SME
Grade 10 Math · Problem type
Area from two sides and the angle between them
Use this for “Two sides of a triangle are 8 and 11 with a 40° angle between them. Find its area.”
Diagram
Two sides and the angle between them fix the triangle: the law of cosines gives the third side.
Law of sines: a/sin A = b/sin B = c/sin C ≈ 11.02
This is an example. Type your own values to start a new problem.
K
=
1
2
×
a
×
b
×
sin(
C
°)
Formulas
K
=
a
×
b
×
sin(
C
°)
2
28.2827
=
8
×
11
×
sin(40°)
2
Assumptions
•
The height to side a is b × sin C, so the area is ½ × a × b × sin C.
•
C must be the angle between the two sides.
Step-by-step
WE KNOW
a = 8, b = 11, C = 40°
FIND
area (K)
Step 1 · Find area (K)
Use K = a × b × sin(C°) ÷ 2
Half
the
base
times
the
height,
and
the
height
is
b
×
sin
C
.
K
=
8
×
11
×
sin(40°)
2
K
=
8
×
11
×
0.6428
2
K
=
88
×
0.6428
2
K
=
56.5664
2
K
=
28.2827
Check
Put the numbers back in. Both sides should match.
28.2827
≈
8
×
11
×
sin(40°)
2
✓
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The law of sines and the law of cosines
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