Worked examples
The question
Question: find the area of a rhombus with diagonals 10 and 6.
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 10 across and 6 up and down: the same as the diagonals.
The box is a rectangle.
Its area is 10 × 6.
box = 10 × 6
10 times 6 is 60.
box = 60
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!
60 is the whole box.
The rhombus is only half of it.
Careful: not A = 10 × 6 = 60 ✗
Half of the box: A = ½ × 60.
A = 12 × 60
Half of 60 is 30 square units.
A = 30
3 Check
Let's check another way.
Cut along the dashed line from top to bottom: two triangles.
Each has base 6 and height half of 10: 5.
Two of them: 2 times half is 1.
2 × 12 × 6 × 5 = 6 × 5 = 30
We get 30 square units again.
So the area is right!
Let's look back.
First: draw the box.
Then: take half.
Last: check.
▶ Watch it solved
The question
Question: find the area of a kite with diagonals 12 cm and 8 cm.
1 Draw the box
A kite has two pairs of equal sides, side by side.
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 kite.
It is 12 cm across and 8 cm up and down: the same as the diagonals.
The box is a rectangle.
Its area is 12 × 8.
box = 12 × 8
12 times 8 is 96.
box = 96 cm2
2 Take half
The diagonals cut the kite into 4 triangles.
The 4 corners of the box outside are the same 4 triangles again.
So the kite is exactly half of the box: A = ½ × d₁ × d₂.
Careful!
96 is the whole box.
The kite is only half of it.
Careful: not A = 12 × 8 = 96 ✗
Half of the box: A = ½ × 96.
A = 12 × 96
Half of 96 is 48 cm².
A = 48 cm2
3 Check
Let's check another way.
Cut along the dashed line from top to bottom: two triangles.
Each has base 8 cm and height half of 12: 6.
Two of them: 2 times half is 1.
2 × 12 × 8 × 6 = 8 × 6 = 48
We get 48 cm² again.
So the area is right!
Let's look back.
First: draw the box.
Then: take half.
Last: check.
▶ Watch it solved
The question
Question: find the other diagonal d of a rhombus with area 24 square units and one diagonal 8.
1 Write the rule
A rhombus is half of the box around it.
The box is the two diagonals.
So A = ½ × d₁ × d₂.
Now put the numbers in: A = 24 and d₁ = 8.
12 × 8 × d = 24
Careful!
Do not forget the half.
The rhombus is only half of the box.
Careful: not d = 24 ÷ 8 ✗
2 Half of the diagonal
Half of 8 is 4.
82 = 4
So 4 × d = 24.
4 × d = 24
3 Remove the 4
4d means 4 times d.
We want just d.
To remove times 4, we divide by 4.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
4d4 = 244
4 divided by 4 is 1.
So just d is left.
24 divided by 4 is 6.
244 = 6
So d = 6!
d = 6
So the other diagonal is 6.
4 Check
Let's check: half of 8 times 6.
12 × 8 × 6 = 4 × 6 = 24
We get 24 square units, the area.
So d = 6 is right!
Let's look back.
First: write the rule.
Then: half of the diagonal.
Then: remove the 4.
Last: check.
▶ Watch it solved
