Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=25.3; b=23.61881257376; c=42.1 and a=25.3; b=47.94328390119; c=42.1.

#1 Obtuse scalene triangle.

Sides: a = 25.3   b = 23.61881257376   c = 42.1

Area: T = 261.9822158372
Perimeter: p = 91.01881257376
Semiperimeter: s = 45.50990628688

Angle ∠ A = α = 31.8° = 31°48' = 0.55550147021 rad
Angle ∠ B = β = 29.46771998726° = 29°28'2″ = 0.51442996591 rad
Angle ∠ C = γ = 118.7332800127° = 118°43'58″ = 2.07222782923 rad

Height: ha = 20.71100520452
Height: hb = 22.18548389904
Height: hc = 12.44657082362

Median: ma = 31.70331612253
Median: mb = 32.66218437043
Median: mc = 12.48440070361

Inradius: r = 5.75767029918
Circumradius: R = 24.0065808662

Vertex coordinates: A[42.1; 0] B[0; 0] C[22.02771275136; 12.44657082362]
Centroid: CG[21.37657091712; 4.14985694121]
Coordinates of the circumscribed circle: U[21.05; -11.5440205783]
Coordinates of the inscribed circle: I[21.89109371312; 5.75767029918]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.2° = 148°12' = 0.55550147021 rad
∠ B' = β' = 150.5332800127° = 150°31'58″ = 0.51442996591 rad
∠ C' = γ' = 61.26771998726° = 61°16'2″ = 2.07222782923 rad




How did we calculate this triangle?

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

a = 25.3 ; ; c = 42.1 ; ; alpha = 31.8° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 25.3**2 = 42.1**2 + b**2 - 2 * 42.1 * b * cos 31° 48' ; ; ; ; ; ; b**2 -71.561b +1132.32 =0 ; ; p=1; q=-71.561; r=1132.32 ; ; D = q**2 - 4pr = 71.561**2 - 4 * 1 * 1132.32 = 591.691675878 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 71.56 ± sqrt{ 591.69 } }{ 2 } ; ; b_{1,2} = 35.78048237 ± 12.1623566372 ; ; b_{1} = 47.9428390072 ; ; b_{2} = 23.6181257328 ; ;
 ; ; text{ Factored form: } ; ; (b -47.9428390072) (b -23.6181257328) = 0 ; ; ; ; b > 0 ; ; ; ; b = 47.943 ; ;
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 = 25.3 ; ; b = 23.62 ; ; c = 42.1 ; ;

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

p = a+b+c = 25.3+23.62+42.1 = 91.02 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 91.02 }{ 2 } = 45.51 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.51 * (45.51-25.3)(45.51-23.62)(45.51-42.1) } ; ; T = sqrt{ 68634.65 } = 261.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 261.98 }{ 25.3 } = 20.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 261.98 }{ 23.62 } = 22.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 261.98 }{ 42.1 } = 12.45 ; ;

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{ 23.62**2+42.1**2-25.3**2 }{ 2 * 23.62 * 42.1 } ) = 31° 48' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 25.3**2+42.1**2-23.62**2 }{ 2 * 25.3 * 42.1 } ) = 29° 28'2" ; ; gamma = 180° - alpha - beta = 180° - 31° 48' - 29° 28'2" = 118° 43'58" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 261.98 }{ 45.51 } = 5.76 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 25.3 }{ 2 * sin 31° 48' } = 24.01 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.62**2+2 * 42.1**2 - 25.3**2 } }{ 2 } = 31.703 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.1**2+2 * 25.3**2 - 23.62**2 } }{ 2 } = 32.662 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 23.62**2+2 * 25.3**2 - 42.1**2 } }{ 2 } = 12.484 ; ;







#2 Acute scalene triangle.

Sides: a = 25.3   b = 47.94328390119   c = 42.1

Area: T = 531.8022082111
Perimeter: p = 115.3432839012
Semiperimeter: s = 57.6711419506

Angle ∠ A = α = 31.8° = 31°48' = 0.55550147021 rad
Angle ∠ B = β = 86.93328001274° = 86°55'58″ = 1.51772635902 rad
Angle ∠ C = γ = 61.26771998726° = 61°16'2″ = 1.06993143613 rad

Height: ha = 42.04396902855
Height: hb = 22.18548389904
Height: hc = 25.26437568699

Median: ma = 43.30663552641
Median: mb = 25.13220720767
Median: mc = 32.03443628977

Inradius: r = 9.22112414168
Circumradius: R = 24.0065808662

Vertex coordinates: A[42.1; 0] B[0; 0] C[1.35437314427; 25.26437568699]
Centroid: CG[14.48545771476; 8.421125229]
Coordinates of the circumscribed circle: U[21.05; 11.5440205783]
Coordinates of the inscribed circle: I[9.7298580494; 9.22112414168]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.2° = 148°12' = 0.55550147021 rad
∠ B' = β' = 93.06771998726° = 93°4'2″ = 1.51772635902 rad
∠ C' = γ' = 118.7332800127° = 118°43'58″ = 1.06993143613 rad

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How did we calculate this triangle?

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

a = 25.3 ; ; c = 42.1 ; ; alpha = 31.8° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 25.3**2 = 42.1**2 + b**2 - 2 * 42.1 * b * cos 31° 48' ; ; ; ; ; ; b**2 -71.561b +1132.32 =0 ; ; p=1; q=-71.561; r=1132.32 ; ; D = q**2 - 4pr = 71.561**2 - 4 * 1 * 1132.32 = 591.691675878 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 71.56 ± sqrt{ 591.69 } }{ 2 } ; ; b_{1,2} = 35.78048237 ± 12.1623566372 ; ; b_{1} = 47.9428390072 ; ; b_{2} = 23.6181257328 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (b -47.9428390072) (b -23.6181257328) = 0 ; ; ; ; b > 0 ; ; ; ; b = 47.943 ; ; : 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 = 25.3 ; ; b = 47.94 ; ; c = 42.1 ; ;

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

p = a+b+c = 25.3+47.94+42.1 = 115.34 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 115.34 }{ 2 } = 57.67 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.67 * (57.67-25.3)(57.67-47.94)(57.67-42.1) } ; ; T = sqrt{ 282813.45 } = 531.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 531.8 }{ 25.3 } = 42.04 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 531.8 }{ 47.94 } = 22.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 531.8 }{ 42.1 } = 25.26 ; ;

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{ 47.94**2+42.1**2-25.3**2 }{ 2 * 47.94 * 42.1 } ) = 31° 48' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 25.3**2+42.1**2-47.94**2 }{ 2 * 25.3 * 42.1 } ) = 86° 55'58" ; ; gamma = 180° - alpha - beta = 180° - 31° 48' - 86° 55'58" = 61° 16'2" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 531.8 }{ 57.67 } = 9.22 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 25.3 }{ 2 * sin 31° 48' } = 24.01 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 47.94**2+2 * 42.1**2 - 25.3**2 } }{ 2 } = 43.306 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.1**2+2 * 25.3**2 - 47.94**2 } }{ 2 } = 25.132 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 47.94**2+2 * 25.3**2 - 42.1**2 } }{ 2 } = 32.034 ; ;
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