Area of a Triangle

Use Area $=\frac{1}{2}\times \text{base}\times \text{height}$.

$10\times 5=50$, and half of $50$ is $25$.

The area is $25$ cm². The halving step is what makes the triangle exactly half of the $10$ cm by $5$ cm rectangle around it.

Use Area $=\frac{1}{2}\times \text{base}\times \text{height}$.

$12\times 4=48$, and half of $48$ is $24$.

The area is $24$ cm², exactly half of the $12$ cm by $4$ cm rectangle around the triangle.

Use Area $=\frac{1}{2}\times \text{base}\times \text{height}$.

The height must be the perpendicular distance from the base to the opposite vertex: that is the $5$ cm line drawn down to the base and marked with a right angle.

$12\times 5=60$, and half of $60$ is $30$. The area is $30$ cm².

In a right-angled triangle the two sides that meet at the right angle are the base and the height, so use $8$ cm and $6$ cm. The $10$ cm hypotenuse is a slanted side and is never used in the area formula.

Area $=\frac{1}{2}\times 8\times 6=\frac{1}{2}\times 48=24$.

The area is $24$ cm².

The height must be perpendicular to the base, but it does not have to sit inside the triangle. Here it is drawn outside, meeting the extended base. It is still a valid height.

Area $=\frac{1}{2}\times 15\times 8=\frac{1}{2}\times 120=60$.

The area is $60$ cm².

The two perpendicular sides can be used as base and height, so the area is $\frac{1}{2}\times 12\times 16=\frac{1}{2}\times 192=96$ cm².

The area is the same whichever side you call the base, so with the $20\text{cm}$ side as the base: $\frac{1}{2}\times 20\times h=96$.

That gives $10h=96$, so $h=96÷10=9.6$ cm.