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Quadrilaterals

By Vighnesh Hemnani

In learning Geometry, one of the essential shapes that we see in everyday life are quadrilaterals. In this Thinkster Math Article, you’ll get to know the definition of a quadrilateral, its different types, how to find its area and example problems.

Why is this concept useful?

Understanding quadrilaterals are very useful in real life; almost all construction such as buildings, bridges, tunnels, require the knowledge of shapes including quadrilaterals. Engineers use these shapes on a daily basis in order figure out what patterns and styles work the best to provide safe and creative construction solutions. But quadrilaterals also show up in the most basic of daily objects: tables, picture frames, packaging – you name it. Knowing the properties of a quadrilateral can help not only you but also engineers to find other quantities such as areas, angles or lengths etc.

Where does this concept fit into the curriculum?

Grade 3 Geometry

What are quadrilaterals?

To put it simply, a quadrilateral is defined as basically any 2-dimensional closed shape that has four sides (hence “quad”-rilaterals) and that is made up of only straight lines.

Due to the fact that a quadrilateral has only 4 sides, there are some important properties that you can use to solve for different quantities such as angles or lengths. Every quadrilateral has:

  1. Four edges/sides
  2. Four points/vertices
  3. Interior Angles (which sum to 360° in total)

As you are probably guessing, if only having 4 sides is the condition for a shape to be a quadrilateral, then there must be many different ways to draw them – you are right! Because there are different ways to draw them, there are some popular quadrilaterals. These types are drawn as below:

How to use this concept?

One can use the concept of quadrilaterals to find either lengths, angles (interior or exterior) and areas of quadrilaterals. Because of certain properties of different quadrilaterals, we can determine some of the missing lengths of sides of a quadrilateral or missing angles of a quadrilateral.

In another instance, we could use the concept in order to find the area of a quadrilateral. Now, as we mentioned previously, quadrilaterals can be drawn in all kinds of ways, therefore, there is no standard formula for all quadrilaterals. However, since we know there are some popular types of quadrilaterals, as covered previously, we can show the formulas for calculating these popular types:

Square:

In a square, the most important concept is that all sides are equal and all interior angles are the same – 90°. To find its area (A), we just need to multiply the length of one side (s) times itself:

        A = (s)2

Rectangle:

Similar to the square, here all the interior angles are the same – 90°, however, the opposite sides in this shape have the same length. So two of the sides are of equal length, and so are the two others. When it comes to find its area (A), we multiply the width (w) by the height (h) of the rectangle:

        A = w • h

Parallelogram:

To find the area of a parallelogram, you need to know the height from the bottom base to the top and the length of the base or the top (length of the base and the top are the same in a parallelogram). The formula to find the area of a parallelogram is as below:
        A = b • h

Rhombus/Kite:

For the rhombus, the two most important dimensions that define the shape of a rhombus are the two diagonals. Note, a kite is just a type of rhombus, so the area to find a kite is the same way as you find it for a rhombus. To find the area, you need to multiply the lengths of the two diagonals and divide by two:

A=vert−diagonal•horz−diagonal2A = \frac{vert-diagonal • horz-diagonal}{2}

Trapezoid:

For the trapezoid, to find the area, you need add the two bases (top and bottom), divide by two and then multiple what you have with the height of the trapezoid:

A=base+base2∗heightA = \frac{ base + base}{2} * height

Sample Math Problems

Question

For each of the shapes below, label which type of quadrilateral they are:

Answer

Shapes with the labels are as below:

Question

Given that you know that the two diagonals in a rhombus are 5cm and 8cm, then what is the area of the rhombus?

Answer

The area of a rhombus is calculated by multiplying the two diagonals and then dividing by two, so the answer to this problem is:

A=5cm∗8cm2=20cm2A = \frac{5cm*8cm}{2} = 20cm^{2}

Question

If we know that the length of a rectangle is 10 cm and that the area of the rectangle is 20 cm2, then is it possible to find the missing side, i.e. the width of the rectangle? If yes, find the value. If no, explain why.

Answer

Yes, it is possible to find the missing width of the rectangle. To find the area of a rectangle, you need to know the length and width of it. We already know the length, so to find one missing quantity in the following formula shouldn’t be too hard:

A=height∗weightA = height*weight

So:

20 = 10 * width

2 cm = width

Question

Find the area of the following quadrilateral:

Answer

The above shape is a trapezoid, and like we previously discussed, the area of a trapezoid is as follows:

A=basebottom+basetop2∗heightA = \frac{base bottom + base top}{2} *height

Since we know that the bottom base is 3 m, the top base is 2 m, and the height of the trapezoid is 2 m, then we can find the area as below:

A=3+22∗2=52∗2=5m2A = \frac{3+2}{2} *2 = \frac{5}{2}*2= 5m^{2}

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Practice Math Problems

1. Find x in the following diagram:

2. Find area of the trapezoid, i.e. find x:

3. A room in a house has exactly the shape of a parallelogram. If you know the height of the parallelogram is 4m and base is 9m, then what is the area of the room?

4. An architect split some coffee on his blueprints for a new project. He was able to save most of the blueprints, but he couldn’t save a plan that has a house in the shape of a kite (from an aerial point of view). He only remembers that the area of the house was 550 cm2 and that the length of the longer diagonal of the kite-shaped house is 50 m. Find the shorter diagonal of the kite-shaped house.

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