Here is a triangle on the board. In a moment we are going to see where it comes from.

Watch me fold a paper rectangle from one corner to the opposite corner. It falls into two matching triangles that sit exactly on top of each other. What fraction of the whole rectangle is each triangle?

Three triangles are on the board, each drawn inside the rectangle that surrounds it. For each one, look at the rectangle's area first, then check the triangle covers exactly half. On the last one the tip leans over — say what the base is and what the height is before we work it out.

Now we work through some triangles on the board together. For each one we name the base, name the perpendicular height, find the rectangle's area, then take half.

• base 8 cm, height 3 cm
• base 12 cm, height 5 cm
• one where the tip leans past the base — the slanted side is a trap

Then we turn the rule around and work backwards. A triangle has an area of 20 cm² and a base of 8 cm. What is its height?

The triangle is half a rectangle, so first we double the area to get the whole rectangle back: 2 × 20 = 40 cm². The rectangle is base × height, so we divide by the base: 40 ÷ 8 = 5 cm. That is the height. We will use exactly this move again in the challenge.

Today's challenge triangles, in order:

• base 10 cm, height 6 cm
• base 7 cm, height 8 cm
• a leaning triangle where the slanted side is longer than the true height — use the straight-up height
• reverse puzzle: the area is 30 cm² and the base is 12 cm. Find the height the same way we just practised — double the area to get the whole rectangle back, then divide by the base.