How to find the Area and Perimeter of a Rhombus
How to find the Area and Perimeter of a Rhombus

The perimeter of a rhombus ABCD is 40 cm. Find the area of a rhombus if it’s diagonal BD measures 12 cm.

319.5k+ views

Hint: First we will use the formula of the perimeter of the rhombus, that is, $4s$, where $s$ is the side of the rhombus and then find the value of side. Then since diagonals are perpendicular with each other, we will use the Pythagorean theorem ${h^2} = {a^2} + {b^2}$, where $h$ is the hypotenuse, $a$ is the height and $b$ is the base of the right-angled triangle and then multiply the above side by 2 to find the other diagonal. Then use the formula of area of rhombus, $\dfrac{1}{2}{d_1}{d_2}$, where ${d_1}$ and ${d_2}$ are the diagonals in the rhombus ABCD to find the required value.
We are given that the perimeter of the rhombus ABCD is 40 cm.
We know that the perimeter is the sum of the length of all sides of the rhombus, that is, $4s$, where $s$ is the side of the rhombus.
Finding the side of the rhombus using the above formula, we get
$\Rightarrow 40 = 4s$
Dividing the above equation by 4 on both sides, we get
$\Rightarrow \dfrac{{40}}{4} = \dfrac{{4s}}{4} \\ \Rightarrow 10 = s \\ \Rightarrow s = 10{\text{ cm}} \\$
We are also given that the diagonal BD measure 12 cm, we have We know that the diagonal of the rhombus intersect each other at O is 90 degrees.
We will use the Pythagorean theorem ${h^2} = {a^2} + {b^2}$, where $h$ is the hypotenuse, $a$ is the height and $b$ is the base of the right-angled triangle.
Applying the Pythagorean theorem in the triangle COB, we get
$\Rightarrow {10^2} = {x^2} + {6^2} \\ \Rightarrow 100 = {x^2} + 36 \\$
Subtracting the above equation by 36 on each of the sides, we get
$\Rightarrow 100 – 36 = {x^2} + 36 – 36 \\ \Rightarrow 64 = {x^2} \\ \Rightarrow {x^2} = 64 \\$
Taking the square root on both sides in the above equation, we get
$\Rightarrow x = \sqrt {64} \\ \Rightarrow x = \pm 8 \\$
Since the side can never be negative, the negative value of $x$ is discarded.
Multiplying the above side by 2 to find the complete diagonal, we get
$\Rightarrow 2 \cdot 8 = 16{\text{ cm}}$
Using the formula of area of rhombus, $\dfrac{1}{2}{d_1}{d_2}$, where ${d_1}$ and ${d_2}$ are the diagonals in the rhombus ABCD, we get
$\Rightarrow \dfrac{1}{2} \times 16 \times 12 = 96{\text{ c}}{{\text{m}}^2}$
Thus, the area of the given rhombus is 96 cm$^2$.
Note: In solving this question, students should note here that both diagonals bisect each other at 90 degrees. Students should remember that the perimeter of the rhombus is the sum of all the sides of the rhombus and the unit of the perimeter has cm, not the square cm.

We are given that the perimeter of the rhombus ABCD is 40 cm.

We know that the perimeter is the sum of the length of all sides of the rhombus, that is, $4s$, where $s$ is the side of the rhombus.

Finding the side of the rhombus using the above formula, we get

$\Rightarrow 40 = 4s$

Dividing the above equation by 4 on both sides, we get

$\Rightarrow \dfrac{{40}}{4} = \dfrac{{4s}}{4} \\ \Rightarrow 10 = s \\ \Rightarrow s = 10{\text{ cm}} \\$

We are also given that the diagonal BD measure 12 cm, we have We know that the diagonal of the rhombus intersect each other at O is 90 degrees.

We will use the Pythagorean theorem ${h^2} = {a^2} + {b^2}$, where $h$ is the hypotenuse, $a$ is the height and $b$ is the base of the right-angled triangle.

Applying the Pythagorean theorem in the triangle COB, we get

$\Rightarrow {10^2} = {x^2} + {6^2} \\ \Rightarrow 100 = {x^2} + 36 \\$

Subtracting the above equation by 36 on each of the sides, we get

$\Rightarrow 100 – 36 = {x^2} + 36 – 36 \\ \Rightarrow 64 = {x^2} \\ \Rightarrow {x^2} = 64 \\$

Taking the square root on both sides in the above equation, we get

$\Rightarrow x = \sqrt {64} \\ \Rightarrow x = \pm 8 \\$

Since the side can never be negative, the negative value of $x$ is discarded.

Multiplying the above side by 2 to find the complete diagonal, we get

$\Rightarrow 2 \cdot 8 = 16{\text{ cm}}$

Using the formula of area of rhombus, $\dfrac{1}{2}{d_1}{d_2}$, where ${d_1}$ and ${d_2}$ are the diagonals in the rhombus ABCD, we get

$\Rightarrow \dfrac{1}{2} \times 16 \times 12 = 96{\text{ c}}{{\text{m}}^2}$

Thus, the area of the given rhombus is 96 cm$^2$.

Note: In solving this question, students should note here that both diagonals bisect each other at 90 degrees. Students should remember that the perimeter of the rhombus is the sum of all the sides of the rhombus and the unit of the perimeter has cm, not the square cm.

Last updated date: 16th Aug 2023

Total views: 319.5k

Views today: 5.19k

Recently Updated Pages

Aryans at first settled in A Sindh B Gujarat C Kashmir class 11 social science CBSE Draw a well labelled diagram of the human brain class 12 biology CBSE Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred Define spin of electron A Rotation of electrons about class 12 chemistry CBSE What do you observe when a Ice cold water is filled class 12 chemistry CBSE Why did Gujarat witness a sunrise 2 hours after Arunachal Pradesh? Aryans at first settled in A Sindh B Gujarat C Kashmir class 11 social science CBSE Draw a well labelled diagram of the human brain class 12 biology CBSE Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred You are watching: The perimeter of a rhombus ABCD is 40 cm. Find the area of a rhombus if it’s diagonal BD measures 12 cm.. Info created by Bút Chì Xanh selection and synthesis along with other related topics.