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What is Area in Math? — Definition & Formula

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Area definition in math

In geometry, area is the amount of space a flat shape, like a polygon, circle or ellipse, takes up on a plane. The area of a shape is always measured in square units.

Once you know how square units relate to area, you can find the area of just about any two-dimensional shape.

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How to find the area of a shape

Flat shapes have two dimensions:

Width
Length

A square, for instance, has a width equal to its length because all side lengths are the same. An ellipse has width and length, too.

We can easily see how the square could be divided up into small, square units like on a coordinate plane. You cannot easily see how an ellipse could be made up of little squares, but it can be.

Since it has width and length, it covers a space, and that space, even with the curving sides of the ellipse, can be divided up into square units:

Counting square area of square and rectangle

Counting the square units in the square is easy: one, two, three, etc..

But, how can you count all the square units in the ellipse? How do you decide what part of a square is under the top curve? What about the curves at the left and right ends?

Fortunately, mathematics has a fast way to add up all the square units without actually counting them.

Square units are the measurement unit for area because plane figures or flat shapes can always be divided into squares of known dimensions, like these:

${mm}^{2}$
${cm}^{2}$
${ft}^{2}$
${yd}^{2}$
${km}^{2}$
${mi}^{2}$

Whether you are finding the of area of a quadrilateral like a trapezoid and a rhombus, or any other closed figure, the area will always be squared.

Area formula

The area formula you use depends on which shape you are trying to find the area for.

Area of squares and rectangles

To find the area of simple shapes like a square or the area of a rectangle, you only need its width, w, and length, l (or base, b). The area is length times width:

A = l x w

The area is always squared. You will always express area as square units, derived from the linear units.

Here is a rectangle 90 meters wide and 120 meters long (the largest size of a FIFA soccer field). What is its area of this rectangle?

A=l\times w

A=120m\times 90m

A=\mathrm{10,800}{m}^{2}

Because the soccer field is measured in linear meters, its area is square meters. The area of the rectangle is 10,800 meters squared.

The area of a square formula is actually even easier than writing out length × width because all sides are equal:

A={s}^{2}

Here is a square with sides 15 inches long, the same size as the bases on an MLB baseball field. Calculating area for this square looks like this:

A={s}^{2}

A={15}^{2}

A=225{in}^{2}

Area of other shapes

All the other polygons do not easily divide into square units. Take a look at a parallelogram.

The two sides cut right across many square units. Of course, a parallelogram is just a knocked-over rectangle.

So, mathematically, if we could cut off one end and attach it to the other, we would have the area in square units. We can do exactly that, since the area of a parallelogram with a base, b, and width or height, h, is found using this formula:

A=b\times h

That is the same formula as for a square or rectangle!

If you divide a parallelogram along a diagonal, what do you have? Two triangles. That means the area of any triangle is half the area of a parallelogram with the same base length and height. A parallelogram, remember, uses the same formula as a rectangle.

The area of a triangle is, then, half the base, b, multiplied times the height, h:

A=\frac{1}{2}bh

Here is a right triangle, a sail from a 45-foot Morgan sailboat with a base $20\frac{1}{4}$ feet and a height $44\frac{1}{2}$ feet. What is its area?

For convenience in multiplying, you can change the fractions to decimals:

A=\frac{1}{2}bh

A=\frac{1}{2}(20.25ft\times 44.5ft)

A=\frac{1}{2}(901.125f{t}^{2})

A=450.5625{ft}^{2}

The area of the triangle sail is approximately 450.6 square feet.

How about the home plate of an MLB baseball field? We can calculate the area of the home-plate pentagon by considering it as two shapes:

A rectangle 17 inches × 8.5 inches
An isosceles right triangle with legs 12 inches

First, we'll use the formula to find the area of the rectangle, which comes out to $144.5{in}^{2}$ .

Next, we'll use the formula to find the area of the triangle, which comes out to $72{in}^{2}$ .

Then, we add these two areas to find the total area, which $216.5{in}^{2}$

Find the area of a circle

Some two-dimensional shapes are not even polygons, like our ellipse, or a circle. The area of a circle with radius (r) is found using this formula:

A=\pi {r}^{2}

If you have a circle with a radius of 4 cm, you can calculate the area of the circle easily with the formula above:

A=\pi {\left(4\right)}^{2}

A=\pi (16)

A=3.14(16)

A\approx 50.24

The area of the circle is approximately 50.24 square centimeters.

Find the area of an ellipse

An ellipse's area is found using its two axes, the major axis (length from the center) usually designated as aa, and the minor axis (width from the center), usually designated as bb, with this formula:

A=\pi ab

Whether you are dealing with a regular polygon or an irregular plane figure, you can find the area!