Triangle calculator

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

You have entered side b, c and angle β.

Triangle has two solutions: a=522.0288097964; b=3623.57; c=4000 and a=5497.29111732; b=3623.57; c=4000.

#1 Obtuse scalene triangle.

Sides: a = 522.0288097964   b = 3623.57   c = 4000

Area: T = 687708.8122029
Perimeter: p = 8145.598809796
Semiperimeter: s = 4072.799904898

Angle ∠ A = α = 5.44552086439° = 5°26'43″ = 0.09550368193 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 133.3554791356° = 133°21'17″ = 2.32774801825 rad

Height: ha = 2634.758784047
Height: hb = 379.5755287371
Height: hc = 343.8544406015

Median: ma = 3807.493280218
Median: mb = 2203.110957088
Median: mc = 1643.589949862

Inradius: r = 168.8544098559
Circumradius: R = 2750.591054334

Vertex coordinates: A[4000; 0] B[0; 0] C[392.782172377; 343.8544406015]
Centroid: CG[1464.261057459; 114.6188135338]
Coordinates of the circumscribed circle: U[2000; -1888.319891827]
Coordinates of the inscribed circle: I[449.2299048982; 168.8544098559]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 174.5554791356° = 174°33'17″ = 0.09550368193 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 46.64552086439° = 46°38'43″ = 2.32774801825 rad




How did we calculate this triangle?

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

b = 3623.57 ; ; c = 4000 ; ; beta = 41.2° ; ;

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 3623.57**2 = 4000**2 + a**2 - 2 * 4000 * a * cos(41° 12') ; ; ; ; ; ; a**2 -6019.319a +2869740.455 =0 ; ; a=1; b=-6019.319; c=2869740.455 ; ; D = b**2 - 4ac = 6019.319**2 - 4 * 1 * 2869740.455 = 24753242.6678 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 6019.32 ± sqrt{ 24753242.67 } }{ 2 } ; ; a_{1,2} = 3009.65963558 ± 2487.63153762 ; ; a_{1} = 5497.2911732 ; ;
a_{2} = 522.028097961 ; ; ; ; (a -5497.2911732) (a -522.028097961) = 0 ; ; ; ; a > 0 ; ; ; ; a = 5497.291 ; ;


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 = 522.03 ; ; b = 3623.57 ; ; c = 4000 ; ;

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

p = a+b+c = 522.03+3623.57+4000 = 8145.6 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8145.6 }{ 2 } = 4072.8 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4072.8 * (4072.8-522.03)(4072.8-3623.57)(4072.8-4000) } ; ; T = sqrt{ 472943410143 } = 687708.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 687708.81 }{ 522.03 } = 2634.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 687708.81 }{ 3623.57 } = 379.58 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 687708.81 }{ 4000 } = 343.85 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 522.03**2-3623.57**2-4000**2 }{ 2 * 3623.57 * 4000 } ) = 5° 26'43" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3623.57**2-522.03**2-4000**2 }{ 2 * 522.03 * 4000 } ) = 41° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4000**2-522.03**2-3623.57**2 }{ 2 * 3623.57 * 522.03 } ) = 133° 21'17" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 687708.81 }{ 4072.8 } = 168.85 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 522.03 }{ 2 * sin 5° 26'43" } = 2750.59 ; ;





#2 Obtuse scalene triangle.

Sides: a = 5497.29111732   b = 3623.57   c = 4000

Area: T = 7242015.510998
Perimeter: p = 13120.86111732
Semiperimeter: s = 6560.43105866

Angle ∠ A = α = 92.15547913561° = 92°9'17″ = 1.60884045307 rad
Angle ∠ B = β = 41.2° = 41°12' = 0.71990756518 rad
Angle ∠ C = γ = 46.64552086439° = 46°38'43″ = 0.81441124711 rad

Height: ha = 2634.758784047
Height: hb = 3997.17215794
Height: hc = 3621.008775499

Median: ma = 2647.655504016
Median: mb = 4452.813262072
Median: mc = 4204.193253769

Inradius: r = 1103.893332139
Circumradius: R = 2750.591054334

Vertex coordinates: A[4000; 0] B[0; 0] C[4136.244383726; 3621.008775499]
Centroid: CG[2712.081127909; 1207.0032585]
Coordinates of the circumscribed circle: U[2000; 1888.319891827]
Coordinates of the inscribed circle: I[2936.86105866; 1103.893332139]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 87.84552086439° = 87°50'43″ = 1.60884045307 rad
∠ B' = β' = 138.8° = 138°48' = 0.71990756518 rad
∠ C' = γ' = 133.3554791356° = 133°21'17″ = 0.81441124711 rad

Calculate another triangle

How did we calculate this triangle?

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

b = 3623.57 ; ; c = 4000 ; ; beta = 41.2° ; ; : Nr. 1

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

b**2 = c**2 + a**2 - 2c a cos beta ; ; ; ; 3623.57**2 = 4000**2 + a**2 - 2 * 4000 * a * cos(41° 12') ; ; ; ; ; ; a**2 -6019.319a +2869740.455 =0 ; ; a=1; b=-6019.319; c=2869740.455 ; ; D = b**2 - 4ac = 6019.319**2 - 4 * 1 * 2869740.455 = 24753242.6678 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 6019.32 ± sqrt{ 24753242.67 } }{ 2 } ; ; a_{1,2} = 3009.65963558 ± 2487.63153762 ; ; a_{1} = 5497.2911732 ; ; : Nr. 1
a_{2} = 522.028097961 ; ; ; ; (a -5497.2911732) (a -522.028097961) = 0 ; ; ; ; a > 0 ; ; ; ; a = 5497.291 ; ; : 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 = 5497.29 ; ; b = 3623.57 ; ; c = 4000 ; ;

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

p = a+b+c = 5497.29+3623.57+4000 = 13120.86 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13120.86 }{ 2 } = 6560.43 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6560.43 * (6560.43-5497.29)(6560.43-3623.57)(6560.43-4000) } ; ; T = sqrt{ 5.245 * 10**{ 13 } } = 7242015.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7242015.51 }{ 5497.29 } = 2634.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7242015.51 }{ 3623.57 } = 3997.17 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7242015.51 }{ 4000 } = 3621.01 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5497.29**2-3623.57**2-4000**2 }{ 2 * 3623.57 * 4000 } ) = 92° 9'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3623.57**2-5497.29**2-4000**2 }{ 2 * 5497.29 * 4000 } ) = 41° 12' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4000**2-5497.29**2-3623.57**2 }{ 2 * 3623.57 * 5497.29 } ) = 46° 38'43" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7242015.51 }{ 6560.43 } = 1103.89 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5497.29 }{ 2 * sin 92° 9'17" } = 2750.59 ; ;




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