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=8.3; b=10.1; c=2.5944440031 and a=8.3; b=10.1; c=12.76657604741.

#1 Obtuse scalene triangle.

Sides: a = 8.3   b = 10.1   c = 2.5944440031

Area: T = 8.50990177739
Perimeter: p = 20.9944440031
Semiperimeter: s = 10.49772200155

Angle ∠ A = α = 40.5° = 40°30' = 0.70768583471 rad
Angle ∠ B = β = 127.7877173258° = 127°47'14″ = 2.23303069152 rad
Angle ∠ C = γ = 11.7132826742° = 11°42'46″ = 0.20444273914 rad

Height: ha = 2.05503657286
Height: hb = 1.68549540146
Height: hc = 6.55994252881

Median: ma = 6.09549207983
Median: mb = 3.50882844151
Median: mc = 9.15224434023

Inradius: r = 0.81105972592
Circumradius: R = 6.39900415294

Vertex coordinates: A[2.5944440031; 0] B[0; 0] C[-5.08656602216; 6.55994252881]
Centroid: CG[-0.83304067302; 2.1866475096]
Coordinates of the circumscribed circle: U[1.29772200155; 6.2576984176]
Coordinates of the inscribed circle: I[0.39772200155; 0.81105972592]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5° = 139°30' = 0.70768583471 rad
∠ B' = β' = 52.2132826742° = 52°12'46″ = 2.23303069152 rad
∠ C' = γ' = 168.2877173258° = 168°17'14″ = 0.20444273914 rad




How did we calculate this triangle?

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 = 8.3 ; ; b = 10.1 ; ; c = 2.59 ; ;

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

p = a+b+c = 8.3+10.1+2.59 = 20.99 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.99 }{ 2 } = 10.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.5 * (10.5-8.3)(10.5-10.1)(10.5-2.59) } ; ; T = sqrt{ 72.4 } = 8.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8.51 }{ 8.3 } = 2.05 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8.51 }{ 10.1 } = 1.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8.51 }{ 2.59 } = 6.56 ; ;

5. 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{ 8.3**2-10.1**2-2.59**2 }{ 2 * 10.1 * 2.59 } ) = 40° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.1**2-8.3**2-2.59**2 }{ 2 * 8.3 * 2.59 } ) = 127° 47'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 2.59**2-8.3**2-10.1**2 }{ 2 * 10.1 * 8.3 } ) = 11° 42'46" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8.51 }{ 10.5 } = 0.81 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.3 }{ 2 * sin 40° 30' } = 6.39 ; ;





#2 Acute scalene triangle.

Sides: a = 8.3   b = 10.1   c = 12.76657604741

Area: T = 41.86880260382
Perimeter: p = 31.16657604741
Semiperimeter: s = 15.58328802371

Angle ∠ A = α = 40.5° = 40°30' = 0.70768583471 rad
Angle ∠ B = β = 52.2132826742° = 52°12'46″ = 0.91112857384 rad
Angle ∠ C = γ = 87.2877173258° = 87°17'14″ = 1.52334485681 rad

Height: ha = 10.08986809731
Height: hb = 8.29106982254
Height: hc = 6.55994252881

Median: ma = 10.73661455021
Median: mb = 9.50991966139
Median: mc = 6.68664669205

Inradius: r = 2.68767963689
Circumradius: R = 6.39900415294

Vertex coordinates: A[12.76657604741; 0] B[0; 0] C[5.08656602216; 6.55994252881]
Centroid: CG[5.95504735652; 2.1866475096]
Coordinates of the circumscribed circle: U[6.38328802371; 0.30224411122]
Coordinates of the inscribed circle: I[5.48328802371; 2.68767963689]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.5° = 139°30' = 0.70768583471 rad
∠ B' = β' = 127.7877173258° = 127°47'14″ = 0.91112857384 rad
∠ C' = γ' = 92.7132826742° = 92°42'46″ = 1.52334485681 rad

Calculate another triangle

How did we calculate this triangle?

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 = 8.3 ; ; b = 10.1 ; ; c = 12.77 ; ;

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

p = a+b+c = 8.3+10.1+12.77 = 31.17 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.17 }{ 2 } = 15.58 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.58 * (15.58-8.3)(15.58-10.1)(15.58-12.77) } ; ; T = sqrt{ 1752.93 } = 41.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.87 }{ 8.3 } = 10.09 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.87 }{ 10.1 } = 8.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.87 }{ 12.77 } = 6.56 ; ;

5. 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{ 8.3**2-10.1**2-12.77**2 }{ 2 * 10.1 * 12.77 } ) = 40° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.1**2-8.3**2-12.77**2 }{ 2 * 8.3 * 12.77 } ) = 52° 12'46" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.77**2-8.3**2-10.1**2 }{ 2 * 10.1 * 8.3 } ) = 87° 17'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.87 }{ 15.58 } = 2.69 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.3 }{ 2 * sin 40° 30' } = 6.39 ; ; : Nr. 1




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