Learn square formulas for area, perimeter, diagonal and side with simple solved examples, shortcuts and exam-focused questions for maths students.
Answer Box:
A square has four equal sides and four right angles. Its main formulas are: Area = side², Perimeter = 4 × side, and Diagonal = side × √2. If the area, perimeter, or diagonal is given, the side can also be calculated using the corresponding inverse formula.
Square Formulas at a Glance
A square is one of the most frequently used figures in elementary and competitive-exam mathematics. Questions based on squares usually test your understanding of area, perimeter, diagonal, side length, and relationships between these quantities.
For quick revision, remember these formulas:
| Quantity | Formula |
|---|---|
| Side | aa |
| Area | a2a^2 |
| Perimeter | 4a4a |
| Diagonal | a2a\sqrt{2} |
| Side from area | Area\sqrt{\text{Area}} |
| Side from perimeter | Perimeter4\frac{\text{Perimeter}}{4} |
| Side from diagonal | Diagonal2\frac{\text{Diagonal}}{\sqrt{2}} |
| Area using diagonal | d22\frac{d^2}{2} |
Here, a represents the side of the square and d represents its diagonal.
These formulas are useful for school mathematics as well as aptitude and competitive examinations where geometry questions are often combined with arithmetic calculations.
What Is a Square?
A square is a quadrilateral having:
- Four equal sides.
- Four angles of 90° each.
- Opposite sides parallel to each other.
- Two diagonals of equal length.
- Diagonals that bisect each other at right angles.
- Diagonals that divide the square into four equal right-angled triangles.
If the length of each side is represented by a, the basic properties can be expressed mathematically.
For example, if each side is 6 cm, then the square has:
- Side = 6 cm
- Area = 36 cm²
- Perimeter = 24 cm
- Diagonal = 626\sqrt2 cm
Understanding these relationships makes formula-based questions much easier.
Square Area Formula
The area of a square is calculated by multiplying its side by itself.
Formula
Area=a2\boxed{\text{Area}=a^2}
Where:
- a = side of the square
- Area is expressed in square units such as cm², m², etc.
Example 1: Find the Area of a Square
A square has a side of 8 cm. Find its area.
Solution:
Given:
a=8 cma=8\text{ cm}
Using the formula:
Area=a2\text{Area}=a^2 =82=8^2 =64 cm2=64\text{ cm}^2
Answer: 64 cm²
Example 2: Side Given in Metres
Find the area of a square whose side is 12 m.
Area=122\text{Area}=12^2 =144 m2=144\text{ m}^2
Answer: 144 m²
Exam Tip
Do not confuse area with perimeter. Area is measured in square units, while perimeter is measured in ordinary units such as cm or m.
Square Perimeter Formula
The perimeter of a square is the total distance around its boundary.
Since all four sides of a square are equal:
Perimeter=a+a+a+a\text{Perimeter}=a+a+a+a
Therefore,
Perimeter=4a\boxed{\text{Perimeter}=4a}
Example 3: Find the Perimeter
A square has a side of 15 cm. Find its perimeter.
Using:
P=4aP=4a P=4×15P=4\times15 P=60 cmP=60\text{ cm}
Answer: 60 cm
Example 4: Find the Side from Perimeter
The perimeter of a square is 72 cm. Find its side.
Using:
P=4aP=4a
Therefore,
a=P4a=\frac{P}{4} a=724=18 cma=\frac{72}{4}=18\text{ cm}
Answer: 18 cm
Square Diagonal Formula
A square has two equal diagonals. The diagonal joins two opposite vertices.
If the side of a square is a, its diagonal is:
d=a2\boxed{d=a\sqrt2}
This formula comes directly from the Pythagorean theorem.
Consider a right triangle formed by two sides of the square and its diagonal.
d2=a2+a2d^2=a^2+a^2 d2=2a2d^2=2a^2
Taking the square root:
d=a2d=a\sqrt2
Therefore:
d=a2\boxed{d=a\sqrt2}
Example 5: Find the Diagonal
A square has a side of 10 cm. Find its diagonal.
d=a2d=a\sqrt2 d=102 cmd=10\sqrt2\text{ cm}
Answer: 10210\sqrt2 cm
If an approximate decimal value is required, using 2≈1.414\sqrt2\approx1.414:
10×1.414=14.14 cm10\times1.414=14.14\text{ cm}
So the diagonal is approximately 14.14 cm.
How to Find the Side When Diagonal Is Given
Sometimes a question provides the diagonal instead of the side.
Starting with:
d=a2d=a\sqrt2
we get:
a=d2\boxed{a=\frac{d}{\sqrt2}}
An equivalent form is:
a=d22\boxed{a=\frac{d\sqrt2}{2}}
Example 6: Diagonal Given
The diagonal of a square is 14214\sqrt2 cm. Find its side.
Using:
a=d2a=\frac{d}{\sqrt2} a=1422a=\frac{14\sqrt2}{\sqrt2} a=14 cma=14\text{ cm}
Answer: 14 cm
Area of a Square When Diagonal Is Given
A useful formula for competitive examinations is the area formula in terms of diagonal.
