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=890; b=697; c=379.6454871274 and a=890; b=697; c=806.783292577.

#1 Obtuse scalene triangle.

Sides: a = 890   b = 697   c = 379.6454871274

Area: T = 125942.1822272
Perimeter: p = 1966.645487127
Semiperimeter: s = 983.3222435637

Angle ∠ A = α = 107.843312158° = 107°50'35″ = 1.88222175472 rad
Angle ∠ B = β = 48.2° = 48°12' = 0.84112486995 rad
Angle ∠ C = γ = 23.95768784201° = 23°57'25″ = 0.41881264069 rad

Height: ha = 283.0166139937
Height: hb = 361.384359332
Height: hc = 663.4743639718

Median: ma = 341.9721656928
Median: mb = 588.7810828613
Median: mc = 776.4880484577

Inradius: r = 128.0788214945
Circumradius: R = 467.4876545708

Vertex coordinates: A[379.6454871274; 0] B[0; 0] C[593.2143898522; 663.4743639718]
Centroid: CG[324.2866256599; 221.1587879906]
Coordinates of the circumscribed circle: U[189.8222435637; 427.2133194257]
Coordinates of the inscribed circle: I[286.3222435637; 128.0788214945]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 72.15768784201° = 72°9'25″ = 1.88222175472 rad
∠ B' = β' = 131.8° = 131°48' = 0.84112486995 rad
∠ C' = γ' = 156.043312158° = 156°2'35″ = 0.41881264069 rad




How did we calculate this triangle?

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

a = 890 ; ; b = 697 ; ; beta = 48.2° ; ;

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 ; ; ; ; 697**2 = 890**2 + c**2 - 2 * 890 * c * cos 48° 12' ; ; ; ; ; ; c**2 -1186.428c +306291 =0 ; ; a=1; b=-1186.428; c=306291 ; ; D = b**2 - 4ac = 1186.428**2 - 4 * 1 * 306291 = 182446.917599 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 1186.43 ± sqrt{ 182446.92 } }{ 2 } ; ; c_{1,2} = 593.21389852 ± 213.569027248 ; ; c_{1} = 806.782925768 ; ; c_{2} = 379.644871272 ; ;
 ; ; text{ Factored form: } ; ; (c -806.782925768) (c -379.644871272) = 0 ; ; ; ; c > 0 ; ; ; ; c = 806.783 ; ;


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 = 890 ; ; b = 697 ; ; c = 379.64 ; ;

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

p = a+b+c = 890+697+379.64 = 1966.64 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1966.64 }{ 2 } = 983.32 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 983.32 * (983.32-890)(983.32-697)(983.32-379.64) } ; ; T = sqrt{ 15861433275.5 } = 125942.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 125942.18 }{ 890 } = 283.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 125942.18 }{ 697 } = 361.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 125942.18 }{ 379.64 } = 663.47 ; ;

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{ 697**2+379.64**2-890**2 }{ 2 * 697 * 379.64 } ) = 107° 50'35" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 890**2+379.64**2-697**2 }{ 2 * 890 * 379.64 } ) = 48° 12' ; ; gamma = 180° - alpha - beta = 180° - 107° 50'35" - 48° 12' = 23° 57'25" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 125942.18 }{ 983.32 } = 128.08 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 890 }{ 2 * sin 107° 50'35" } = 467.49 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 697**2+2 * 379.64**2 - 890**2 } }{ 2 } = 341.972 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 379.64**2+2 * 890**2 - 697**2 } }{ 2 } = 588.781 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 697**2+2 * 890**2 - 379.64**2 } }{ 2 } = 776.48 ; ;







#2 Acute scalene triangle.

Sides: a = 890   b = 697   c = 806.783292577

Area: T = 267639.6022111
Perimeter: p = 2393.783292577
Semiperimeter: s = 1196.891146289

Angle ∠ A = α = 72.15768784201° = 72°9'25″ = 1.25993751064 rad
Angle ∠ B = β = 48.2° = 48°12' = 0.84112486995 rad
Angle ∠ C = γ = 59.64331215799° = 59°38'35″ = 1.04109688477 rad

Height: ha = 601.4377308115
Height: hb = 767.9765902759
Height: hc = 663.4743639718

Median: ma = 608.5476501639
Median: mb = 774.6277068115
Median: mc = 690.094407161

Inradius: r = 223.6122257595
Circumradius: R = 467.4876545708

Vertex coordinates: A[806.783292577; 0] B[0; 0] C[593.2143898522; 663.4743639718]
Centroid: CG[466.6665608097; 221.1587879906]
Coordinates of the circumscribed circle: U[403.3911462885; 236.2660445461]
Coordinates of the inscribed circle: I[499.8911462885; 223.6122257595]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.843312158° = 107°50'35″ = 1.25993751064 rad
∠ B' = β' = 131.8° = 131°48' = 0.84112486995 rad
∠ C' = γ' = 120.357687842° = 120°21'25″ = 1.04109688477 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 890 ; ; b = 697 ; ; beta = 48.2° ; ; : 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 ; ; ; ; 697**2 = 890**2 + c**2 - 2 * 890 * c * cos 48° 12' ; ; ; ; ; ; c**2 -1186.428c +306291 =0 ; ; a=1; b=-1186.428; c=306291 ; ; D = b**2 - 4ac = 1186.428**2 - 4 * 1 * 306291 = 182446.917599 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 1186.43 ± sqrt{ 182446.92 } }{ 2 } ; ; c_{1,2} = 593.21389852 ± 213.569027248 ; ; c_{1} = 806.782925768 ; ; c_{2} = 379.644871272 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -806.782925768) (c -379.644871272) = 0 ; ; ; ; c > 0 ; ; ; ; c = 806.783 ; ; : 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 = 890 ; ; b = 697 ; ; c = 806.78 ; ;

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

p = a+b+c = 890+697+806.78 = 2393.78 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2393.78 }{ 2 } = 1196.89 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1196.89 * (1196.89-890)(1196.89-697)(1196.89-806.78) } ; ; T = sqrt{ 71630956618.4 } = 267639.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 267639.6 }{ 890 } = 601.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 267639.6 }{ 697 } = 767.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 267639.6 }{ 806.78 } = 663.47 ; ;

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{ 697**2+806.78**2-890**2 }{ 2 * 697 * 806.78 } ) = 72° 9'25" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 890**2+806.78**2-697**2 }{ 2 * 890 * 806.78 } ) = 48° 12' ; ; gamma = 180° - alpha - beta = 180° - 72° 9'25" - 48° 12' = 59° 38'35" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 267639.6 }{ 1196.89 } = 223.61 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 890 }{ 2 * sin 72° 9'25" } = 467.49 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 697**2+2 * 806.78**2 - 890**2 } }{ 2 } = 608.547 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 806.78**2+2 * 890**2 - 697**2 } }{ 2 } = 774.627 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 697**2+2 * 890**2 - 806.78**2 } }{ 2 } = 690.094 ; ;
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