Question

find the area of a rhombus having side 13 cm and one of whose diagonal is 10 cm​

Answer 1

$\underline{\pmb{\sf{Understanding\; the\; question\;:}}}$

Here, the concept of area of quadrilateral has been used. We're given with the side and one of the diagonal of the rhombus. We've been asked to calculate it's area. In order to calculate the area of rhombus we firstly need to find out it's length of another diagonal by using Pythagoras theorem and then by using area of rhombus, we'll calculate the answer.

$\underline{\pmb{\sf{Formula\; used\;:}}}$

$\star\; \underline{\boxed{\pmb{\sf{{H}^{2}={P}^{2}+{B}^{2}}}}}$

$\star\;\underline{\boxed{\pmb{\frak{Area\; of\; rhombus=\dfrac{1}{2}\times {d}_{1}\times {d}_{2}}}}}$

• H is Hypotenuse of triangle
• P is Perpendicular of triangle
• B is Base of triangle  
• d1 is diagonal 1 of rhombus
• d2 is another diagonal of rhombus.

$\underline{\pmb{\sf{Figure\;:}}}$

$\setlength{\unitlength}{1 cm}\begin{picture}(20,15)\thicklines\qbezier(1,1)(1,1)(6,1)\qbezier(1,1)(1,1)(1.6,4)\qbezier(1.6,4)(1.6,4)(6.6,4)\qbezier(6,1)(6,1)(6.6,4)\qbezier(6.6,4)(6.6,4)(1,1)\qbezier(1.6,4)(1.6,4)(6,1)\put(0.7,0.5){\sf A}\put(6,0.5){\sf B}\put(1.4,4.3){\sf D}\put(6.6,4.3){\sf C}\end{picture}$

Given:

⋆ Side of rhombus = s = 13 cm

⋆ One of the diagonal of rhombus = $\sf{{d}_{1}}$ = 10 cm

Now, let's calculate the diagonal of rhombus by Pythagoras theorem. Pythagoras theorem states, the square of hypotenuse is equal to the sum of square of Perpendicular and square of base of triangle.  

$\underline{\pmb{\sf{Solution\;:}}}$

We know, the diagonals of a rhombus bisects each other at right angle so taking ∆AOD. Here,

• AD = 13 cm
• AO = 10/2 = 5 cm

$:\implies\quad\sf{{AD}^{2}={AO}^{2}+{OD}^{2}}$

$:\implies\quad\sf{{13}^{2}={5}^{2}+{OD}^{2}}$

$:\implies\quad\sf{169-25={OD}^{2}}$

$:\implies\quad\sf{OD=\sqrt{144}}$

$:\implies\quad\sf{OD=12\; cm}$

Henceforth, the diagonal BD measures 24 cm.

Now, let's calculate the area of rhombus by its formula.

$:\implies\quad\sf{Area\; of\; rhombus=\dfrac{1}{2}\times {d}_{1}\times {d}_{2}}$

$:\implies\quad\sf{Area\; of\; rhombus=\dfrac{1}{2}\times 10\times 24}$

$:\implies\quad\sf{Area\; of\; rhombus=\dfrac{1}{2}\times 240}$

$:\implies\quad\sf{Area\; of\; rhombus=\cancel{\dfrac{240}{2}}}$

$:\implies\quad\underline{\boxed{\sf{Area\; of\; rhombus=120\; {cm}^{2}}}}$

$\underline{\pmb{\sf{Required\; answer\;:}}}$

$\qquad\qquad\leadsto$ Therefore, the area of rhombus is 120 cm².

$\begin{gathered}\rule{230px}{.2ex}\\\end{gathered}$

Additional information:

• Area of square = side × side
• Area of triangle = 1/2 × b× h
• Area of trapezium = 1/2 (a + b) × h
• Area of parallelogram = b × h
• Area of circle = π × r²
• Perimeter of Rectangle = 2 (l + b)
• Perimeter of Square = 4 × Side