = a * b * π
• Trapezoid surface area formula: A = (a + b) * h / 2
• Parallelogram expanse formulas:
  • A = a * h or
  • A = a * b * sin(angle) or
  • A = e * f * sin(angle)
• Rhombus area formulas:
  • A = a * h or
  • A = (eastward * f) / ii or
  • A = southward² * sin(angle)
• Kite area formulas:
  • A = (e * f) / 2 or
  • A = a * b * sin(γ)
• Pentagon expanse formula: A = a² * √(25 + x√5) / four
• Hexagon area formula: A = 3/2 * √3 * a²
• Octagon area formula: A = 2 * (1 + √2) * a²
• Annulus expanse formula: A = π(R² - r²)
• Quadrilateral area formula: A = e * f * sin(angle)
• Regular polygon area formula: A = n * a² * cot(π/n) / 4

Desire to change the area unit? Simply click on the unit of measurement proper noun and a drop-down list will appear.

Square expanse formula

Did you lot forget what'due south the square surface area formula? Then you're in the right place. The surface 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 nearly bones and nearly oftentimes used formula, although others as well exist. For instance, there are square area formulas that utilize the diagonal, perimeter, circumradius or inradius.

You may also be interested in finding the expanse of the largest foursquare in a circle!

Rectangle surface 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 surface area) to decorating your apartment (how much pigment or wallpaper exercise I need?) to computing how many people your canvas cake can feed.

Triangle area formula

There are many dissimilar 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 / two

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

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

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

• Triangle Surface area = 0.25 * √( (a + b + c) * (-a + b + c) * (a - b + c) * (a + b - c) )

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

• Triangle Surface 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 which class the correct angle. Then, the area of a right triangle may exist expressed as:

Right Triangle Area = a * b / 2

Circle area formula

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

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

In this calculator, we've implemented just that equation, just in our circumvolve calculator you tin summate the area from two different formulas, given:

1. Bore

• Circle Area = πr² = π * (d / 2)²

2. Circumference

• Circle Expanse = c² / 4π

Also, the circle expanse 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 from taking a proportion of a circumvolve. The area of the sector is proportional to its angle, and then knowing circumvolve area formula nosotros can write that:

α / 360° = Sector Area / Circle Area

Angle conversion tells us that 360° = 2π

α / 2π = Sector Surface area / πr²

so:

• Sector Area = r² * α / two

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, only 2 different values: a and b. The simply difference betwixt the circle and ellipse area formula is substitution of r² by the product of the semi-major and semi-minor axes, a * b:

• Ellipsis Area = π * a * b

Trapezoid expanse formula

The area of a trapezoid may exist 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 peak

Besides, the trapezoid surface area formula may be expressed as:

Trapezoid area = one thousand * h, where 1000 is the arithmetic hateful of the lengths of the two parallel sides

Area of a parallelogram formula

Whether you want to summate the area given base and tiptop, sides and angle, or diagonals of a parallelogram and the angle between them, you are in the right place. In our tool yous'll detect 3 formulas for the area of a parallelogram:

1. Base of operations and height

• Parallelogram Area = a * h

2. Sides and an bending between them

• Parallelogram Area = a * b * sin(α)

3. Diagonals and an angle betwixt them

• Parallelogram Area = due east * f * sin(θ)

Surface area of a rhomb formula

We've implemented three useful formulas for the adding of the area of a rhomb. Yous can find the area if you know the:

1. Side and height

• Rhombus Area = a * h

2. Diagonals

• Rhombus Area = (eastward * f) / two

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 2 not-coinciding side lengths and the bending betwixt those two sides

• Kite Area = a * b * sin(α)

Pentagon area formula

The expanse of a pentagon tin can be calculated from the formula:

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

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

Area of a hexagon formula

The bones formula for the expanse of a hexagon is:

• Hexagon Surface area = iii/2 * √3 * a² , where a is the regular hexagon side

So where does the formula come from? You tin think of a regular hexagon as the drove of vi congruent equilateral triangles. To find the hexagon expanse, all we need to do is to detect the area of i triangle and multiply it by six. The formula for a regular triangle area is equal to the squared side times the square root of iii divided past 4:

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

Hexagon Surface area = 6 * Equilateral Triangle Area = 6 *(a² * √iii) / 4 = 3/two * √3 * a²

Area of an octagon formula

To notice the octagon area, all y'all demand to do is know the side length and the formula below:

• Octagon Area = 2 * (1 + √ii) * 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 altitude from the center of the polygon to the mid-indicate of a side. At the aforementioned fourth dimension, information technology'southward the height of a triangle made by taking a line from the vertices of the octagon to its eye. That triangle - one of eight congruent ones - is an isosceles triangle, so it's peak may exist calculated using east.yard., Pythagoras' theorem, from the formula:

h = (1 + √2) * a / 4

So finally nosotros obtain the first equation:

Octagon Surface area = perimeter * apothem / two = (8 * a * (ane + √2) * a / four) / 2 = 2 * (1 + √2) * a²

Surface area of an annulus formula

An annulus is a ring-shaped object - it'due south a region divisional by 2 concentric circles of dissimilar radii. Finding the area of annulus formula is an like shooting fish in a barrel task if you remember the circle area formula. Just have a look: an annulus expanse is a difference in the areas of the larger circle of radius R and the smaller ane of radius r:

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

By the way, accept you seen our ring size converter?

Area of a quadrilateral formula

The quadrilateral formula this area calculator implements uses 2 given diagonals and the angle between them.

• Quadrilateral Expanse = e * f * sin(α) , where e, f are diagonals

We can use any of 2 angles, every bit nosotros calculate their sine. Knowing that two adjacent angles are supplementary, we can state that sin(bending) = sin(180° - angle).

If you're searching for other formulas for the surface area of a quadrilateral, bank check out our dedicated quadrilateral tool, where you'll discover Bretschneider's formula (given iv sides and two reverse angles) and a formula that uses bimedians and the angle between them.

Regular polygon area formula

The formula for regular polygon surface 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.yard., parameters such as the circumradius or perimeter. Y'all tin can find those formulas in a dedicated paragraph of our polygon area calculator.

If you're dealing with an irregular polygon, remember that yous tin can always dissever the shape into simpler figures, e.1000., triangles. Simply summate the area of each of them and at the end sum them up. Decomposition of a polygon into a set up of triangles is chosen polygon triangulation.

FAQ

What quadrilateral has the largest expanse?

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

What shape has the largest surface area given perimeter?

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

How do I summate area of an irregular shape?

To calculate the area of an irregular shape:

1. Divide the shape into several subshapes for which you lot tin do the area calculations easily, like triangles, rectangles, trapezoids, (semi)circles, etc.
2. Calculate the surface area of each of these subshapes.
3. Sum up the areas of subshapes to get the final consequence.

How practice I calculate the area under a curve?

To discover the expanse nether a bend over an interval, you have to compute the definite integral of the office describing this curve between the two points that correspond to the endpoints of the interval in question.

Hanna Pamuła , PhD candidate

Area of a rectangle Area of crescent Eye of mass … eighteen more

Source: https://www.omnicalculator.com/math/area

Posted by: salazarlinut1989.blogspot.com

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