2 Dimensional Figures

What is a two-dimensional figure in math? Maybe the term 2D sounds familiar and makes you think of a video game, like Tetris, or a movie. This concept is similar to the understanding of 2D that we have in math. Two-dimensional (2D) figures are shapes formed by closed lines in a plane, and they have two dimensions: length and width. As they do not exist in three-dimensional (3D) space, these shapes do not have depth.

Definition of 2-dimensional figures

For example, if we drew three lines on a 2D plane surface, like a piece of paper, we could obtain a triangle, which is an example of a 2D shape. We just need one plane to show these 2D figures, as they do not have depth. In math, there are as many 2D shapes as you can imagine, as you just have to link one line with another in a plane.

These lines that form the shapes are called the sides of the plane figure. All sides do not have to be connected, as we can distinguish between closed shapes or open shapes, depending on whether they form vertices or not. We will mainly focus on closed shapes, as they are the most common in math.

Examples of 2-dimensional figures, StudySmarter Originals

Examples of 2-dimensional figures

• Triangle
• Square
• Rectangle
• Circle

Let's consider each of these four two-dimensional shapes in more detail.

Triangle

As a 2D shape, the triangle consists of three sides and three vertices. The summation of all the internal angles in a triangle is equal to 180º. We can distinguish between different types of triangles depending on whether the sides are equal or not. We can also distinguish types of triangles by the angles they form with one another.

For example, the triangles with all sides of the same length are called equilateral triangles, while if they have just two equal sides, they are called isosceles triangles. If none of the sides are the same length, the triangle is called a scalene triangle. On the other hand, an example of a triangle classified by its internal angles is a right triangle, which has an angle of 90º.

Different triangles, StudySmarter Originals

Square

As 2-dimensional figures, squares are formed by four equal sides with four vertices. All of the internal angles formed by the vertices are equal to 90º. We can label a 2D shape as a square only if all four sides are of the same length.

Square, StudySmarter Originals

Rectangle

These shapes are formed by four sides, with each side equal only to its opposite side. Therefore, both of these two pairs have the same length between them. In a rectangle, all internal angles formed by the vertices are equal to 90º, like in the square. If all the sides' lengths were equal as well, the 2-dimensional figure would be a square.

Illustration of a rectangle, StudySmarter Originals

Circle

In a 2D plane, the circle consists of points that are all equally distanced with respect to one point in the shape's center. This means that it has no vertices. In other words, we can also understand a circle as a uniquely curved line that is equally distanced from the center at all of its points.

The distance from the circle's points to its center is called the radius. Also, if we measure from one point of the circle to another, passing through the circle's center, the distance is called the diameter. The diameter is always twice the length of the radius.

Circle, StudySmarter Originals

There are more two-dimensional shapes in math that we can classify based on aspects like the number of sides and vertices as well as their structure.

Perimeter of a 2-dimensional figure

In math, the perimeter of a 2D figure is the total sum of the length of all of its sides. Therefore, if the sides of the plane figure are expressed in the length unit of meters, for example, the perimeter of the shape is also expressed with meters. We can express the perimeter with the following formula:

$P=a_1+a_2+a_3+...+a_n=\underset{i=1}{\overset{n}{\sum a_n}}$

In the perimeter formula, the terms $a_1+a_2+a_3$ (and so on) represent the different sides in the two-dimensional figure. In the second part of the equation is a symbol ($\sum$) which indicates that all these side lengths should be summed up.

Perimeter of a triangle

Let’s take a look at the 2D shapes with the lowest number of sides: the triangles. The triangle has three sides; therefore, the perimeter of the triangle is equal to the sum of those three sides. Let's take a look at an example of the perimeter calculation below.

Isosceles triangle with side lengths, StudySmarter Originals

Perimeter of squares and rectangles

Even though a triangle, a square, and a rectangle are not the same, we can still calculate their perimeters with the same formula given above. And if we have any other 2D figure, this process of summing up all sides remains the same as well.

For squares and rectangles, we have to sum up four sides to calculate the perimeter. The perimeter of the square is $a+a+a+a$, where a is the side length of all four sides. The perimeter of a rectangle is $a+a+b+b$, where a and b are the two different side lengths of the equal opposite pairs. Let's see some examples.

Perimeter of a circle

Now you may be wondering, "But what about the circle?" Calculating a circle's perimeter of course cannot be done with side lengths! We defined the circle as a 2D shape formed by points that are all equally distanced from the center. To calculate the perimeter of a circle in 2D (also called the circumference), we use a different formula:

$P=\hspace{0.17em}2\pi r$

In this formula, r is equal to the radius of the circle and $\pi$ is the number pi, which has a fixed value. From this formula, we see that the perimeter of a circle is proportional to its radius. So, if we increase the radius of a circle, we also increase its perimeter.

Area of 2-dimensional figures

In math, the area of a two-dimensional figure is the quantity of surface delimited by the perimeter of a figure in a plane. In other words, the area in 2D is the space inside the lines we use to draw a figure. We use square units to describe area, like square meters (m²) or square feet (ft²).

Now, take a look out from your computer at the floor of the room. Imagine the walls as lines of a shape in 2D. The surface of the floor you are observing is its area because it is the space inside the perimeter (in this case, the room's walls).

Depending on two-dimensional figure and its shape, we have different formulas to compute area.

Area of a triangle

Starting again with the 2D shape with the lowest number of vertices, the area of the triangle is calculated with the following math formula:

$A=\frac{1}{2}bh$

Isosceles triangle with base and height, StudySmarter Originals

The area of the triangle depends on the base b of the triangle and its height h, which is the distance from the middle of the base to the opposite vertex. The base of the triangle does not need to be its shortest side: it can be any side. However, we then need to measure the height from the side chosen as the base to the opposite vertex.

Area of squares and rectangles

The area measurement for the square and the rectangle are the same, but we will describe the area of the rectangle first, as it is more general with this math formula:

$A=bh$

In this case, b is one side and h is another side with a different value. This area computation works for any 2D figure with four sides that are parallel to each other, called a parallelogram. Therefore, it also works for the square, but as all the sides have the same length in a square, we can also calculate its area as:

$A=b^2=b\times b$

Where b is the length of any side.

Area of a circle

Lastly, we have the area of the circle. As with the perimeter, its area also depends on the radius. The area of a circle can be calculated with the following equation:

$A=\pi r^2$

Again, the r corresponds to the radius of the circle, and π is the number pi. From the formula, we see that if we make the radius bigger and bigger, the area of the circle also grows (in this case, by the power of two).

For example, you could see how this relationship works in real life in a garden. Imagine you attach a rope to some point and spin it in circles around that point. This motion would describe the shape of a 2D circle. If you moved the spinning rope further away from its center point, increasing the radius of the circle, you would see that the area of the spinning rope is now bigger.

Further representations of 2-dimensional figures

We have previously seen some 2D shapes such as the triangle, the square, the rectangle, and the circle. But an infinite number of figures exist that you could describe. In general, we classify two-dimensional figures by their number of sides and vertices as well as their internal angles (formed by the vertices).

If we increased a rectangle's sides by one, it would have five sides, making it a pentagon. With six sides, it'd be a hexagon, and so on.

Polygons with different numbers of sides, StudySmarter Originals

There are also different types of four-sided two-dimensional figures. Apart from the rectangle and the square, if a 2D shape has at least two equal sides and its angles are not 90º, it is a rhombus, with a shape similar to a diamond.

Rhombus, StudySmarter Originals

There are a lot of different 2D shapes, with regular sides, irregular sides, equal angles, etc. Now you just have to use some imagination and try to search for examples of them!