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**.

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Jetzt kostenlos anmeldenWhat 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**.

**Two-dimensional figures** are the flat plane shapes or figures that have two dimensions (length and width) in the same plane.

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.

Now, think about the popular game, Tetris, which is played in 2D. All of the shapes we can see in this game are two-dimensional figures that have lengths and widths. In Tetris, there are numerous two-dimensional shapes, but in math, there are four distinguished two-dimensional figures we work often with:

**Triangle****Square****Rectangle****Circle**

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

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.

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.

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.

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.

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.

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.

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.

In the picture above, we have an **isosceles triangle** in 2D. This type of triangle has two sides of the same length and a third side with a different length. If we compute the perimeter of this 2D figure, we obtain:

$P=a+b+c\phantom{\rule{0ex}{0ex}}=\hspace{0.17em}3m+3m+1m\phantom{\rule{0ex}{0ex}}=7m$

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.

Eva has a whiteboard that measures 46 cm by 60 cm. what is the perimeter of this board?

**Solution:** Two different side lengths are given, and we know that a whiteboard has four sides. So, the figure will be a rectangle. The perimeter of this rectangle $=46+46+60+60=212cm$

Find the perimeter of the given figure.

**Solution:** The perimeter of the above square figure is:

$Perimeter=a+a+a+a\phantom{\rule{0ex}{0ex}}=25+25+25+25\phantom{\rule{0ex}{0ex}}=100cm$

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.

The diameter of a circle is given as 14 cm. What is the perimeter or circumference of this circle?

**Solution:** The circle's diameter was given as $d=14cm.$To calculate the perimeter, we need to find the radius. And we know that the diameter is twice the length of the radius.

$\Rightarrow d=2r\phantom{\rule{0ex}{0ex}}\Rightarrow r=\frac{d}{2}\phantom{\rule{0ex}{0ex}}=\frac{14}{2}\phantom{\rule{0ex}{0ex}}=7cm$

So, the perimeter of a circle is:

$P=2\pi r\phantom{\rule{0ex}{0ex}}=2\times \mathrm{\pi}\times 7\phantom{\rule{0ex}{0ex}}=44cm$

Hence, the perimeter of the circle is **44 cm**.

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 (m^{2}) or square feet (ft^{2}).

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.

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

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

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.

A triangle has a base of 13 inches and a height of 6 inches. What is the area of this triangle?

**Solution:** Here, base $b=13inches$ and height $h=6inches.$ So the area is:

$A=\frac{1}{2}\times b\times h\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\times 13\times 6\phantom{\rule{0ex}{0ex}}=13\times 3\phantom{\rule{0ex}{0ex}}=39$

So, the area of the given triangle is **39 inches**^{2}.

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.

We have a tablecloth of size 70 inches by 70 inches. What is its area?

**Solution:** Here, both sides are the same length, so it is a square with length $b=70inches$. The area of the square tablecloth is:

$A={b}^{2}\phantom{\rule{0ex}{0ex}}={\left(70\right)}^{2}\phantom{\rule{0ex}{0ex}}=4900$

The area of the table cloth is **4900 inches**^{2}.

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.

Find the area of a circle with radius $r=5.2cm$ and round it to the nearest tenth.

**Solution:** The area of the circle is:

$A=\pi {r}^{2}\phantom{\rule{0ex}{0ex}}=3.14\times {\left(5.2\right)}^{2}\phantom{\rule{0ex}{0ex}}=3.14\times 5.2\times 5.2\phantom{\rule{0ex}{0ex}}=84.9056\phantom{\rule{0ex}{0ex}}\mathbf{\approx}\mathbf{84}\mathbf{.}\mathbf{9}\mathbf{}\mathit{c}{\mathit{m}}^{\mathbf{2}}$

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.

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.

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!

- In math, two-dimensional figures consist of figures with two dimensions: length and width. They are also called polygons.
- We can classify two-dimensional figures by the number of sides and vertices, the sides' lengths, and the internal angles they form.
- Some of the most-used shapes in math are the triangle, the square, the rectangle, and the circle.
- The circle consists of points that are all equally distanced with respect to one point in the shape's center. This means it has no vertices.
- The perimeter is the sum of all side lengths of the shape. For the circle, it is directly proportional to its radius.
- The area of the figure is the 2D surface delimited by its sides. Depending on the figure, we use different math formulas to compute its area.
- There are shapes with five sides called pentagons, six sides called hexagons, and more. Also, there are more examples of shapes with four sides, such as the rhombus.

**area**. You can compute the area with different math formulas, depending on which shape you have.

What dimensions do 2-dimensional figures have?

Length and width

Which is the name of the lines that form the 2-dimensional figures?

Sides

Are open figures a type of 2-dimensional figures?

Yes, although we cannot determine its perimeter and area

How much do the angles of a triangle sum?

180º

Which one of these shapes have always all of their sides equal?

Squares

Which is the name of the distance from the center of a circle to any of its points?

Radius

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