# What is the relationship between the area of a rectangle and its side?

Mensuration is the branch of mathematics that deals with the measurement of geometric figures and their parameters like volume, length, shape, surface area, lateral surface area, etc. the shape can exist in 2-Dimension or 3-Dimension form. Here basic terms are studied and the formula to calculate the parameters of various geometric figures and find the relationship between the area of a rectangle and its side.

### Basic terms used in mensuration

Basic terms used in mensuration are Area, Perimeter, Volume, Curved surface area, Lateral surface area, Total surface area. Let’s learn about these parameters in detail,

**Area**

The area is defined as the surface which is covered by the closed shape is known as its area. Generally, the area is denoted by A, and the unit of area is meter^{2 }(m^{2}) or cm^{2}.

**Perimeter**

The perimeter of the figure is defined as the measure of the continuous line along the boundary of the given figure is called its Perimeter. Generally, it is denoted by P, and the unit of perimeter is meter (m) or centimeter (cm).

**Volume**

The volume is defined as space occupied by a 3-Dimensional shape is called a Volume. it is denoted by V, its unit is m^{3} or cm^{3}.

**Curved Surface Area**

Curved surface area is defined as, if there is any curved surface in a figure, then the total area is called a Curved Surface area, denoted by CSA and unit is m^{2} or cm^{2} Example: Sphere has a curved surface so its total area is called curved surface area.

**Lateral Surface area**

Lateral surface area is define as the total area of all the lateral surfaces that surrounds the given figure (cylinder, hemisphere, etc) is called the Lateral Surface area. It is denoted by LSA and its unit is m^{2} or cm^{2}.

**Total Surface Area**

Total surface area is define as the sum of all the curved and lateral surface areas is called the Total Surface area. It is denoted by TSA and its unit is m^{2} or cm^{2}.

TSA = LSA + CSA

### Mensuration formula for some common shapes

If l is the length, b is the breadth, h is the height of the shape then the Mensuration formula for some common shapes are as,

**Square**

Area of square = l x b

Perimeter of square = perimeter is the sum of all sides in square = l + l + l + l = 4 x l

**Rectangle**

Area of Rectangle = l x b

Perimeter of rectangle = perimeter is the sum of all sides in rectangle = l + b + l + b = 2(l + b).

**Circle**

Area of circle = πr^{2}

The perimeter of circle = 2πr

**Triangle**

Triangle has mainly three different types. They are equilateral triangles in which all sides and all angles are equal, isosceles triangle, has 2 sides and their angles are equal and scalene triangle, has all three sides and all three angles unequal.

Type of triangle | Area | |

Scalene triangle (all unequal sides) | √[s(s−l)(s−b)(s−h)], Where, s = (l+b+h)/2 | l + b + h |

Isosceles Triangle (two side are equal) | 1/2 x b x h | 2 x l + b |

Equilateral triangle (all side are equal) | (√3/4) × l^{2} | 3 x l |

### What is the relationship between the area of a rectangle and its side?

Let’s derive the formula of the area of a rectangle with the help of congruence . take a rectangle of ABCD. Now draw a diagonal AD in the rectangle ABCD. the diagonal AD divides the rectangle ABCD into two equal parts or congruent triangles. Then, the area of the rectangle is the sum of the area of these two triangles.

Area of Rectangle ABCD = Area of Triangle ACD + Area of Triangle ADB

= 2 × Area of Triangle ACD

= 2 × (1/2 × Base (CD) × Height (AC))

= CD × AC

= Length × Breadth

This is the required relationship between the area of a rectangle and its side.

### Sample Problems

**Question 1: Calculate the area of the rectangle if the length is 12 and its breadth is 5.**

**Solution:**

Given: Length of rectangle = 12units

breadth of rectangle = 5 units

Area of rectangle = length × breadth

= 12 × 5

= 60 sq unit.

**Question 2: Find the perimeter of rectangle if one of its side is 6cm and area is 48cm ^{2}.**

**Solution:**

Given: length of rectangle l = 6 cm

Let breadth of rectangle = b cm

Area of rectangle = l × b

48 = 6 × b

b = 48/6

b = 8cm

Perimeter of rectangle = 2 × (l + b)

= 2 × (6 + b)

= 2 × (6 + 8)

= 2 × 14 = 28cm

**Question 3: What will be the area of an equilateral triangle if its side is 16 cm?**

**Solution:**

Area of equilateral triangle (A) = (√3/4) × side

^{2}A = (√3/4) × 16 × 16

A = √3 × 16 × 4

A = 64√3 cm

^{2}

**Question 4: The area of the rectangle is 6912 cm ^{2} and the ratio of the length and the breadth of a rectangle is 4 : 3, Find the original length and breadth of the rectangle.**

**Solution:**

Let the length and the breadth of the rectangle be 4b cm and 3b respectively.

Area of rectangle = l × b

(4b)(3b) = 6912

12b

^{2}= 6912b

^{2}= 576 = 4 × 144 = 22 × 122b = 2 × 12 = 24

length of rectangle = 4 × b = 4 × 24 = 96cm

Breadth of rectangle = 3 × 24 = 72cm

**Question 5: Calculate the area of the isosceles triangle if one side is 4cm and the height is 12cm.**

**Solution:**

Area of isosceles triangle = 1/2 × b × h

= 1/2 × 4 × 12

= 24 cm

^{2}

**Question 6: In the rectangle if its length is thrice to its breadth. If its area is 867 m ^{2}, then what is the breadth of the rectangle?**

**Solution:**

Let, length of rectangle be l and breadth be b

According to question,

l = 3 × b ⇢ (i)

Area of rectangle = 867

l x b = 867

3 x b x b = 867 from eq (i)

3 x b

^{2}= 867b

^{2 }= 867/3b = √289 = 17m

**Question 7: Calculate the side of the square if its area is 64cm ^{2}**

**Solution:**

Area of square = 64cm

^{2}s

^{2}= 64s = 8 cm

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