# Area and two angles

Calculate the size of all sides and internal angles of a triangle ABC, if it is given by area S = 501.9; and two internal angles α = 15°28' and β = 45°.

Correct result:

C =  119.533 °
a =  20.859
b =  55.309
c =  68.056

#### Solution:

$A=15+\dfrac{ 28 }{ 60 }=\dfrac{ 232 }{ 15 } \doteq 15.4667 \ ^\circ \ \\ B=45 \ ^\circ \ \\ C=180-A-B=180-15.4667-45=\dfrac{ 1793 }{ 15 }=119.533 ^\circ =119^\circ 32'$

Try calculation via our triangle calculator.

$S=501.9 \ \\ a=k \cdot \ u \ \\ b=k \cdot \ v \ \\ c=k \cdot \ w \ \\ \ \\ w=1 \ \\ u:w=\sin A: \sin C \ \\ u=w \cdot \ \sin(A) / \sin(C)=1 \cdot \ \sin(15.4667^\circ ) / \sin(119.5333^\circ ) \doteq 0.3065 \ \\ \ \\ v:w=\sin B: \sin C \ \\ v=w \cdot \ \sin(B) / \sin(C)=1 \cdot \ \sin(45^\circ ) / \sin(119.5333^\circ ) \doteq 0.8127 \ \\ \ \\ s=(u+v+w)/2=(0.3065+0.8127+1)/2=1.0596 \ \\ T=\sqrt{ s \cdot \ (s-u) \cdot \ (s-v) \cdot \ (s-w) }=\sqrt{ 1.0596 \cdot \ (1.0596-0.3065) \cdot \ (1.0596-0.8127) \cdot \ (1.0596-1) } \doteq 0.1084 \ \\ k=\sqrt{ S/T }=\sqrt{ 501.9/0.1084 } \doteq 68.0562 \ \\ a=k \cdot \ u=68.0562 \cdot \ 0.3065=20.859$
$b=k \cdot \ v=68.0562 \cdot \ 0.8127=55.309$
$c=k \cdot \ w=68.0562 \cdot \ 1=68.056$

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