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=35; b=54; c=21.78221496582 and a=35; b=54; c=77.63223745147.

#1 Obtuse scalene triangle.

Sides: a = 35   b = 54   c = 21.78221496582

Area: T = 229.7966025756
Perimeter: p = 110.7822149658
Semiperimeter: s = 55.39110748291

Angle ∠ A = α = 23° = 0.4011425728 rad
Angle ∠ B = β = 142.9266264341° = 142°55'35″ = 2.49545339003 rad
Angle ∠ C = γ = 14.0743735659° = 14°4'25″ = 0.24656330253 rad

Height: ha = 13.13112014717
Height: hb = 8.51109639169
Height: hc = 21.09994809384

Median: ma = 37.26990356981
Median: mb = 10.9887766919
Median: mc = 44.18801368158

Inradius: r = 4.14986110617
Circumradius: R = 44.78878316418

Vertex coordinates: A[21.78221496582; 0] B[0; 0] C[-27.92551124282; 21.09994809384]
Centroid: CG[-2.04876542567; 7.03331603128]
Coordinates of the circumscribed circle: U[10.89110748291; 43.44334615592]
Coordinates of the inscribed circle: I[1.39110748291; 4.14986110617]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157° = 0.4011425728 rad
∠ B' = β' = 37.0743735659° = 37°4'25″ = 2.49545339003 rad
∠ C' = γ' = 165.9266264341° = 165°55'35″ = 0.24656330253 rad


How did we calculate this triangle?

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

a = 35 ; ; b = 54 ; ; alpha = 23° ; ;

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 ; ; ; ; 35**2 = 54**2 + c**2 - 2 * 54 * c * cos 23° ; ; ; ; ; ; c**2 -99.415c +1691 =0 ; ; p=1; q=-99.415; r=1691 ; ; D = q**2 - 4pr = 99.415**2 - 4 * 1 * 1691 = 3119.24761652 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 99.41 ± sqrt{ 3119.25 } }{ 2 } ; ;
c_{1,2} = 49.70726209 ± 27.9251124282 ; ; c_{1} = 77.6323745147 ; ; c_{2} = 21.7821496582 ; ; ; ; text{ Factored form: } ; ; (c -77.6323745147) (c -21.7821496582) = 0 ; ; ; ; c > 0 ; ; ; ; c = 77.632 ; ;
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 = 35 ; ; b = 54 ; ; c = 21.78 ; ;

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

p = a+b+c = 35+54+21.78 = 110.78 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 110.78 }{ 2 } = 55.39 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 55.39 * (55.39-35)(55.39-54)(55.39-21.78) } ; ; T = sqrt{ 52806.21 } = 229.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 229.8 }{ 35 } = 13.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 229.8 }{ 54 } = 8.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 229.8 }{ 21.78 } = 21.1 ; ;

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{ 54**2+21.78**2-35**2 }{ 2 * 54 * 21.78 } ) = 23° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+21.78**2-54**2 }{ 2 * 35 * 21.78 } ) = 142° 55'35" ; ;
 gamma = 180° - alpha - beta = 180° - 23° - 142° 55'35" = 14° 4'25" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 229.8 }{ 55.39 } = 4.15 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 23° } = 44.79 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 54**2+2 * 21.78**2 - 35**2 } }{ 2 } = 37.269 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21.78**2+2 * 35**2 - 54**2 } }{ 2 } = 10.988 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 54**2+2 * 35**2 - 21.78**2 } }{ 2 } = 44.18 ; ;



#2 Obtuse scalene triangle.

Sides: a = 35   b = 54   c = 77.63223745147

Area: T = 819.0011403138
Perimeter: p = 166.6322374515
Semiperimeter: s = 83.31661872573

Angle ∠ A = α = 23° = 0.4011425728 rad
Angle ∠ B = β = 37.0743735659° = 37°4'25″ = 0.64770587533 rad
Angle ∠ C = γ = 119.9266264341° = 119°55'35″ = 2.09331081724 rad

Height: ha = 46.88000801793
Height: hb = 30.33333853014
Height: hc = 21.09994809384

Median: ma = 64.53879174315
Median: mb = 53.82327905853
Median: mc = 23.74545489914

Inradius: r = 9.83300393969
Circumradius: R = 44.78878316418

Vertex coordinates: A[77.63223745147; 0] B[0; 0] C[27.92551124282; 21.09994809384]
Centroid: CG[35.1865828981; 7.03331603128]
Coordinates of the circumscribed circle: U[38.81661872573; -22.34439806208]
Coordinates of the inscribed circle: I[29.31661872573; 9.83300393969]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157° = 0.4011425728 rad
∠ B' = β' = 142.9266264341° = 142°55'35″ = 0.64770587533 rad
∠ C' = γ' = 60.0743735659° = 60°4'25″ = 2.09331081724 rad

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How did we calculate this triangle?

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

a = 35 ; ; b = 54 ; ; alpha = 23° ; ; : 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 ; ; ; ; 35**2 = 54**2 + c**2 - 2 * 54 * c * cos 23° ; ; ; ; ; ; c**2 -99.415c +1691 =0 ; ; p=1; q=-99.415; r=1691 ; ; D = q**2 - 4pr = 99.415**2 - 4 * 1 * 1691 = 3119.24761652 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 99.41 ± sqrt{ 3119.25 } }{ 2 } ; ; : Nr. 1
c_{1,2} = 49.70726209 ± 27.9251124282 ; ; c_{1} = 77.6323745147 ; ; c_{2} = 21.7821496582 ; ; ; ; text{ Factored form: } ; ; (c -77.6323745147) (c -21.7821496582) = 0 ; ; ; ; c > 0 ; ; ; ; c = 77.632 ; ; : 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 = 35 ; ; b = 54 ; ; c = 77.63 ; ;

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

p = a+b+c = 35+54+77.63 = 166.63 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 166.63 }{ 2 } = 83.32 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 83.32 * (83.32-35)(83.32-54)(83.32-77.63) } ; ; T = sqrt{ 670763.3 } = 819 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 819 }{ 35 } = 46.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 819 }{ 54 } = 30.33 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 819 }{ 77.63 } = 21.1 ; ;

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{ 54**2+77.63**2-35**2 }{ 2 * 54 * 77.63 } ) = 23° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 35**2+77.63**2-54**2 }{ 2 * 35 * 77.63 } ) = 37° 4'25" ; ;
 gamma = 180° - alpha - beta = 180° - 23° - 37° 4'25" = 119° 55'35" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 819 }{ 83.32 } = 9.83 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 35 }{ 2 * sin 23° } = 44.79 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 54**2+2 * 77.63**2 - 35**2 } }{ 2 } = 64.538 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 77.63**2+2 * 35**2 - 54**2 } }{ 2 } = 53.823 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 54**2+2 * 35**2 - 77.63**2 } }{ 2 } = 23.745 ; ;
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