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.
Table of Contents
ToggleA 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.
A square is a quadrilateral having:
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:
Understanding these relationships makes formula-based questions much easier.
The area of a square is calculated by multiplying its side by itself.
Area=a2\boxed{\text{Area}=a^2}
Where:
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
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
Do not confuse area with perimeter. Area is measured in square units, while perimeter is measured in ordinary units such as cm or m.
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}
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
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
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}
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.
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}}
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
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}}
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.
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.
| 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 |
The area of a square is 169 cm². Find its side.
a=169a=\sqrt{169} a=13 cma=13\text{ cm}
Answer: 13 cm
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
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:
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
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
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.
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.
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.
For side aa:
A=a2A=a^2
but
P=4aP=4a
They are not interchangeable.
Area should be written in units such as:
Perimeter should be written in units such as:
The expression:
a2a\sqrt2
is the diagonal, not the area.
If area is given as 225 cm²:
a=225=15 cma=\sqrt{225}=15\text{ cm}
Do not take 225 as the side.
For area from diagonal:
A=d22A=\frac{d^2}{2}
not d/2d/2.
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.
Try these questions without looking at the answers first.
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
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
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
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
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
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
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:
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.
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.
The area of a square is:
A=a2A=a^2
where aa is the length of one side.
The perimeter is:
P=4aP=4a
because all four sides of a square are equal.
The diagonal is:
d=a2d=a\sqrt2
where aa is the side length.
Take the square root of the area:
a=Aa=\sqrt A
For example, if the area is 144 cm², the side is 12 cm.
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.
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.
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.
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.
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