Arectangle is a four-sided figure with opposite sides parallel and equaland all angles equal. It can also be defined as a parallelogram with alangles equal.

Arectangle has a unique length and breadth and all angles are equal to 90 degrees. If the length and breadth of the rectangle becomes equal, the rectangle will become a square

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Before we learn how to find the dimensions of a rectangle let us look at the properties of a rectangle in detail.

- A rectangle has opposites equal
- A rectangle has every angle equal to 90
^{0} - A rectangle is completely defined with the length and breadth.

Let us look at how to calculate the area and perimeter of a rectangle

Perimeter:The perimeter of a rectangle is the sum of the length of all four sidesof the rectangle. If the length of the rectangle is l and breadth of the rectangle is b, the the perimeter can be calculated as

Perimeter = Sum of the length of the four sides = length + breadth + Length + breadth.

Opposite sides of a rectangle are equal

So perimeter = 2 Length + 2 breadth = 2 (length + breadth) = 2(l+b)

Perimeter has the same units as that of length like millimeter (mm), centimeter (cm) or meter (m)

Area: Area is the measure of the space enclosed by the rectangle. Area of a rectangle is calculated by the formula

Area = length x breadth

A = l x b

The area of the rectangle has the units of square of distance like millimeter squared (mm^{2}) or centimeter squared (cm^{2}) or meter squared (m^{2})

Giventhe area and perimeter of a rectangle, we can find the dimensions of a rectangle. By dimensions we mean th length and the breadth. Let us solvesome examples on this

**Ex 1:** Find the dimensions of a rectangle if the perimeter of a rectangle is 14 cm and the area of the rectangle is 10 cm^{2.}.

**Sol:** Let the length of the rectangle = l

And let the breadth of the rectangle = b

**Step 1:** Perimeter = 2(l+b) = 14

**Step 2:** Area = l x b = 10

**Step 3**: l+b = 14/2 = 7

**Step 4**: lb =10

**Step 5:** Find two numbers whose sum is 7 or product is 10

By trial and error we get numbers as 5 and 2

Length = 5cm and Breadth = 2cm

Dimensions of the rectangle is 5 cm x 2 cm

Note: To solve the set of equation

**Step 6**: I+b = 7 and lb = 10 mathematically, we can use the identity

**Step 7:** (l-b)^{2} = (l+b)^{2} – 4lb

**Step 8:** (l-b)^{2} = 7^{2} – 4 x 10 = 49-40 =9

**Step 9:** l-b = `sqrt(9)` = 3 or –3

If l-b = 3 and l+b = 7 then l = 5 and b="2

If l-b = -3 and l+b = 7 then l = 2 and b="5

Weget the same set of values. By convention length is assumed to be greater than the breadth hence we choose length to be 5 cm and breadth to be 2 cm

**Ex 2:** Find the dimensions of a rectangle whose length is 3cm more than the breadth and the perimeter is 34 cms.

**Sol:** Let the length of the rectangle = l

And let the breadth of the rectangle = b

**Step 1:** Perimeter = 2(l+b) = 34

**Step 2:** l+b = 17

**Step 3:** Given, Length = breadth + 3 or l="b+3

b+3+b = 17

**Step 4:** 2b+3=17

**Step 5:** 2b="17-3" = 10

**Step 6:** b="10/2" = 5

**Step 7**: l="b+3" = 7+3= 10

Length of the rectangle = 10 cm and breadth of the rectangle = 7 cm

Dimensions of the rectangle is 10 cm x 7 cm

Note : Always mention the units appropriately else the answer is incomplete.

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Let us summarize the properties of a rectangle. A rectangle has

- Opposite sides parallel and equal
- All angles equal and each angle is equal to 90
^{0} - Length and breadth as its dimensions
- Twice the sum of length and breadth as the perimeter
- Product of length and breadth as the area

Let us try some exercises on how to find dimensions of a rectangle

**Pro 1:** Find the dimensions of a rectangle if the breadth of a rectangle is half that of the length and the area of the rectangle is 50 cm^{2.}

**Ans:** 10 cm x 5cm

**Pro 2:** Find the dimensions of some rectangles with area of 100 cm^{2. }Whichof these will have the minimum perimeter? (Hint: If the perimeter is minimized for a given are, the rectangle will become a square with equallength and breadth)

**Ans:** 10 cm x 10 cm

**Pro 3:** If the length of rectangle is doubled and the breadth is halved, how will the area and perimeter get affected?

**Ans:** Area will remain the same , perimeter will increase assuming length is greater than the breadth.