Step by step
The question
Question: find the area of a rhombus with diagonals 8 and 4.
1 Draw the box
A rhombus has four equal sides.
The marks show them.
The dashed lines join opposite corners.
They are the diagonals.
The diagonals cross at a right angle.
Draw a box around the rhombus.
It is 8 across and 4 up and down: the same as the diagonals.
The box is a rectangle.
Its area is 8 × 4.
8 times 4 is 32.
2 Take half
The diagonals cut the rhombus into 4 triangles.
The 4 corners of the box outside are the same 4 triangles again.
So the rhombus is exactly half of the box: A = ½ × d₁ × d₂.
Careful!
32 is the whole box.
The rhombus is only half of it.
Half of the box: A = ½ × 32.
Half of 32 is 16 square units.
3 Check
Let's check another way.
Cut along the dashed line from top to bottom: two triangles.
Each has base 4 and height half of 8: 4.
Two of them: 2 times half is 1.
We get 16 square units again.
So the area is right!
Let's look back.
First: draw the box.
Then: take half.
Last: check.