Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle α.

Triangle has two solutions: a=8.3; b=10.1; c=2.5944440031 and a=8.3; b=10.1; c=12.76657604741.

#1 Obtuse scalene triangle.

Sides: a = 8.3   b = 10.1   c = 2.5944440031

Area: T = 8.50990177739
Perimeter: p = 20.9944440031
Semiperimeter: s = 10.49772200155

Angle ∠ A = α = 40.5° = 40°30' = 0.70768583471 rad
Angle ∠ B = β = 127.7877173258° = 127°47'14″ = 2.23303069152 rad
Angle ∠ C = γ = 11.7132826742° = 11°42'46″ = 0.20444273914 rad

Height: ha = 2.05503657286
Height: hb = 1.68549540146
Height: hc = 6.55994252881

Median: ma = 6.09549207983
Median: mb = 3.50882844151
Median: mc = 9.15224434023

Inradius: r = 0.81105972592
Circumradius: R = 6.39900415294

Vertex coordinates: A[2.5944440031; 0] B[0; 0] C[-5.08656602216; 6.55994252881]
Centroid: CG[-0.83304067302; 2.1866475096]
Coordinates of the circumscribed circle: U[1.29772200155; 6.2576984176]
Coordinates of the inscribed circle: I[0.39772200155; 0.81105972592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5° = 139°30' = 0.70768583471 rad
∠ B' = β' = 52.2132826742° = 52°12'46″ = 2.23303069152 rad
∠ C' = γ' = 168.2877173258° = 168°17'14″ = 0.20444273914 rad


How did we calculate this triangle?

1. Input data entered: side a, b and angle α.

a = 8.3 ; ; b = 10.1 ; ; alpha = 40.5° ; ;

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 8.3**2 = 10.1**2 + c**2 - 2 * 10.1 * c * cos 40° 30' ; ; ; ; ; ; c**2 -15.36c +33.12 =0 ; ; p=1; q=-15.36; r=33.12 ; ; D = q**2 - 4pr = 15.36**2 - 4 * 1 * 33.12 = 103.455759558 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.36 ± sqrt{ 103.46 } }{ 2 } ; ;
c_{1,2} = 7.68010025 ± 5.08566022158 ; ; c_{1} = 12.7657604741 ; ; c_{2} = 2.59444003098 ; ; ; ; text{ Factored form: } ; ; (c -12.7657604741) (c -2.59444003098) = 0 ; ; ; ; c > 0 ; ; ; ; c = 12.766 ; ;
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 = 8.3 ; ; b = 10.1 ; ; c = 2.59 ; ;

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

p = a+b+c = 8.3+10.1+2.59 = 20.99 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.99 }{ 2 } = 10.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.5 * (10.5-8.3)(10.5-10.1)(10.5-2.59) } ; ; T = sqrt{ 72.4 } = 8.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.51 }{ 8.3 } = 2.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.51 }{ 10.1 } = 1.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.51 }{ 2.59 } = 6.56 ; ;

7. 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{ 10.1**2+2.59**2-8.3**2 }{ 2 * 10.1 * 2.59 } ) = 40° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+2.59**2-10.1**2 }{ 2 * 8.3 * 2.59 } ) = 127° 47'14" ; ;
 gamma = 180° - alpha - beta = 180° - 40° 30' - 127° 47'14" = 11° 42'46" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.51 }{ 10.5 } = 0.81 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 40° 30' } = 6.39 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.1**2+2 * 2.59**2 - 8.3**2 } }{ 2 } = 6.095 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.59**2+2 * 8.3**2 - 10.1**2 } }{ 2 } = 3.508 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.1**2+2 * 8.3**2 - 2.59**2 } }{ 2 } = 9.152 ; ;



#2 Acute scalene triangle.

Sides: a = 8.3   b = 10.1   c = 12.76657604741

Area: T = 41.86880260382
Perimeter: p = 31.16657604741
Semiperimeter: s = 15.58328802371

Angle ∠ A = α = 40.5° = 40°30' = 0.70768583471 rad
Angle ∠ B = β = 52.2132826742° = 52°12'46″ = 0.91112857384 rad
Angle ∠ C = γ = 87.2877173258° = 87°17'14″ = 1.52334485681 rad

Height: ha = 10.08986809731
Height: hb = 8.29106982254
Height: hc = 6.55994252881

Median: ma = 10.73661455021
Median: mb = 9.50991966139
Median: mc = 6.68664669205

Inradius: r = 2.68767963689
Circumradius: R = 6.39900415294

Vertex coordinates: A[12.76657604741; 0] B[0; 0] C[5.08656602216; 6.55994252881]
Centroid: CG[5.95504735652; 2.1866475096]
Coordinates of the circumscribed circle: U[6.38328802371; 0.30224411122]
Coordinates of the inscribed circle: I[5.48328802371; 2.68767963689]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5° = 139°30' = 0.70768583471 rad
∠ B' = β' = 127.7877173258° = 127°47'14″ = 0.91112857384 rad
∠ C' = γ' = 92.7132826742° = 92°42'46″ = 1.52334485681 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle α.

a = 8.3 ; ; b = 10.1 ; ; alpha = 40.5° ; ; : Nr. 1

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 8.3**2 = 10.1**2 + c**2 - 2 * 10.1 * c * cos 40° 30' ; ; ; ; ; ; c**2 -15.36c +33.12 =0 ; ; p=1; q=-15.36; r=33.12 ; ; D = q**2 - 4pr = 15.36**2 - 4 * 1 * 33.12 = 103.455759558 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.36 ± sqrt{ 103.46 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 7.68010025 ± 5.08566022158 ; ; c_{1} = 12.7657604741 ; ; c_{2} = 2.59444003098 ; ; ; ; text{ Factored form: } ; ; (c -12.7657604741) (c -2.59444003098) = 0 ; ; ; ; c > 0 ; ; ; ; c = 12.766 ; ; : 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 = 8.3 ; ; b = 10.1 ; ; c = 12.77 ; ;

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

p = a+b+c = 8.3+10.1+12.77 = 31.17 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.17 }{ 2 } = 15.58 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.58 * (15.58-8.3)(15.58-10.1)(15.58-12.77) } ; ; T = sqrt{ 1752.93 } = 41.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.87 }{ 8.3 } = 10.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.87 }{ 10.1 } = 8.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.87 }{ 12.77 } = 6.56 ; ;

7. 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{ 10.1**2+12.77**2-8.3**2 }{ 2 * 10.1 * 12.77 } ) = 40° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+12.77**2-10.1**2 }{ 2 * 8.3 * 12.77 } ) = 52° 12'46" ; ;
 gamma = 180° - alpha - beta = 180° - 40° 30' - 52° 12'46" = 87° 17'14" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.87 }{ 15.58 } = 2.69 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 40° 30' } = 6.39 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.1**2+2 * 12.77**2 - 8.3**2 } }{ 2 } = 10.736 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.77**2+2 * 8.3**2 - 10.1**2 } }{ 2 } = 9.509 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.1**2+2 * 8.3**2 - 12.77**2 } }{ 2 } = 6.686 ; ;
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