We know:
d=a2d=a\sqrt2
Squaring both sides:
d2=2a2d^2=2a^2
Since:
a2=Areaa^2=\text{Area}
we get:
Area=d22\boxed{\text{Area}=\frac{d^2}{2}}
Example 7: Find Area from Diagonal
The diagonal of a square is 20 cm. Find its area.
Using:
Area=d22\text{Area}=\frac{d^2}{2} =2022=\frac{20^2}{2} =4002=\frac{400}{2} =200 cm2=200\text{ cm}^2
Answer: 200 cm²
This formula can save time when the diagonal is directly given.
Important Relationship Between Side, Area, Perimeter and Diagonal
For a square with side a:
Area=a2\boxed{\text{Area}=a^2} Perimeter=4a\boxed{\text{Perimeter}=4a} Diagonal=a2\boxed{\text{Diagonal}=a\sqrt2}
These three formulas are the foundation of most square-based questions.
If the area is given:
a=Area\boxed{a=\sqrt{\text{Area}}}
If the perimeter is given:
a=P4\boxed{a=\frac{P}{4}}
If the diagonal is given:
a=d2\boxed{a=\frac{d}{\sqrt2}}
A good strategy is to first identify what has been given and what the question asks you to find. Then choose the shortest formula connecting the two quantities.
Square Formulas Using Different Given Values
| Given | To Find | Formula |
|---|---|---|
| Side | Area | A=a2A=a^2 |
| Side | Perimeter | P=4aP=4a |
| Side | Diagonal | d=a2d=a\sqrt2 |
| Area | Side | a=Aa=\sqrt A |
| Perimeter | Side | a=P/4a=P/4 |
| Diagonal | Side | a=d/2a=d/\sqrt2 |
| Diagonal | Area | A=d2/2A=d^2/2 |
Solved Questions on Square Formulas
Example 8: Area Given, Find Side
The area of a square is 169 cm². Find its side.
a=169a=\sqrt{169} a=13 cma=13\text{ cm}
Answer: 13 cm
Example 9: Area Given, Find Perimeter
The area of a square is 196 m². Find its perimeter.
First find the side:
a=196=14 ma=\sqrt{196}=14\text{ m}
Now:
P=4aP=4a P=4×14=56 mP=4\times14=56\text{ m}
Answer: 56 m
Example 10: Side Given, Find Area and Perimeter
The side of a square is 25 cm. Find its area and perimeter.
Area:
A=252=625 cm2A=25^2=625\text{ cm}^2
Perimeter:
P=4×25=100 cmP=4\times25=100\text{ cm}
Answer:
- Area = 625 cm²
- Perimeter = 100 cm
Example 11: Perimeter Given, Find Area
The perimeter of a square is 80 cm. Find its area.
First calculate the side:
a=804a=\frac{80}{4} a=20 cma=20\text{ cm}
Now calculate area:
A=202A=20^2 A=400 cm2A=400\text{ cm}^2
Answer: 400 cm²
Example 12: Diagonal Given, Find Area
The diagonal of a square is 10210\sqrt2 cm. Find the area.
Using:
A=d22A=\frac{d^2}{2} A=(102)22A=\frac{(10\sqrt2)^2}{2} =100×22=\frac{100\times2}{2} =100 cm2=100\text{ cm}^2
Answer: 100 cm²
Square Formula Questions Based on Change in Side
Questions sometimes ask what happens to the area or perimeter when the side of a square changes.
Suppose the original side is a.
Original area:
A=a2A=a^2
If the side becomes 2a, the new area is:
(2a)2=4a2(2a)^2=4a^2
Therefore, doubling the side makes the area four times the original area.
For perimeter:
Original:
P=4aP=4a
New perimeter:
P=4(2a)=8aP=4(2a)=8a
So, doubling the side doubles the perimeter.
This difference is frequently useful in aptitude-style mathematics.
Example 13: Side Doubled
The side of a square is doubled. What happens to its area?
Original area:
a2a^2
New side:
2a2a
New area:
(2a)2=4a2(2a)^2=4a^2
Therefore, the new area is four times the original area.
Square vs Rectangle: Basic Difference
A square and rectangle both have four sides and four right angles, but their side relationships are different.
| Property | Square | Rectangle |
|---|---|---|
| Number of sides | 4 | 4 |
| Angles | All 90° | All 90° |
| Equal sides | All four | Opposite sides |
| Area | a2a^2 | l×bl\times b |
| Perimeter | 4a4a | 2(l+b)2(l+b) |
| Diagonal | a2a\sqrt2 | l2+b2\sqrt{l^2+b^2} |
A common mistake is to use the rectangle’s area or perimeter formula without checking whether all four sides are equal.
Common Mistakes in Square Formula Questions
1. Confusing Area and Perimeter
For side aa:
A=a2A=a^2
but
P=4aP=4a
They are not interchangeable.
