SME
Grade 10 Math · Problem type
Rhombus from its diagonals
Use this for “A rhombus has diagonals 12 and 16. Find its side and its area.”
Diagram
A rhombus: four equal sides, opposite angles equal; the diagonals cut each other in half at right angles.
Diagonal AC: 12
Diagonal BD: 16
Area: 96
This is an example. Type your own values to start a new problem.
K
=
×
2
AO
Half of AC (AO)
worked out (not typed)
BO
Half of BD (BO)
worked out (not typed)
s
Side
worked out (not typed)
Formulas
AO
=
p
2
6
=
12
2
BO
=
q
2
8
=
16
2
s
=
√
AO² + BO²
10
=
√
6
2
+
8
2
K
=
p
×
q
2
96
=
12
×
16
2
Assumptions
•
The diagonals of a rhombus bisect each other at right angles.
•
Half of each diagonal makes the legs of a right triangle whose hypotenuse is a side.
•
The area is half the product of the diagonals: K = pq ÷ 2.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
p = 12, q = 16
FIND
half of AC (AO) (AO), half of BD (BO) (BO), side (s), area (K)
Step 1 · Find half of AC (AO): AO
Use AO = p ÷ 2
The diagonals bisect each other.
AO
=
12
2
AO = 6
Step 2 · Find half of BD (BO): BO
Use BO = q ÷ 2
The diagonals bisect each other.
BO
=
16
2
BO = 8
Step 3 · Find side (s)
Use s = √(AO² + BO²)
The half diagonals meet at a right angle: the side is the hypotenuse.
s
=
√
6
2
+
8
2
s
=
√
36 + 64
s
=
√
100
s
=
10
Step 4 · Find area (K)
Use K = p × q ÷ 2
The
diagonals
cut
the
rhombus
into
four
right
triangles:
half
of
p
×
q
.
K
=
12
×
16
2
K
=
192
2
K
=
96
Check
Put the numbers back in. Both sides should match.
6
=
12
2
✓
8
=
16
2
✓
10
=
√
6
2
+
8
2
✓
96
=
12
×
16
2
✓
Topic check
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