# What are area and volume?

Area is the size of a surface!

These shapes all have the same area of 9:

It helps to imagine how much paint would cover the shape.

There are special formulas for certain shapes:

• The formula is:
• Area = w × h w = width
• h = height
• The width is 5, and the height is 3, so we know w = 5 and h = 3:
• Area = 5 × 3 = 15

## Area by Counting Squares

1. We can also put the shape on a grid and count the number of squares:
2. The rectangle has an area of 15
3. Example: When each square is 1 meter on a side, then the area is 15 m2 (15 square meters)

### Square Meter vs Meter Square

• The basic unit of area in the metric system is the square meter, which is a square that has 1 meter on each side:
• 1 square meter
• Be careful to say “square meters” not “meters squared”:

There are also “square mm”, “square cm” etc, learn more at Metric Area.

### Approximate Area by Counting Squares

Sometimes the squares don't match the shape exactly, but we can get an “approximate” answer.

### One way is:

• more than half a square counts as 1
• less than half a square counts as

## Difference Between Area and Volume (with Comparison Chart)

As you can see, there are many objects around us, have certain area or volume, though we do not recognise it. While the area is the region covered by the closed plane figure, the volume is the amount of space occupied by an object. The measurement of area is done in suqare metres, whereas the measurement of volume is done in cubic metres.

The terms area and volume are two important concepts of mensurations have a wide usage not only in mathematics but also in our day to day life. The article makes an attempt to shed light on the significant differences between area and volume. Take a look at it.

### Content: Area Vs Volume

1. Comparison Chart
2. Definition
3. Key Differences
4. Conclusion

### Comparison Chart

Basis for ComparisonAreaVolume
Meaning Area refers to the region or space of the plane figure or object. Volume refers to the quantity of space contained by an object.
Figure
Shapes Plane figures Solids
What is it? Amount of space enclosed Capacity of the solid
Measured in Square unit Cubic unit
Deal with 2 Dimensional shapes 3 Dimensional shapes

### Definition of Area

In geometry, the area of an object is nothing but its size, i.e. it is the two-dimensional space or region, which a closed figure covers. It measures the extent of space taken up by a plane object, calculated by multiplying the dimensions of the shape.

Area helps us to determine how many squares of fixed size, the shape would take to cover it. The standard unit of area, as per International System of Units (SI), is the square meters (expressed as m2). Below you can find the formula for area of various objects:

Formula:

• Area of Square = side × side
• Area of Rectangle = l × w
• Area of Parallelogram = b × h
• Area of Triangle = (b × h) / 2
• Area of Circle = πr2

where, l is the length
w is the width
h is the height
b is the base

## How to Calculate Area, Perimeter and Volume

••• Hemera Technologies/Photos.com/Getty Images

Updated July 22, 2019

By Sarah Celebi

The measurement of area, perimeter, and volume is crucial to construction projects, crafts, and other applications.

Area is the space inside the boundary of a two-dimensional shape. Perimeter is the distance around a two-dimensional shape such as a square or circle. Volume is a measure of the three-dimensional space taken up by an object, such as a cube. If you know the object's dimensions, then measurement of area and volume is easy.

Surface area and volume formulas for all everyday geometric shapes can easily be found online, although it's not a bad idea to review how to derive these on your own should the need arise.

You can also often get one of these from another; for example, if you know the formula for the area of a circle, you may be able to figure out that the volume of a cylinder is just the area of the associated circle(s) at the end times the cylinder's height.

Record the length (l) and width (w) of a square or rectangle. Substitute your measurements into the formula

to solve for area (A). In this example, a rectangular garden measures 5m by 7m.

Calculating the area of the garden, we get:

The area of the garden is 35 meters squared or 35 square meters.

Measure the base (b) and height (h) of the triangle. Use the formula

to find the area of a triangle. A triangle with a height of 7m and a base of 3m has an area of

A = ½(7m × 3m) = ½ (21m2) = 10.5m2.

The area (A) of the triangle is 10.5 meters squared or 10.5 square meters.

Measure the radius (r) of the circle. Multiply π (3.14) by the square of the radius to solve for the area (A) of a circle.

For example, a circle with a radius (r) of 5 inches will have an area of

A = π × (5 × 5) = 78.5 square inches

The area (A) of a circle with a radius of 5 inches is 78.5 square inches.

Record the lengths of all sides of the square, rectangle, or triangle.

Add the measurements to get the value of the perimeter (P). For example, a rectangular garden measures 5m by 7m has two sides measuring 5m and two sides measuring 7m. The perimeter (P) is:

P = 5 + 5 + 7 + 7 = 24 meters

The perimeter of the rectangular garden is 24 meters.

to find the perimeter, or circumference, of a circle. For example, a circle with a radius of 3 inches has a circumference of

P = π × (2 × 3) = 18.8 inches.

You can also find the circumference of a circle using the diameter (d). The diameter of a circle is two times the radius. The formula to calculate the circumference using a circle's diameter is

Volume: The volume (V) of most objects can be found by multiplying the base area (A) by height (h).

Record the length (l), width (w), and height (h) of a square or rectangle. Use the formula

to solve for the volume (V). In this formula, the base area (A) can be found by multiplying the length (l) by the width (w). For example, a box measuring 3 feet long, 1 foot wide and 5 feet high has a volume of

V = (3 × 1) × 5 = 15 cubic feet.

The box is 15 cubic feet.

to find the volume of a pyramid. For example, for a pyramid with a base area (A) of 25m2 and a height of 7m

V = (1/3) × 25 × 7 = 58.3 m3

The volume of the pyramid is 58.3 cubic meters or 58.3 meters cubed.

