- C++ Basics
- C++ Home
- C++ Overview
- C++ Environment Setup
- C++ Basic Syntax
- C++ Comments
- C++ Data Types
- C++ Variable Types
- C++ Variable Scope
- C++ Constants/Literals
- C++ Modifier Types
- C++ Storage Classes
- C++ Operators
- C++ Loop Types
- C++ Decision Making
- C++ Functions
- C++ Numbers
- C++ Arrays
- C++ Strings
- C++ Pointers
- C++ References
- C++ Date & Time
- C++ Basic Input/Output
- C++ Data Structures

- C++ Object Oriented
- C++ Classes & Objects
- C++ Inheritance
- C++ Overloading
- C++ Polymorphism
- C++ Abstraction
- C++ Encapsulation
- C++ Interfaces

# Count number of squares in a rectangle in C++

We are given with a rectangle of length L and breadth B, such that L>=B. The goal is to find the number of squares that a rectangle of size LXB can accommodate.

Above figure shows a rectangle of size 3 X 2. It has 2, 2X2 squares and 6,1X1 squares in it.

Total squares= 6+2=8.

Every rectangle of size LXB has L*B number of 1X1 squares.

Biggest squares are of size BXB.

For L=B=1, squares = 1.

For L=B=2, squares = 1 + 4 = 5. ( 1 of 2X2, 4 of 1X1 )

For L=B=3, squares = 1 + 4 + 9 = 14. ( 1 of 3X3, 4 of 2X2, 9 of 1X1 )

For L=B=4, squares = 1 + 4 + 9 + 16 = 30 ( 1 of 4X4, 4 of 3X3, 9 of 2X2, 16 of 1X1 )

……………..

For every BXB rectangle number of squares is

for ( i=1 to B ) No. of squares + = i*i.

When L>B. More squares will be added. When L=B+1 ( 1 extra column than B ). Then squares added will be L + ( L-1) + ….+3+2+1 = L(L+1)/2

So for extra L-B columns squares added will be ( L-B ) * (B)(B+1)/2

Total squares will be squares in BXB + (L-B) * (L)(L+1)/2.

You can also use formula B(B+1)(2B+1)/6 for the series (1 + 4 + 9 +......BXB) in step 8.

Let’s understand with examples.

**Input** − L=4, B=2

**Output** − Count of squares in rectangle − 11

**Explanation** − 8 squares of 1X1 and 3 of 2X2.

**Input** − L=3, B=3

**Output** − Count of squares in rectangle − 14

**Explanation** − 9 squares of 1X1 , 4 of 2X2 and 1 of 3X3.

## Approach used in the below program is as follows

We take an integers length and breadth for dimensions of rectangle.

Function numofSquares(int l, int b) takes dimensions and returns the number of squares in the rectangle of size lXb.

For biggest squares bXb. Use for loop from 1 to b and add each i*i to squares.

Now if l>b. New added squares will be (l-b)(b)(b+1)/2. Add this to squares.

Return squares as desired result.

**Note** − keeping length>=breadth

## Example

#include<iostream> using namespace std; int numofSquares(int l, int b){ int squares=0; for(int i=1;i<=b;i++) //squares inside biggest square of size breadth X breadth{ squares += i*i; } squares+=(l-b)*b*(b+1)/2; return squares; } int main(){ int length = 5, breadth = 4; //keep length always >= breadth cout << "Count of squares is :"<< numofSquares(length,breadth); }

## Output

If we run the above code it will generate the following output −

Count of squares is :40

- Related Articles
- Possible number of Rectangle and Squares with the given set of elements in C++
- Tiling a Rectangle with the Fewest Squares in C++
- Count Magic squares in a grid in C++
- Program to count number of perfect squares are added up to form a number in C++
- Count the number of rhombi possible inside a rectangle of given size in C++
- Count the total number of squares that can be visited by Bishop in one move in C++
- Program to find number of squares in a chessboard in C++
- Sum of perimeter of all the squares in a rectangle using JavaScript
- Count squares with odd side length in Chessboard in C++
- See the figure and find the ratio of(a) Number of triangles to the number of circles inside the rectangle.(b) Number of squares to all the figures inside the rectangle.(c) Number of circles to all the figures inside the rectangle."
- Count number of trees in a forest in C++
- Count unset bits of a number in C++
- Count Number of Teams in C++
- Count number of ways to divide a number in parts in C++
- Count number of Distinct Substring in a String in C++