Question

HELLO MADAM, my doubt is, a rhombus has one of its sides as 10cm and one angle as 40 degree. Find the area of the rhombus

Answer 1

Answer on Question#39373 - Math - Geometry

A rhombus has one of its sides as $10 \, \text{cm}$ and one angle as $40 \, \text{degree}$ . Find the area of the rhombus.

Solution:

If a quadrilateral is a rhombus, then:

- all four sides are equal

- the diagonals bisect each other at right angles

- the diagonals bisect each vertex angles

So $\Delta AOB, \Delta COB, \Delta AOD$ and $\Delta COD$ are right triangles and $\Delta AOB = \Delta COB = \Delta AOD = \Delta COD$

The area of the rhombus is:

$$A _ {A B C D} = 4 A _ {\Delta A O B}$$ $$A _ {\Delta A O B} = \frac {1}{2} A O \cdot B O$$ $$A O = A B \cdot \cos \angle O A B$$ $$B O = A B \cdot \sin \angle O A B$$ $$\angle O A B = \angle O A D = \frac {1}{2} \angle B A D = \frac {1}{2} \cdot 4 0 {}^ {\circ} = 2 0 {}^ {\circ}$$ $$A O = 1 0 \cdot \cos 2 0 {}^ {\circ} = 9. 4 0$$ $$B O = 1 0 \cdot \sin 2 0 {}^ {\circ} = 3. 4 2$$ $$A _ {A B C D} = 4 \cdot \frac {1}{2} \cdot 9. 4 0 \cdot 3. 4 2 = 6 4. 3 0 \mathrm {c m} ^ {2}$$

Answer: The area of the rhombus is $64.30 \, \text{cm}^2$