4. Square Formulas
A square is a quadrilateral with four equal sides and four right angles. Each interior angle of a square is 90°.
The side of a square is represented by a, while its diagonal is represented by d.
The square is an important topic in mensuration because its formulas are also useful in questions involving diagonals, circles, areas and perimeters.
What Is a Square?
All four sides are equal: a = a = a = a
d = Diagonal
1. Area of a Square
The area of a square is obtained by multiplying its side by itself.
The PDF also gives the area in terms of the diagonal:
Therefore:
Example 1: Area from Side
Find the area of a square whose side is 8 cm.
Formula:
Area = 8²
Area = 8 × 8
Area = 64 cm²
Answer: 64 cm²
Example 2: Area from Diagonal
The diagonal of a square is 14 cm. Find its area.
Formula:
Area = ½ × 14²
Area = ½ × 196
Area = 98 cm²
Answer: 98 cm²
2. Perimeter of a Square
A square has four equal sides. Therefore, its perimeter is the sum of its four sides.
Example
A square has a side of 12 cm. Find its perimeter.
P = 4 × 12
P = 48 cm
Answer: 48 cm
If a question asks for the total boundary, fencing or wire required around a square, use:
3. Diagonal of a Square
A diagonal joins two opposite corners of a square. The two diagonals of a square are equal and intersect at the centre.
The PDF gives the diagonal formula as:
This can also be written as:
Example 1: Find Diagonal from Side
Find the diagonal of a square whose side is 7 cm.
d = 7√2
Answer: 7√2 cm
Approximately:
d ≈ 9.90 cm
Example 2: Find Side from Diagonal
The diagonal of a square is 10√2 cm. Find its side.
From:
Therefore:
a = 10√2 / √2
a = 10 cm
Answer: 10 cm
4. Inradius of a Square
The incircle of a square is a circle that touches all four sides of the square.
The radius of this circle is called the inradius.
The PDF gives:
Why?
The diameter of the incircle is equal to the side of the square. Therefore:
Since:
we get:
Example
A square has a side of 20 cm. Find the radius of its incircle.
r = 20/2
r = 10 cm
Answer: 10 cm
5. Circumradius of a Square
The circumcircle of a square passes through all four vertices of the square.
The radius of this circle is called the circumradius.
The PDF gives the formula:
Since the diagonal of the square is:
and the circumradius is half the diagonal:
we obtain:
Example
A square has a side of 10 cm. Find its circumradius.
R = 10/√2
R = 5√2 cm
Approximately:
R ≈ 7.07 cm
Answer: 5√2 cm
6. Important Relationships in a Square
| Quantity | Formula |
|---|---|
| Side | a |
| Area | a² |
| Area using diagonal | ½d² |
| Perimeter | 4a |
| Diagonal | a√2 |
| Inradius | a/2 |
| Circumradius | a/√2 |
7. Finding Area When Perimeter Is Given
If the perimeter of a square is known, first find its side using:
Then calculate:
Example
The perimeter of a square is 40 cm. Find its area.
Step 1: Find the side.
a = P/4
a = 40/4
a = 10 cm
Step 2: Find area.
Area = a²
Area = 10²
Area = 100 cm²
Answer: 100 cm²
8. Finding Side When Area Is Given
From:
we get:
Example
The area of a square is 144 cm². Find its side.
a = √144
a = 12 cm
Answer: 12 cm
9. Common Mistakes in Square Questions
Mistake 1: Confusing Area and Perimeter
Area is measured in square units, while perimeter is measured in ordinary length units.
Area → cm², m²
Perimeter → cm, m
Mistake 2: Forgetting √2 in the diagonal formula
Mistake 3: Confusing inradius and circumradius
For a square:
Mistake 4: Using the diagonal as the side
The diagonal is longer than the side.
10. Real-Life Example
Suppose a square garden has a side of 20 m.
To find the land covered by the garden:
= 20²
= 400 m²
To put a fence around the garden:
= 4 × 20
= 80 m
11. Practice Questions
Question 1: Find the area of a square whose side is 15 cm.
Question 2: Find the perimeter of a square whose side is 18 cm.
Question 3: Find the diagonal of a square whose side is 10 cm.
Question 4: The diagonal of a square is 20√2 cm. Find its side.
Question 5: Find the inradius of a square whose side is 24 cm.
Question 6: Find the circumradius of a square whose side is 14 cm.
Question 7: The area of a square is 225 cm². Find its side and perimeter.
12. Square Formula Quick Revision
13. Mensuration Exam Tip
For quick revision, remember the following chain:
These five formulas cover the main square formulas shown in the reference sheet.
In Part 5 — Rectangle Formulas, we will cover the area, perimeter, diagonal and other important rectangle relationships with diagrams, formula explanations, solved examples and exam tips.