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=71.64404339892; b=35; c=105 and a=136.7944257855; b=35; c=105.

#1 Obtuse scalene triangle.

Sides: a = 71.64404339892   b = 35   c = 105

Area: T = 458.3665564205
Perimeter: p = 211.6440433989
Semiperimeter: s = 105.8220216995

Angle ∠ A = α = 14.44550067074° = 14°26'42″ = 0.25221129275 rad
Angle ∠ B = β = 7° = 0.12221730476 rad
Angle ∠ C = γ = 158.5554993293° = 158°33'18″ = 2.76773066784 rad

Height: ha = 12.79662810575
Height: hb = 26.19223179546
Height: hc = 8.73107726515

Median: ma = 69.5843849092
Median: mb = 88.16113628019
Median: mc = 20.55330019969

Inradius: r = 4.3321550031
Circumradius: R = 143.5966408342

Vertex coordinates: A[105; 0] B[0; 0] C[71.10664370579; 8.73107726515]
Centroid: CG[58.7022145686; 2.91102575505]
Coordinates of the circumscribed circle: U[52.5; -133.6555072813]
Coordinates of the inscribed circle: I[70.82202169946; 4.3321550031]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.5554993293° = 165°33'18″ = 0.25221129275 rad
∠ B' = β' = 173° = 0.12221730476 rad
∠ C' = γ' = 21.44550067074° = 21°26'42″ = 2.76773066784 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 = 71.64 ; ; b = 35 ; ; c = 105 ; ;

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

p = a+b+c = 71.64+35+105 = 211.64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 211.64 }{ 2 } = 105.82 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 105.82 * (105.82-71.64)(105.82-35)(105.82-105) } ; ; T = sqrt{ 210098.99 } = 458.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 458.37 }{ 71.64 } = 12.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 458.37 }{ 35 } = 26.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 458.37 }{ 105 } = 8.73 ; ;

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{ 71.64**2-35**2-105**2 }{ 2 * 35 * 105 } ) = 14° 26'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-71.64**2-105**2 }{ 2 * 71.64 * 105 } ) = 7° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 105**2-71.64**2-35**2 }{ 2 * 35 * 71.64 } ) = 158° 33'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 458.37 }{ 105.82 } = 4.33 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 71.64 }{ 2 * sin 14° 26'42" } = 143.6 ; ;





#2 Obtuse scalene triangle.

Sides: a = 136.7944257855   b = 35   c = 105

Area: T = 875.2298885288
Perimeter: p = 276.7944257855
Semiperimeter: s = 138.3977128928

Angle ∠ A = α = 151.5554993293° = 151°33'18″ = 2.64551336308 rad
Angle ∠ B = β = 7° = 0.12221730476 rad
Angle ∠ C = γ = 21.44550067074° = 21°26'42″ = 0.37442859752 rad

Height: ha = 12.79662810575
Height: hb = 50.01330791593
Height: hc = 16.67110263864

Median: ma = 38.03772548227
Median: mb = 120.6765533938
Median: mc = 84.92769361929

Inradius: r = 6.32440393213
Circumradius: R = 143.5966408342

Vertex coordinates: A[105; 0] B[0; 0] C[135.7754614201; 16.67110263864]
Centroid: CG[80.25882047337; 5.55770087955]
Coordinates of the circumscribed circle: U[52.5; 133.6555072813]
Coordinates of the inscribed circle: I[103.3977128928; 6.32440393213]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 28.44550067074° = 28°26'42″ = 2.64551336308 rad
∠ B' = β' = 173° = 0.12221730476 rad
∠ C' = γ' = 158.5554993293° = 158°33'18″ = 0.37442859752 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 = 136.79 ; ; b = 35 ; ; c = 105 ; ;

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

p = a+b+c = 136.79+35+105 = 276.79 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 276.79 }{ 2 } = 138.4 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 138.4 * (138.4-136.79)(138.4-35)(138.4-105) } ; ; T = sqrt{ 766025.6 } = 875.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 875.23 }{ 136.79 } = 12.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 875.23 }{ 35 } = 50.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 875.23 }{ 105 } = 16.67 ; ;

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{ 136.79**2-35**2-105**2 }{ 2 * 35 * 105 } ) = 151° 33'18" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 35**2-136.79**2-105**2 }{ 2 * 136.79 * 105 } ) = 7° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 105**2-136.79**2-35**2 }{ 2 * 35 * 136.79 } ) = 21° 26'42" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 875.23 }{ 138.4 } = 6.32 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 136.79 }{ 2 * sin 151° 33'18" } = 143.6 ; ;




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