Diagonals of the rhombus

How long are the diagonals e, and f in the diamond if its side is 5 cm long and its area is 20 cm²?

Final Answer:

e = 4.4721 cm
f = 8.9443 cm

a = 5 cm S = 20 cm^{2} S = a h h = S / a = 20 / 5 = 4 cm h = a \cdot sin (A) A = \frac{1 8 0 °}{π} \cdot arcsin (h / a) = \frac{1 8 0 °}{π} \cdot arcsin (4 / 5) ≐ 53.1301^{\circ} e = a \cdot 2 - 2 \cdot cos A = a \cdot 2 - 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 - 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 - 2 \cdot 0.6 = 4.472 = 4.4721 cm

f = a \cdot 2 + 2 \cdot cos A = a \cdot 2 + 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 + 2 \cdot cos 53.130102354156 ° = 5 \cdot 2 + 2 \cdot 0.6 = 8.94427 = 8.9443 cm Verifying Solution: S = E F / 2 (E / 2)^{2} + (F / 2)^{2} = a^{2} (E / 2)^{2} + (2 \cdot S / E / 2)^{2} = a^{2} E^{2} + 4 \cdot S^{2} / E^{2} = 4 a^{2} z = E^{2} z + 4 \cdot S^{2} / z = 4 a^{2} z^{2} + 4 \cdot S^{2} = 4 \cdot a^{2} \cdot z z^{2} + 4 \cdot 2 0^{2} = 4 \cdot 5^{2} \cdot z z^{2} - 100 z + 1600 = 0 a = 1; b = - 100; c = 1600 D = b^{2} - 4 a c = 10 0^{2} - 4 \cdot 1 \cdot 1600 = 3600 D > 0 z_{1, 2} = \frac{- b \pm D}{2 a} = \frac{1 0 0 \pm 3 6 0 0}{2} z_{1, 2} = \frac{1 0 0 \pm 6 0}{2} z_{1, 2} = 50 \pm 30 z_{1} = 80 z_{2} = 20 z = z_{1} = 80 E = z = 80 = 45 cm ≐ 8.9443 cm F = 2 \cdot S / E = 2 \cdot 20 / 8.9443 = 25 cm ≐ 4.4721 cm

Help us improve the example. If you find a mistake, let us know. Thank you!

Tips for related online calculators

Are you looking for help with calculating roots of a quadratic equation?
Do you have a linear equation or system of equations and are looking for a solution? Or do you have a quadratic equation?
See also our right triangle calculator.
See also our trigonometric triangle calculator.
Try conversion angle units angle degrees, minutes, seconds, radians, grads.

algebra

arithmetic

planimetrics

goniometry and trigonometry

Units of physical quantities

Grade of the word problem

We encourage you to watch this tutorial video on this math problem: video1 video2 video3