2. Forgetting Square Units
Area should be written in units such as:
- cm²
- m²
- km²
Perimeter should be written in units such as:
- cm
- m
- km
3. Using a2a\sqrt2 for Area
The expression:
a2a\sqrt2
is the diagonal, not the area.
4. Missing the Square Root
If area is given as 225 cm²:
a=225=15 cma=\sqrt{225}=15\text{ cm}
Do not take 225 as the side.
5. Forgetting to Square the Diagonal
For area from diagonal:
A=d22A=\frac{d^2}{2}
not d/2d/2.
Quick Revision: Square Formulas
Before an examination, remember these formulas:
Area
A=a2\boxed{A=a^2}
Perimeter
P=4a\boxed{P=4a}
Diagonal
d=a2\boxed{d=a\sqrt2}
Side from Area
a=A\boxed{a=\sqrt A}
Side from Perimeter
a=P4\boxed{a=\frac P4}
Side from Diagonal
a=d2\boxed{a=\frac d{\sqrt2}}
Area from Diagonal
A=d22\boxed{A=\frac{d^2}{2}}
These formulas are enough to solve a large number of direct square-based problems.
Practice Questions on Square Formulas
Try these questions without looking at the answers first.
Question 1
A square has a side of 9 cm. What is its area?
A. 18 cm²
B. 36 cm²
C. 81 cm²
D. 90 cm²
Answer: C. 81 cm²
Question 2
The perimeter of a square is 48 cm. What is its side?
A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm
Answer: B. 12 cm
Question 3
What is the diagonal of a square having side 7 cm?
A. 77 cm
B. 1414 cm
C. 727\sqrt2 cm
D. 49249\sqrt2 cm
Answer: C. 727\sqrt2 cm
Question 4
The area of a square is 256 cm². Find its side.
A. 14 cm
B. 15 cm
C. 16 cm
D. 18 cm
Answer: C. 16 cm
Question 5
The diagonal of a square is 12 cm. What is its area?
A. 36 cm²
B. 48 cm²
C. 72 cm²
D. 144 cm²
Answer: C. 72 cm²
Question 6
If the side of a square is increased from aa to 3a3a, the area becomes:
A. 3a23a^2
B. 6a26a^2
C. 9a29a^2
D. 12a212a^2
Answer: C. 9a29a^2
Why Square Formulas Matter for Teacher Job Exams
For teacher recruitment and teaching-related examinations, basic geometry is an important part of mathematical problem-solving. Square questions are generally useful because they test more than simple formula memorization.
A question may provide:
- Side and ask for area.
- Perimeter and ask for side.
- Area and ask for perimeter.
- Diagonal and ask for area.
- A change in side and ask for the change in area.
- A word problem involving fencing, flooring or a square-shaped region.
The safest approach is to understand the relationship among the formulas rather than memorizing each one separately.
For broader exam preparation, students can also revise mathematics concepts through relevant JEE Main syllabus and weightage resources where applicable to their level.
Internal Links and Relevant Resources
- JEE Main Syllabus and Class 11–12 Weightage
- NDA Exam Pattern and Marking Scheme
- SSC CGL Form Not Submitting: Fix
Practice Square Formulas Online
You have covered the formulas and solved examples. The next useful step is to check whether you can identify the correct formula quickly under exam conditions.
[Start Square Formula Online Test]
Practice is especially useful for improving calculation speed and avoiding common mistakes between area, perimeter and diagonal formulas.
FAQs on Square Formulas
What is the formula for the area of a square?
The area of a square is:
A=a2A=a^2
where aa is the length of one side.
What is the formula for the perimeter of a square?
The perimeter is:
P=4aP=4a
because all four sides of a square are equal.
What is the diagonal formula of a square?
The diagonal is:
d=a2d=a\sqrt2
where aa is the side length.
How do you find the side of a square from its area?
Take the square root of the area:
a=Aa=\sqrt A
For example, if the area is 144 cm², the side is 12 cm.
How do you find the area of a square from its diagonal?
Use:
A=d22A=\frac{d^2}{2}
For example, a square with a diagonal of 10 cm has an area of 100/2=50100/2=50 cm².
What happens to the area if the side of a square is doubled?
If the side changes from aa to 2a2a, the area changes from a2a^2 to 4a24a^2. Therefore, the area becomes four times the original area.
Are square area and perimeter formulas the same?
No. The area is:
a2a^2
while the perimeter is:
4a4a
Area measures the surface enclosed by the square, whereas perimeter measures the distance around it.
Conclusion
The most important square formulas to remember are Area = a2a^2, Perimeter = 4a4a and Diagonal = a2a\sqrt2. From these, you can derive the formulas for finding the side, area from the diagonal, and other related quantities.
For teacher-job and other mathematics examinations, focus on recognizing which value is given, which value is required, and which formula connects them. After revising the formulas, solve mixed questions rather than practicing only direct substitutions. That is the quickest way to turn formula knowledge into exam-ready problem-solving skill.