Area of Parallelograms and Triangles

The area of a parallelogram is base × perpendicular height.

Cutting the triangle off one end of a parallelogram and sliding it to the other end makes a rectangle with the same base and the same height, so there is no halving. The height must be measured perpendicular to the base, not along the sloping side.

The height of a triangle is the perpendicular distance from the base to the opposite vertex, so it always meets the base (or the base extended) at a right angle.

It does not have to point straight up the page, and it is usually not one of the sides: any of the three sides can be chosen as the base, and the height is then measured perpendicular to that side.

Area of a $\text{parallelogram}=$ base × perpendicular height.

The $4\text{cm}$ is the dashed line marked with a right angle, so it is the perpendicular height.

$\text{Area}=6\times 4=24{\text{cm}}^2$.

Area of a $\text{parallelogram}=$ base × perpendicular height.

Here the base is $8\text{cm}$ and the perpendicular height is $5\text{cm}$. The height is drawn outside the shape, against the extended base, but it is still the perpendicular height.

$\text{Area}=8\times 5=40{\text{cm}}^2$.

Area of a $\text{triangle}=\frac{1}{2}\times$ base × height.

Here the base is $10\text{cm}$ and the perpendicular height is $5\text{cm}$. The height is drawn outside the triangle, against the extended base, but it is still the perpendicular height.

$\text{Area}=\frac{1}{2}\times 10\times 5=25{\text{cm}}^2$.

Area of a $\text{parallelogram}=$ base × perpendicular height.

The diagram gives the base as $6\text{cm}$ and a sloping side as $14\text{cm}$. The $14\text{cm}$ side is not the perpendicular height, because it does not meet the base at a right angle, and no perpendicular height is marked.

Without the perpendicular height the area cannot be determined from this diagram.