Triangle calculator SSA

Please enter two sides and a non-included angle
°


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?

1. Use Law of Cosines

a = 2.25 ; ; b = 2.15 ; ; beta = 65° 18' ; ; ; ; b**2 = a**2 + c**2 - 2ac cos beta ; ; 2.15**2 = 2.25**2 + c**2 -2 * 2.25 * c * cos (65° 18') ; ; ; ; c**2 -1.88c +0.44 =0 ; ; p=1; q=-1.88; 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} = 0.94020092 ± 0.666316563313 ; ; c_{1} = 1.60651748331 ; ; c_{2} = 0.273884356687 ; ; ; ; text{ Factored form: } ; ; (c -1.60651748331) (c -0.273884356687) = 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 = 1.61 ; ;

2. 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 ; ;

3. Semiperimeter of the triangle

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

4. 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 ; ;

5. 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 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 2.15**2+1.61**2-2.25**2 }{ 2 * 2.15 * 1.61 } ) = 71° 56'45" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.25**2+1.61**2-2.15**2 }{ 2 * 2.25 * 1.61 } ) = 65° 18' ; ; gamma = 180° - alpha - beta = 180° - 71° 56'45" - 65° 18' = 42° 45'15" ; ;

7. Inradius

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

8. Circumradius

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

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.15**2+2 * 1.61**2 - 2.25**2 } }{ 2 } = 1.528 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1.61**2+2 * 2.25**2 - 2.15**2 } }{ 2 } = 1.633 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.15**2+2 * 2.25**2 - 1.61**2 } }{ 2 } = 2.049 ; ;







#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 - 2ac cos beta ; ; 2.15**2 = 2.25**2 + c**2 -2 * 2.25 * c * cos (65° 18') ; ; ; ; c**2 -1.88c +0.44 =0 ; ; p=1; q=-1.88; 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} = 0.94020092 ± 0.666316563313 ; ; c_{1} = 1.60651748331 ; ; c_{2} = 0.273884356687 ; ; ; ; text{ Factored form: } ; ; (c -1.60651748331) (c -0.273884356687) = 0 ; ; ; ; c>0 ; ; : Nr. 1


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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 2.15**2+0.27**2-2.25**2 }{ 2 * 2.15 * 0.27 } ) = 108° 3'15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.25**2+0.27**2-2.15**2 }{ 2 * 2.25 * 0.27 } ) = 65° 18' ; ; gamma = 180° - alpha - beta = 180° - 108° 3'15" - 65° 18' = 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 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.15**2+2 * 0.27**2 - 2.25**2 } }{ 2 } = 1.041 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.27**2+2 * 2.25**2 - 2.15**2 } }{ 2 } = 1.189 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.15**2+2 * 2.25**2 - 0.27**2 } }{ 2 } = 2.196 ; ;
Calculate another triangle

Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.