Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=37; b=31; c=8.84402674969 and a=37; b=31; c=46.15224495884.

#1 Obtuse scalene triangle.

Sides: a = 37   b = 31   c = 8.84402674969

Area: T = 109.4332930686
Perimeter: p = 76.84402674969
Semiperimeter: s = 38.42201337485

Angle ∠ A = α = 1276.999595765° = 126°59'59″ = 2.21765610948 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 111.0004042355° = 11°1″ = 0.1921993273 rad

Height: ha = 5.91552935506
Height: hb = 7.06601890765
Height: hc = 24.75878324353

Median: ma = 13.31663495264
Median: mb = 21.98546574845
Median: mc = 33.84546807881

Inradius: r = 2.84883224812
Circumradius: R = 23.16443865229

Vertex coordinates: A[8.84402674969; 0] B[0; 0] C[27.49663585427; 24.75878324353]
Centroid: CG[12.11222086799; 8.25326108118]
Coordinates of the circumscribed circle: U[4.42201337485; 22.73987603142]
Coordinates of the inscribed circle: I[7.42201337485; 2.84883224812]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 533.0004042355° = 53°1″ = 2.21765610948 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 1698.999595765° = 168°59'59″ = 0.1921993273 rad


How did we calculate this triangle?

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

a = 37 ; ; b = 31 ; ; beta = 42° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 31**2 = 37**2 + c**2 - 2 * 37 * c * cos 42° ; ; ; ; ; ; c**2 -54.993c +408 =0 ; ; p=1; q=-54.993; r=408 ; ; D = q**2 - 4pr = 54.993**2 - 4 * 1 * 408 = 1392.19893243 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 54.99 ± sqrt{ 1392.2 } }{ 2 } ; ;
c_{1,2} = 27.49635854 ± 18.6560910457 ; ; c_{1} = 46.1524495884 ; ; c_{2} = 8.84026749693 ; ; ; ; text{ Factored form: } ; ; (c -46.1524495884) (c -8.84026749693) = 0 ; ; ; ; c > 0 ; ; ; ; c = 46.152 ; ;
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 = 37 ; ; b = 31 ; ; c = 8.84 ; ;

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

p = a+b+c = 37+31+8.84 = 76.84 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 76.84 }{ 2 } = 38.42 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 38.42 * (38.42-37)(38.42-31)(38.42-8.84) } ; ; T = sqrt{ 11975.57 } = 109.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 109.43 }{ 37 } = 5.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 109.43 }{ 31 } = 7.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 109.43 }{ 8.84 } = 24.76 ; ;

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{ 31**2+8.84**2-37**2 }{ 2 * 31 * 8.84 } ) = 126° 59'59" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37**2+8.84**2-31**2 }{ 2 * 37 * 8.84 } ) = 42° ; ;
 gamma = 180° - alpha - beta = 180° - 126° 59'59" - 42° = 11° 1" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 109.43 }{ 38.42 } = 2.85 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37 }{ 2 * sin 126° 59'59" } = 23.16 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 8.84**2 - 37**2 } }{ 2 } = 13.316 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.84**2+2 * 37**2 - 31**2 } }{ 2 } = 21.985 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 37**2 - 8.84**2 } }{ 2 } = 33.845 ; ;



#2 Acute scalene triangle.

Sides: a = 37   b = 31   c = 46.15224495884

Area: T = 571.3177306694
Perimeter: p = 114.1522449588
Semiperimeter: s = 57.07662247942

Angle ∠ A = α = 533.0004042355° = 53°1″ = 0.92550315588 rad
Angle ∠ B = β = 42° = 0.73330382858 rad
Angle ∠ C = γ = 854.9995957645° = 84°59'59″ = 1.4843522809 rad

Height: ha = 30.8822016578
Height: hb = 36.8599181077
Height: hc = 24.75878324353

Median: ma = 34.68882444281
Median: mb = 38.84993796798
Median: mc = 25.14993111088

Inradius: r = 10.0109724868
Circumradius: R = 23.16443865229

Vertex coordinates: A[46.15224495884; 0] B[0; 0] C[27.49663585427; 24.75878324353]
Centroid: CG[24.55496027104; 8.25326108118]
Coordinates of the circumscribed circle: U[23.07662247942; 2.0199072121]
Coordinates of the inscribed circle: I[26.07662247942; 10.0109724868]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 1276.999595765° = 126°59'59″ = 0.92550315588 rad
∠ B' = β' = 138° = 0.73330382858 rad
∠ C' = γ' = 955.0004042355° = 95°1″ = 1.4843522809 rad

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How did we calculate this triangle?

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

a = 37 ; ; b = 31 ; ; beta = 42° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 31**2 = 37**2 + c**2 - 2 * 37 * c * cos 42° ; ; ; ; ; ; c**2 -54.993c +408 =0 ; ; p=1; q=-54.993; r=408 ; ; D = q**2 - 4pr = 54.993**2 - 4 * 1 * 408 = 1392.19893243 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 54.99 ± sqrt{ 1392.2 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 27.49635854 ± 18.6560910457 ; ; c_{1} = 46.1524495884 ; ; c_{2} = 8.84026749693 ; ; ; ; text{ Factored form: } ; ; (c -46.1524495884) (c -8.84026749693) = 0 ; ; ; ; c > 0 ; ; ; ; c = 46.152 ; ; : 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 = 37 ; ; b = 31 ; ; c = 46.15 ; ;

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

p = a+b+c = 37+31+46.15 = 114.15 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.15 }{ 2 } = 57.08 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.08 * (57.08-37)(57.08-31)(57.08-46.15) } ; ; T = sqrt{ 326403.46 } = 571.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 571.32 }{ 37 } = 30.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 571.32 }{ 31 } = 36.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 571.32 }{ 46.15 } = 24.76 ; ;

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{ 31**2+46.15**2-37**2 }{ 2 * 31 * 46.15 } ) = 53° 1" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 37**2+46.15**2-31**2 }{ 2 * 37 * 46.15 } ) = 42° ; ;
 gamma = 180° - alpha - beta = 180° - 53° 1" - 42° = 84° 59'59" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 571.32 }{ 57.08 } = 10.01 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 37 }{ 2 * sin 53° 1" } = 23.16 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 46.15**2 - 37**2 } }{ 2 } = 34.688 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 46.15**2+2 * 37**2 - 31**2 } }{ 2 } = 38.849 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 31**2+2 * 37**2 - 46.15**2 } }{ 2 } = 25.149 ; ;
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