Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=3.894395228; b=45; c=42 and a=67.02770155446; b=45; c=42.

#1 Obtuse scalene triangle.

Sides: a = 3.894395228   b = 45   c = 42

Area: T = 53.94105191246
Perimeter: p = 90.894395228
Semiperimeter: s = 45.447697614

Angle ∠ A = α = 3.27222167114° = 3°16'20″ = 0.05771109555 rad
Angle ∠ B = β = 138.7287783289° = 138°43'40″ = 2.42112565824 rad
Angle ∠ C = γ = 38° = 0.66332251158 rad

Height: ha = 27.70547663897
Height: hb = 2.39773564055
Height: hc = 2.56985961488

Median: ma = 43.48222870133
Median: mb = 19.57988516563
Median: mc = 24.06441108745

Inradius: r = 1.18768890673
Circumradius: R = 34.11096541551

Vertex coordinates: A[42; 0] B[0; 0] C[-2.92766325672; 2.56985961488]
Centroid: CG[13.02444558109; 0.85661987163]
Coordinates of the circumscribed circle: U[21; 26.87987742761]
Coordinates of the inscribed circle: I[0.447697614; 1.18768890673]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 176.7287783289° = 176°43'40″ = 0.05771109555 rad
∠ B' = β' = 41.27222167114° = 41°16'20″ = 2.42112565824 rad
∠ C' = γ' = 142° = 0.66332251158 rad


How did we calculate this triangle?

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

b = 45 ; ; c = 42 ; ; gamma = 38° ; ;

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 42**2 = 45**2 + a**2 - 2 * 45 * a * cos 38° ; ; ; ; ; ; a**2 -70.921a +261 =0 ; ; p=1; q=-70.921; r=261 ; ; D = q**2 - 4pr = 70.921**2 - 4 * 1 * 261 = 3985.78367718 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 70.92 ± sqrt{ 3985.78 } }{ 2 } ; ; a_{1,2} = 35.46048391 ± 31.5665316323 ; ; a_{1} = 67.0270155423 ; ; a_{2} = 3.89395227767 ; ; ; ;
 text{ Factored form: } ; ; (a -67.0270155423) (a -3.89395227767) = 0 ; ; ; ; a > 0 ; ; ; ; a = 67.027 ; ;
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 = 3.89 ; ; b = 45 ; ; c = 42 ; ;

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

p = a+b+c = 3.89+45+42 = 90.89 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.89 }{ 2 } = 45.45 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.45 * (45.45-3.89)(45.45-45)(45.45-42) } ; ; T = sqrt{ 2909.58 } = 53.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.94 }{ 3.89 } = 27.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.94 }{ 45 } = 2.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.94 }{ 42 } = 2.57 ; ;

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{ 45**2+42**2-3.89**2 }{ 2 * 45 * 42 } ) = 3° 16'20" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.89**2+42**2-45**2 }{ 2 * 3.89 * 42 } ) = 138° 43'40" ; ; gamma = 180° - alpha - beta = 180° - 3° 16'20" - 138° 43'40" = 38° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.94 }{ 45.45 } = 1.19 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.89 }{ 2 * sin 3° 16'20" } = 34.11 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 42**2 - 3.89**2 } }{ 2 } = 43.482 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 3.89**2 - 45**2 } }{ 2 } = 19.579 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 3.89**2 - 42**2 } }{ 2 } = 24.064 ; ;





#2 Obtuse scalene triangle.

Sides: a = 67.02770155446   b = 45   c = 42

Area: T = 928.484390373
Perimeter: p = 154.0277015545
Semiperimeter: s = 77.01435077723

Angle ∠ A = α = 100.7287783289° = 100°43'40″ = 1.75880314666 rad
Angle ∠ B = β = 41.27222167114° = 41°16'20″ = 0.72203360712 rad
Angle ∠ C = γ = 38° = 0.66332251158 rad

Height: ha = 27.70547663897
Height: hb = 41.26659512769
Height: hc = 44.21435192252

Median: ma = 27.77330948365
Median: mb = 51.20660582979
Median: mc = 53.08330519696

Inradius: r = 12.05661175641
Circumradius: R = 34.11096541551

Vertex coordinates: A[42; 0] B[0; 0] C[50.37664382479; 44.21435192252]
Centroid: CG[30.79221460826; 14.73878397417]
Coordinates of the circumscribed circle: U[21; 26.87987742761]
Coordinates of the inscribed circle: I[32.01435077723; 12.05661175641]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 79.27222167114° = 79°16'20″ = 1.75880314666 rad
∠ B' = β' = 138.7287783289° = 138°43'40″ = 0.72203360712 rad
∠ C' = γ' = 142° = 0.66332251158 rad

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How did we calculate this triangle?

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

b = 45 ; ; c = 42 ; ; gamma = 38° ; ; : Nr. 1

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

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 42**2 = 45**2 + a**2 - 2 * 45 * a * cos 38° ; ; ; ; ; ; a**2 -70.921a +261 =0 ; ; p=1; q=-70.921; r=261 ; ; D = q**2 - 4pr = 70.921**2 - 4 * 1 * 261 = 3985.78367718 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 70.92 ± sqrt{ 3985.78 } }{ 2 } ; ; a_{1,2} = 35.46048391 ± 31.5665316323 ; ; a_{1} = 67.0270155423 ; ; a_{2} = 3.89395227767 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (a -67.0270155423) (a -3.89395227767) = 0 ; ; ; ; a > 0 ; ; ; ; a = 67.027 ; ; : 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 = 67.03 ; ; b = 45 ; ; c = 42 ; ;

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

p = a+b+c = 67.03+45+42 = 154.03 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 154.03 }{ 2 } = 77.01 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 77.01 * (77.01-67.03)(77.01-45)(77.01-42) } ; ; T = sqrt{ 862082.36 } = 928.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 928.48 }{ 67.03 } = 27.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 928.48 }{ 45 } = 41.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 928.48 }{ 42 } = 44.21 ; ;

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{ 45**2+42**2-67.03**2 }{ 2 * 45 * 42 } ) = 100° 43'40" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 67.03**2+42**2-45**2 }{ 2 * 67.03 * 42 } ) = 41° 16'20" ; ; gamma = 180° - alpha - beta = 180° - 100° 43'40" - 41° 16'20" = 38° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 928.48 }{ 77.01 } = 12.06 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 67.03 }{ 2 * sin 100° 43'40" } = 34.11 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 42**2 - 67.03**2 } }{ 2 } = 27.773 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 67.03**2 - 45**2 } }{ 2 } = 51.206 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 67.03**2 - 42**2 } }{ 2 } = 53.083 ; ;
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