How to find the area of a rhombus from its vertices

Expert intro: A lot of students try to force the Pythagorean theorem onto every quadrilateral. For a rhombus, the diagonals are often the cleanest path, especially when the vertices are already given.

Detailed walkthrough

We have the vertices

$G(1,-9),\ H(7,-2),\ I(9,7),\ J(3,0).$

Since the figure is a rhombus, its area can be found from the diagonals:

$A=\frac{1}{2}d_1d_2.$

1) Find the diagonals

The diagonals are $GI$ and $HJ$.

Diagonal $GI$

=\sqrt{8^2+16^2} =\sqrt{64+256} =\sqrt{320}=8\sqrt{5}.$$ #### Diagonal $HJ$ $$HJ=\sqrt{(3-7)^2+(0-(-2))^2} =\sqrt{(-4)^2+2^2} =\sqrt{16+4} =\sqrt{20}=2\sqrt{5}.$$ ### 2) Compute the area $$A=\frac{1}{2}(8\sqrt{5})(2\sqrt{5}) =\frac{1}{2}(16\cdot 5) =40.$$ ### Answer $$\boxed{40}$$ So the area of the rhombus is 40 square units. ### 💡 Pitfall guide A common error is to try to find the area by splitting the shape into triangles without checking the diagonals first. Another trap is using side lengths in the rhombus formula for area, which only works if you know the included angle. Here the diagonals are much simpler and less error-prone. ### 🔄 Real-world variant If the vertices had been listed in a different order, the area would still be the same as long as the same four points formed the same rhombus. If one coordinate were changed, you would need to verify the quadrilateral is still a rhombus before using the diagonal formula. ### 🔍 Related terms diagonals of a rhombus, coordinate geometry, area formula