Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=100; b=67; c=54.72441969737 and a=100; b=67; c=100.7054995318.

#1 Obtuse scalene triangle.

Sides: a = 100   b = 67   c = 54.72441969737

Area: T = 1721.953265197
Perimeter: p = 221.7244196974
Semiperimeter: s = 110.8622098487

Angle ∠ A = α = 110.068831319° = 110°4'6″ = 1.92110544673 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 30.93216868105° = 30°55'54″ = 0.5439859778 rad

Height: ha = 34.43990530394
Height: hb = 51.40215717005
Height: hc = 62.9322039105

Median: ma = 35.24401598635
Median: mb = 73.31552021562
Median: mc = 80.59766225496

Inradius: r = 15.53223837044
Circumradius: R = 53.23220269237

Vertex coordinates: A[54.72441969737; 0] B[0; 0] C[77.71545961457; 62.9322039105]
Centroid: CG[44.14662643731; 20.97773463683]
Coordinates of the circumscribed circle: U[27.36220984868; 45.66114088351]
Coordinates of the inscribed circle: I[43.86220984868; 15.53223837044]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 69.93216868105° = 69°55'54″ = 1.92110544673 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 149.068831319° = 149°4'6″ = 0.5439859778 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 = 100 ; ; b = 67 ; ; c = 54.72 ; ;

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

p = a+b+c = 100+67+54.72 = 221.72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 221.72 }{ 2 } = 110.86 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 110.86 * (110.86-100)(110.86-67)(110.86-54.72) } ; ; T = sqrt{ 2965120.94 } = 1721.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1721.95 }{ 100 } = 34.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1721.95 }{ 67 } = 51.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1721.95 }{ 54.72 } = 62.93 ; ;

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{ 100**2-67**2-54.72**2 }{ 2 * 67 * 54.72 } ) = 110° 4'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 67**2-100**2-54.72**2 }{ 2 * 100 * 54.72 } ) = 39° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 54.72**2-100**2-67**2 }{ 2 * 67 * 100 } ) = 30° 55'54" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1721.95 }{ 110.86 } = 15.53 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 110° 4'6" } = 53.23 ; ;





#2 Acute scalene triangle.

Sides: a = 100   b = 67   c = 100.7054995318

Area: T = 3168.78553517
Perimeter: p = 267.7054995318
Semiperimeter: s = 133.8522497659

Angle ∠ A = α = 69.93216868105° = 69°55'54″ = 1.22105381863 rad
Angle ∠ B = β = 39° = 0.68106784083 rad
Angle ∠ C = γ = 71.06883131895° = 71°4'6″ = 1.2440376059 rad

Height: ha = 63.3765707034
Height: hb = 94.59106075135
Height: hc = 62.9322039105

Median: ma = 69.39219883054
Median: mb = 94.59765012089
Median: mc = 68.62330717727

Inradius: r = 23.67437110411
Circumradius: R = 53.23220269237

Vertex coordinates: A[100.7054995318; 0] B[0; 0] C[77.71545961457; 62.9322039105]
Centroid: CG[59.47331971545; 20.97773463683]
Coordinates of the circumscribed circle: U[50.35224976589; 17.27106302699]
Coordinates of the inscribed circle: I[66.85224976589; 23.67437110411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110.068831319° = 110°4'6″ = 1.22105381863 rad
∠ B' = β' = 141° = 0.68106784083 rad
∠ C' = γ' = 108.932168681° = 108°55'54″ = 1.2440376059 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 = 100 ; ; b = 67 ; ; c = 100.7 ; ;

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

p = a+b+c = 100+67+100.7 = 267.7 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 267.7 }{ 2 } = 133.85 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.85 * (133.85-100)(133.85-67)(133.85-100.7) } ; ; T = sqrt{ 10041200.61 } = 3168.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3168.79 }{ 100 } = 63.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3168.79 }{ 67 } = 94.59 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3168.79 }{ 100.7 } = 62.93 ; ;

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{ 100**2-67**2-100.7**2 }{ 2 * 67 * 100.7 } ) = 69° 55'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 67**2-100**2-100.7**2 }{ 2 * 100 * 100.7 } ) = 39° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 100.7**2-100**2-67**2 }{ 2 * 67 * 100 } ) = 71° 4'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3168.79 }{ 133.85 } = 23.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 100 }{ 2 * sin 69° 55'54" } = 53.23 ; ;




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