A parallelogram has adjacent sides of 18 cm and 10 cm, and the angle between them is 60°. Find the exact area of the parallelogram and the height corresponding to the longer side.
90√3 cm2, 5√3 cm
The area of a parallelogram with adjacent sides \(a\) and \(b\) and included angle \(\theta\) is \(A = ab\sin\theta\). This comes from splitting the figure into two triangles, each of area \(\tfrac12 ab\sin\theta\).
Here \(a = 18\ \text{cm}\), \(b = 10\ \text{cm}\) and \(\theta = 60\degree\), with \(\sin 60\degree = \dfrac{\sqrt{3}}{2}\).
\(A = 18 \times 10 \times \dfrac{\sqrt{3}}{2} = 180 \times \dfrac{\sqrt{3}}{2} = 90\sqrt{3}\ \text{cm}^2\).
The height on any side is found from area = base × height, so \(h = \dfrac{A}{\text{base}}\). The longer side is 18 cm, so we divide by 18.
\(h = \dfrac{90\sqrt{3}}{18} = 5\sqrt{3}\ \text{cm}\).
Key idea: use \(ab\sin\theta\) for area, then treat the parallelogram as base × height to recover the height on the longer side.
Hence the area is \(90\sqrt{3}\ \text{cm}^2\) and the height on the longer side is \(5\sqrt{3}\ \text{cm}\), i.e. 90√3 cm2, 5√3 cm.
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