Triangle calculator SSA

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Triangle has two solutions with side c=1.60765174794 and with side c=0.27438843527

#1 Acute scalene triangle.

Sides: a = 2.25   b = 2.15   c = 1.60765174794

Area: T = 1.64219760508
Perimeter: p = 6.00765174794
Semiperimeter: s = 3.00332587397

Angle ∠ A = α = 71.94659111748° = 71°56'45″ = 1.25656930333 rad
Angle ∠ B = β = 65.3° = 65°18' = 1.14397000016 rad
Angle ∠ C = γ = 42.75440888252° = 42°45'15″ = 0.74661996187 rad

Height: ha = 1.46595342673
Height: hb = 1.52774195821
Height: hc = 2.04441433994

Median: ma = 1.52884221294
Median: mb = 1.63328117484
Median: mc = 2.04987253103

Inradius: r = 0.54767314651
Circumradius: R = 1.18332584743

Vertex coordinates: A[1.60765174794; 0] B[0; 0] C[0.94402009161; 2.04441433994]
Centroid: CG[0.84989061318; 0.68113811331]
Coordinates of the circumscribed circle: U[0.80332587397; 0.8698836011]
Coordinates of the inscribed circle: I[0.85332587397; 0.54767314651]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.0544088825° = 108°3'15″ = 1.25656930333 rad
∠ B' = β' = 114.7° = 114°42' = 1.14397000016 rad
∠ C' = γ' = 137.2465911175° = 137°14'45″ = 0.74661996187 rad




How did we calculate this triangle?

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 2.25 ; ; b = 2.15 ; ; c = 1.61 ; ;

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 2.25+2.15+1.61 = 6.01 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6.01 }{ 2 } = 3 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3 * (3-2.25)(3-2.15)(3-1.61) } ; ; T = sqrt{ 2.7 } = 1.64 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.64 }{ 2.25 } = 1.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.64 }{ 2.15 } = 1.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.64 }{ 1.61 } = 2.04 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 2.25**2-2.15**2-1.61**2 }{ 2 * 2.15 * 1.61 } ) = 71° 56'45" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.15**2-2.25**2-1.61**2 }{ 2 * 2.25 * 1.61 } ) = 65° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.61**2-2.25**2-2.15**2 }{ 2 * 2.15 * 2.25 } ) = 42° 45'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.64 }{ 3 } = 0.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.25 }{ 2 * sin 71° 56'45" } = 1.18 ; ;





#2 Obtuse scalene triangle.

Sides: a = 2.25   b = 2.15   c = 0.27438843527

Area: T = 0.28799294459
Perimeter: p = 4.67438843527
Semiperimeter: s = 2.33769421764

Angle ∠ A = α = 108.0544088825° = 108°3'15″ = 1.88658996202 rad
Angle ∠ B = β = 65.3° = 65°18' = 1.14397000016 rad
Angle ∠ C = γ = 6.64659111748° = 6°38'45″ = 0.11659930318 rad

Height: ha = 0.24988261742
Height: hb = 0.26603994846
Height: hc = 2.04441433994

Median: ma = 1.04107359508
Median: mb = 1.18987520008
Median: mc = 2.19663029937

Inradius: r = 0.12197844982
Circumradius: R = 1.18332584743

Vertex coordinates: A[0.27438843527; 0] B[0; 0] C[0.94402009161; 2.04441433994]
Centroid: CG[0.40546950896; 0.68113811331]
Coordinates of the circumscribed circle: U[0.13769421764; 1.17553073884]
Coordinates of the inscribed circle: I[0.18769421764; 0.12197844982]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 71.94659111748° = 71°56'45″ = 1.88658996202 rad
∠ B' = β' = 114.7° = 114°42' = 1.14397000016 rad
∠ C' = γ' = 173.3544088825° = 173°21'15″ = 0.11659930318 rad

Calculate another triangle

How did we calculate this triangle?

1. Use Law of Cosines

a = 2.25 ; ; b = 2.15 ; ; beta = 65° 18' ; ; ; ; b**2 = a**2 + c**2 - 2bc cos( beta ) ; ; 2.15**2 = 2.25**2 + c**2 -2 * 2.15 * c * cos (65° 18') ; ; ; ; c**2 -1.88c +0.44 =0 ; ; p=1; q=-1.8804018321; r=0.44 ; ; D = q**2 - 4pr = 1.88**2 - 4 * 1 * 0.44 = 1.77591105018 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1.88 ± sqrt{ 1.78 } }{ 2 } ; ; c_{1,2} = 0.940200916052 ± 0.666316563313 ; ;
c_{1} = 1.60651747937 ; ; c_{2} = 0.273884352739 ; ; ; ; (c -1.60651747937) (c -0.273884352739) = 0 ; ; ; ; c>0 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 2.25 ; ; b = 2.15 ; ; c = 0.27 ; ;

2. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 2.25+2.15+0.27 = 4.67 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 4.67 }{ 2 } = 2.34 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.34 * (2.34-2.25)(2.34-2.15)(2.34-0.27) } ; ; T = sqrt{ 0.08 } = 0.28 ; ;

5. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.28 }{ 2.25 } = 0.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.28 }{ 2.15 } = 0.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.28 }{ 0.27 } = 2.04 ; ;

6. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 2.25**2-2.15**2-0.27**2 }{ 2 * 2.15 * 0.27 } ) = 108° 3'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2.15**2-2.25**2-0.27**2 }{ 2 * 2.25 * 0.27 } ) = 65° 18' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.27**2-2.25**2-2.15**2 }{ 2 * 2.15 * 2.25 } ) = 6° 38'45" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.28 }{ 2.34 } = 0.12 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.25 }{ 2 * sin 108° 3'15" } = 1.18 ; ;




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