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=149; b=176; c=27.54658410637 and a=149; b=176; c=318.5659886399.

#1 Obtuse scalene triangle.

Sides: a = 149   b = 176   c = 27.54658410637

Area: T = 441.745511228
Perimeter: p = 352.5465841064
Semiperimeter: s = 176.2732920532

Angle ∠ A = α = 10.5° = 10°30' = 0.18332595715 rad
Angle ∠ B = β = 167.5699331838° = 167°34'10″ = 2.92546365659 rad
Angle ∠ C = γ = 1.93106681621° = 1°55'50″ = 0.03436965162 rad

Height: ha = 5.92994645944
Height: hb = 5.02198308214
Height: hc = 32.07334524866

Median: ma = 101.5733306926
Median: mb = 61.12219001664
Median: mc = 162.4777095801

Inradius: r = 2.5066029349
Circumradius: R = 408.8121617816

Vertex coordinates: A[27.54658410637; 0] B[0; 0] C[-145.5077022668; 32.07334524866]
Centroid: CG[-39.3220393868; 10.69111508289]
Coordinates of the circumscribed circle: U[13.77329205318; 408.5879546137]
Coordinates of the inscribed circle: I[0.27329205318; 2.5066029349]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.5° = 169°30' = 0.18332595715 rad
∠ B' = β' = 12.43106681621° = 12°25'50″ = 2.92546365659 rad
∠ C' = γ' = 178.0699331838° = 178°4'10″ = 0.03436965162 rad


How did we calculate this triangle?

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

a = 149 ; ; b = 176 ; ; alpha = 10.5° ; ;

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 ; ; ; ; 149**2 = 176**2 + c**2 - 2 * 176 * c * cos 10° 30' ; ; ; ; ; ; c**2 -346.106c +8775 =0 ; ; p=1; q=-346.106; r=8775 ; ; D = q**2 - 4pr = 346.106**2 - 4 * 1 * 8775 = 84689.1745824 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 346.11 ± sqrt{ 84689.17 } }{ 2 } ; ; c_{1,2} = 173.05286373 ± 145.507022668 ; ; c_{1} = 318.559886398 ; ; c_{2} = 27.5458410624 ; ;
 ; ; text{ Factored form: } ; ; (c -318.559886398) (c -27.5458410624) = 0 ; ; ; ; c > 0 ; ; ; ; c = 318.56 ; ;
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 = 149 ; ; b = 176 ; ; c = 27.55 ; ;

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

p = a+b+c = 149+176+27.55 = 352.55 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 352.55 }{ 2 } = 176.27 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 176.27 * (176.27-149)(176.27-176)(176.27-27.55) } ; ; T = sqrt{ 195138.74 } = 441.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 441.75 }{ 149 } = 5.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 441.75 }{ 176 } = 5.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 441.75 }{ 27.55 } = 32.07 ; ;

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{ 176**2+27.55**2-149**2 }{ 2 * 176 * 27.55 } ) = 10° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 149**2+27.55**2-176**2 }{ 2 * 149 * 27.55 } ) = 167° 34'10" ; ; gamma = 180° - alpha - beta = 180° - 10° 30' - 167° 34'10" = 1° 55'50" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 441.75 }{ 176.27 } = 2.51 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 149 }{ 2 * sin 10° 30' } = 408.81 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 176**2+2 * 27.55**2 - 149**2 } }{ 2 } = 101.573 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27.55**2+2 * 149**2 - 176**2 } }{ 2 } = 61.122 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 176**2+2 * 149**2 - 27.55**2 } }{ 2 } = 162.477 ; ;





#2 Obtuse scalene triangle.

Sides: a = 149   b = 176   c = 318.5659886399

Area: T = 5108.658769028
Perimeter: p = 643.5659886399
Semiperimeter: s = 321.7879943199

Angle ∠ A = α = 10.5° = 10°30' = 0.18332595715 rad
Angle ∠ B = β = 12.43106681621° = 12°25'50″ = 0.21769560877 rad
Angle ∠ C = γ = 157.0699331838° = 157°4'10″ = 2.74113769945 rad

Height: ha = 68.57325864467
Height: hb = 58.05329282986
Height: hc = 32.07334524866

Median: ma = 246.3298947976
Median: mb = 232.5876974294
Median: mc = 34.90655825677

Inradius: r = 15.87662464791
Circumradius: R = 408.8121617816

Vertex coordinates: A[318.5659886399; 0] B[0; 0] C[145.5077022668; 32.07334524866]
Centroid: CG[154.6898969689; 10.69111508289]
Coordinates of the circumscribed circle: U[159.2879943199; -376.506609365]
Coordinates of the inscribed circle: I[145.7879943199; 15.87662464791]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.5° = 169°30' = 0.18332595715 rad
∠ B' = β' = 167.5699331838° = 167°34'10″ = 0.21769560877 rad
∠ C' = γ' = 22.93106681621° = 22°55'50″ = 2.74113769945 rad

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How did we calculate this triangle?

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

a = 149 ; ; b = 176 ; ; alpha = 10.5° ; ; : 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 ; ; ; ; 149**2 = 176**2 + c**2 - 2 * 176 * c * cos 10° 30' ; ; ; ; ; ; c**2 -346.106c +8775 =0 ; ; p=1; q=-346.106; r=8775 ; ; D = q**2 - 4pr = 346.106**2 - 4 * 1 * 8775 = 84689.1745824 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 346.11 ± sqrt{ 84689.17 } }{ 2 } ; ; c_{1,2} = 173.05286373 ± 145.507022668 ; ; c_{1} = 318.559886398 ; ; c_{2} = 27.5458410624 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -318.559886398) (c -27.5458410624) = 0 ; ; ; ; c > 0 ; ; ; ; c = 318.56 ; ; : 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 = 149 ; ; b = 176 ; ; c = 318.56 ; ;

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

p = a+b+c = 149+176+318.56 = 643.56 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 643.56 }{ 2 } = 321.78 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 321.78 * (321.78-149)(321.78-176)(321.78-318.56) } ; ; T = sqrt{ 26098383.4 } = 5108.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5108.66 }{ 149 } = 68.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5108.66 }{ 176 } = 58.05 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5108.66 }{ 318.56 } = 32.07 ; ;

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{ 176**2+318.56**2-149**2 }{ 2 * 176 * 318.56 } ) = 10° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 149**2+318.56**2-176**2 }{ 2 * 149 * 318.56 } ) = 12° 25'50" ; ; gamma = 180° - alpha - beta = 180° - 10° 30' - 12° 25'50" = 157° 4'10" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5108.66 }{ 321.78 } = 15.88 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 149 }{ 2 * sin 10° 30' } = 408.81 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 176**2+2 * 318.56**2 - 149**2 } }{ 2 } = 246.329 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 318.56**2+2 * 149**2 - 176**2 } }{ 2 } = 232.587 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 176**2+2 * 149**2 - 318.56**2 } }{ 2 } = 34.906 ; ;
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