For a cylinder with a circular base, use the formula

to solve for the volume of a cylinder. For example, a cylinder with a radius of 2 meters and a height of 5 meters will have a volume of

V = π x (2 x 2) x 5 = 62.8 m3

The volume of the cylinder is 62.8 cubic meters or 62.8 meters cubed.

Calculating Area, Perimeter, and Volume

Calculating the area, perimeter, and volume of simple geometric shapes can be found by applying some basic formulas. It is a good idea to learn and understand what they are and commit those formulas to memory.

## Area

Extent of a two-dimensional surface
AreaCommon symbolsASI unitSquare metre [m2]In SI base units1 m2Dimension

L

2

{displaystyle {mathsf {L}}^{2}}

The combined area of these three shapes is approximately 15.57 squares.

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat.[1] It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size.

[2] In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long.

[3] A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles.

[4] For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

[5]

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area.[1][6][7] Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

[8] In analysis, the area of a subset of the plane is defined using Lebesgue measure,[9] though not every subset is measurable.[10] In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.[1]

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

### Formal definition

An approach to defining what is meant by “area” is through axioms. “Area” can be defined as a function from a collection M of special kind of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:

• For all S in M, a(S) ≥ 0.
• If S and T are in M then so are ST and ST, and also a(ST) = a(S) + a(T) − a(ST).
• If S and T are in M with ST then TS is in M and a(TS) = a(T) − a(S).
• If a set S is in M and S is congruent to T then T is also in M and a(S) = a(T).
• Every rectangle R is in M. If the rectangle has length h and breadth k then a(R) = hk.
• Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. SQT. If there is a unique number c such that a(S) ≤ c ≤ a(T) for all such step regions S and T, then a(Q) = c.

It can be proved that such an area function actually exists.[11]

### Units

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth.[12] Algebraically, these units can be thought of as the squares of the corresponding length units.

The SI unit of area is the square metre, which is considered an SI derived unit.[3]

### Conversions

Although there are 10 mm in 1 cm, there are 100 mm2 in 1 cm2.

• Calculation of the area of a square whose length and width are 1 metre would be:
• 1 metre × 1 metre = 1 m2
• and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as:

3 metres × 2 metres = 6 m2. This is equivalent to 6 million square millimetres. Other useful conversions are:

• 1 square kilometre = 1,000,000 square metres
• 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres
• 1 square centimetre = 100 square millimetres.

### Non-metric units

In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units.

1 foot = 12 inches,

the relationship between square feet and square inches is

1 square foot = 144 square inches,

where 144 = 122 = 12 × 12. Similarly:

• 1 square yard = 9 square feet
• 1 square mile = 3,097,600 square yards = 27,878,400 square feet

• 1 square inch = 6.4516 square centimetres
• 1 square foot = 0.09290304 square metres
• 1 square yard = 0.83612736 square metres
• 1 square mile = 2.589988110336 square kilometres

### Other units including historical

There are several other common units for area. The are was the original unit of area in the metric system, with:

• 1 are = 100 square metres

## What Are Area and Volume?

The last episode about how to use math to send encrypted messages ended with a bit of a cliffhanger. The good news is that you will soon find out how the story ends. But, as I warned last time, the bad news is that that isn’t going to happen quite yet. Because, in order to escape, the protagonist of our story first needs to learn a few things about measuring “sizes” in one, two, and three-dimensional space—which means that today we’re talking about area and volume.

### How to Calculate Length

In the article about 1D, 2D, and 3D coordinates, we learned that a one-dimensional object exists along a line.

We also talked about using the number line first mentioned in the article on negative numbers and integers as an example of this one-dimensional object.

Now, let’s imagine that the number line is instead a long straight road, and that the various tick marks representing the locations of positive and negative integers are actually mile markers—with the zero marker being located at the initial position of your car.

If you were to drive down the road and stop at mile marker ten, you could figure out the number of miles between that marker and mile marker zero (which is the origin where you started) by counting off how many one mile long “unit” measuring sticks you could lay down end-to-end between the two points. Of course, this gives the same result as simply calculating: 10 – 0 = 10.

So, “size” in one dimension is a measure of the length of a line segment. Of course, it’s also a quantity you calculate all the time in the real world using a tape measure: How wide is a room? How tall are you? How far is it to your destination? And numerous other similar things too.

### How to Calculate Area

Okay, that’s 1D. What about 2D? Well, first of all, as we talked about in the article on 1D, 2D, and 3D coordinates, a two-dimensional object is something that exists in a plane. As an example of a 2D object, imagine a square with sides that are each 5 meters long.

While we’re at it, let’s also imagine another square with sides that are each 1 meter long. This second square is what we’ll call the “unit” square.

In the exact same way that we used unit measuring sticks to figure out the length of lines in one-dimension, we can use unit squares to figure out the two-dimensional size—called the area—of our big square.

How?

Well, the area of a square is just the number of unit squares that fit inside it. So, in the case of our big 5 meter by 5 meter square, 5 unit squares will fit inside of it from left-to-right (since the unit squares are only 1 meter wide), and 5 unit squares will fit across it in the other direction from top-to-bottom.

That means a total of 5 x 5 = 25 unit squares will fit inside our big square. Since each unit square has an area of 1 meter x 1 meter = 1 square meter, the area of the big square is 25 square meters.

Don’t be confused by the “square meters”—it just indicates that something with a length in meters has been multiplied by something else with a length in meters—and that this is therefore a 2D quantity.