Area of Square: Formula, Examples and Questions

Area of Square is the region enclosed by all four equal sides of a square. Its area is found by multiplying the side by itself. This page covers the area of square formula, unit conversions, relations with perimeter and diagonal, solved examples, and common aptitude problems.

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What Is the Area of a Square?

The area of a square is the surface enclosed within its four equal sides. If the side length is a, then its area is a² square units.

A square has four equal sides and four right angles. To calculate its area, multiply the length of one side by itself. For example, a square with side 8 cm has area 8 × 8 = 64 cm². The unit of area is always the square of the unit used for the side.

Area of Square Video Lessons

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15 Lessons
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Square Area and Perimeter

Learn how to calculate the area and perimeter of a square using the relevant side-length formulas, with clear application of both mensuration concepts.

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Area of Square Formula & Tricks

Important Formulas

Area using side
A = a²

A is the area and a is the length of one side of the square.

Side using area
a = √A

The side is the square root of the given area.

Area using perimeter
A = (P/4)²

Since the side of a square is P/4, substitute it in A = a².

Area using diagonal
A = d²/2

The diagonal d divides the square into two right triangles, giving this relation.

Perimeter of a square
P = 4a

The perimeter is the sum of its four equal sides.

Quick Tricks

Use the side ratio directly

If the sides of two squares are in the ratio m:n, their areas are in the ratio m²:n².

Example: If the sides are in the ratio 2:3, the areas are in the ratio 4:9.
Apply percentage change to the square

If the side changes by r%, the new area is A(1 ± r/100)². The plus sign is used for an increase and the minus sign for a decrease.

Example: If the side increases by 10%, the area becomes A × 1.1² = 1.21A, so the area increases by 21%.
Convert units before squaring

Convert the side into one unit first, then square it. For area units, 1 m² = 10,000 cm².

Example: A side of 2 m gives area 4 m² = 40,000 cm².

Area of Square Concepts

Finding Area from the Side

Multiply the side length by itself to find the area: A = a².

The two dimensions of a square are equal, so length × breadth becomes a × a. If the side is 15 m, the area is 15² = 225 m².

Example: For a square of side 12 cm, A = 12² = 144 cm².

Finding the Side from the Area

Take the positive square root of the area to find the side: a = √A.

A length cannot be negative, so only the positive square root is used. If the area is 625 m², the side is √625 = 25 m.

Example: If A = 196 cm², then a = √196 = 14 cm.

Area from Perimeter

When the perimeter is known, divide it by 4 to get the side and then square it: A = (P/4)².

All four sides of a square are equal. For a perimeter of 48 cm, the side is 48/4 = 12 cm, so the area is 12² = 144 cm².

Example: If P = 80 m, then A = (80/4)² = 20² = 400 m².

Area from the Diagonal

If the diagonal is d, the area is d²/2.

The diagonal of a square satisfies d = a√2. Squaring gives d² = 2a², so A = a² = d²/2.

Example: For a diagonal of 10 cm, the area is 10²/2 = 50 cm².

Effect of Changing the Side

The area changes as the square of the side, not in the same ratio as the side.

If the side is multiplied by k, the area is multiplied by k². Thus, doubling the side makes the area four times, while halving the side makes the area one-fourth.

Example: A square with side 6 cm has area 36 cm². A side of 12 cm gives area 144 cm², which is four times larger.

Area of Square Visual Guide

Use the diagram or chart below to understand the concept clearly.

Square with equal sides

Visual guide: Square with equal sides
Each side: a Area: a²
A square has four equal sides of length a, and its area is a².

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Practice Area of Square Questions

Practise published questions related to this topic.

1The Least Common Multiple of two numbers is 150. Which of the following can be the Highest Common Factor of the two numbers?→ 2Which of the following is the CORRECT comparison?→ 3A dealer marks the price of a computer at ₹52,900. He offers a discount of 9% on the marked price. After the sale, the salesperson is given a commission of 5% on the final selling price. What is the amount of commission earned by the salesperson (in ₹)?→ 4A shopkeeper sold an article for ₹864 at a 20% profit. What was its cost price?→ 5With Rs. 1,000, one can buy 4 pairs of trousers or 8 shirts. What is the amount required to buy 3 pairs of trousers and 3 shirts?→ 6A clothing retailer applies two successive discounts of 15% and 5% on a jacket originally priced at ₹ 1500. What is the price of the jacket after both discounts?→ 7Rakesh's income is 25% more than Rohan's income. By what percentage is Rohan's income less than Rakesh's income?→ 8A trader buys an article at ₹3,500 and marks it up by 35% . He gives two successive discounts of 20% and 10% . What is his net profit or loss percentage?→ 9The current market value of a specialized industrial machine is ₹1,50,000. It is expected to depreciate at an annual rate of 10%. What will be the future valuation of the machine after a period of 3 years?→ 10Which of the following numbers is divisible by 11?→

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Area of Square Quick Quiz

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Quick Revision Notes

Area of Square Revision Points

Key formulas and rules for quick revision.

  • Area of a square = side².
  • Side = √area.
  • Perimeter = 4 × side.
  • Area from perimeter = (perimeter/4)².
  • Area from diagonal = diagonal²/2.
  • If the side is multiplied by k, the area is multiplied by k².
  • Always express area in square units.
  • 1 m² = 10,000 cm² and 1 cm² = 100 mm².

Area of Square FAQs

What is the area of a square with side 9 cm?

Area = 9² = 81 cm².

How do you find the area of a square when its perimeter is 36 cm?

Side = 36/4 = 9 cm. Therefore, area = 9² = 81 cm².

What is the area of a square with diagonal 14 cm?

Use A = d²/2. Therefore, area = 14²/2 = 196/2 = 98 cm².

If the area of a square is 441 m², what is its side?

Side = √441 = 21 m.

If the side of a square increases by 20%, by what percentage does its area increase?

New area factor = 1.2² = 1.44. Therefore, the area increases by 44%.

What happens to the area if the side of a square is reduced by 25%?

The new area factor is 0.75² = 0.5625. Thus, the area becomes 56.25% of the original and decreases by 43.75%.

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