Square Definition
| Definition of a Square | A parallelogram with all sides of the same length. |
| Definition of a Parallelogram | A parallelogram is any shape that has four sides where each opposite side is parallel to each other. |
Properties of a Square
In order to solve any problems involving squares, you should know what the properties of squares are. The three most important properties are listed in the table below.
| Property 1 | A square has four right angles. Recall that right angles are 90 degrees. |
| Property 2 | A square four sides that are congruent. Congruent means that line segments are the same length. |
| Property 3 | The diagonals of the square are perpendicular bisectors, meaning they are perpendicular and cross at exactly their midpoints. |
In order to solve for any problem dealing with squares, you should know the most important formulas when it comes to squares. They’re listed in the table below.
| Perimeter of a Square | P = s+s+s+s = 4s |
| Area of a Square | A =  |
| Diagonal of a Square | D =  |
Problem 1
You have to build a fence around your garden. You want the garden to be square. What are the dimensions of this fence given that the perimeter is 28 meters?
Problem 2
You want to build a square pool in one corner of your garden. You want to be able to have at least 49 square meters of space left after the pool is built. Do the following dimensions satisfy these conditions?
Problem 3
You and your friend have a bet. Your friend has stated that the diagonal of a square is the same size as the length of the square. Since you know the properties of squares, you bet that you can guess the exact length of the diagonal of a square sandwich by only knowing the length of one side. What is the diagonal of the sandwich below?
Problem 4
Determine the area of the shaded region below. Remember to use what you know about the properties of squares in addition to what you know about the properties of triangles.
Problem 5
You have 2 squares that have a length of 2cm, while you have 8 squares that have the length of 1cm. Try to come up with a way to position these squares so that the two squares have a larger perimeter than the 8 squares. Next, try the other way around: position the 8 squares so that together they have a larger perimeter than the two squares.
Solution Problem 1
You have to build a fence around your garden. In this problem you needed to find the dimensions of this fence given that the perimeter of the square garden is 28 meters. Let’s take a look at the perimeter formula.
| Side of a square | s |
| Perimeter of a square | P = s+s+s+s = 4*s |
In order to find the dimensions, we simply need to follow the steps below.
| Step 1 | Side of a square | s |
| Step 2 | Formula of the perimeter | P = 4*s |
| Step 3 | Solve for one side | 28 = 4s s = 7 |
The length and width of the square are 7.
Solution Problem 2
You want to build a square pool in one corner of your garden. You want to be able to have at least 49 square meters of space left after the pool is built. Let’s check whether the dimensions satisfy these conditions.
| Garden area | Square pool area | Area left over |
| A | B | C |
In order to find the area left over, simply follow these steps.
| Step 1 | Area of the garden | A = 15*15 = 225 |
| Step 2 | Area of the pool | A = 6*6 = 36 |
| Step 3 | Area left over | A = 225 – 36 = 189 |
Solution Problem 3
You and your friend have a bet. Your friend has stated that the diagonal of a square is the same size as the length of the square. You needed to guess the diagonal of the sandwich in this problem. To do that, simply follow the formula for the diagonal of a square.
| Formula | |
| Diagonal of a square |  |
To solve this, simply plug the length of the side into the formula.
\[
d = \sqrt{2*10^{2}} = 14.14
\]
Solution Problem 4
In order to determine the area of the shaded region, we actually only need one formula. However, we need to know some of the properties of a square:
| Property 1 | A square has 4 right angles |
| Property 2 | The diagonal of a square forms two right triangles |
Because of this, we can easily find the area of the triangle by using the formula for the area of a triangle.
| Area of a triangle | A = ½ b*h |
| Shaded region | A = ½ 25*25 = 312 |
Solution Problem 5
One way you can position these squares so that the two squares have a larger perimeter than the 8 squares is the following.
| A | B |
| P = 14 | P = 12 |
Next here is one way you can position the 8 squares so that together they have a larger perimeter than the two squares.
| A | B |
| P = 12 | P = 18 |
Summarise with AI:
Did you like this article? Rate it!
Loading...
Nice emma really precise information.. I appreciate u
Hey Raptor! Thanks very much for the positive feedback, very much appreciated!
explains everything for a child or a confused student in their ap maths class who is lost in the more detailed mathematics
Draw all types of polygon and its area and perimeter class 7th NCERT course
I’m just curious if the area between the polygon and the circumscribed circle has a name.
https://www.superprof.co.uk/resources/academic/maths/geometry/plane/orthocenter-centroid-circumcenter-and-incenter-of-a-triangle.html
Draw all types of polygon and its area and perimeter