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=77.73303539385; b=123; c=165 and a=155.6154883853; b=123; c=165.

#1 Obtuse scalene triangle.

Sides: a = 77.73303539385   b = 123   c = 165

Area: T = 4534.502198085
Perimeter: p = 365.7330353939
Semiperimeter: s = 182.8655176969

Angle ∠ A = α = 26.54223402826° = 26°32'32″ = 0.46332512291 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 108.4587659717° = 108°27'28″ = 1.89329432611 rad

Height: ha = 116.6732618896
Height: hb = 73.73217395261
Height: hc = 54.9643660374

Median: ma = 140.2377291828
Median: mb = 113.3633371341
Median: mc = 61.47656371395

Inradius: r = 24.79769682145
Circumradius: R = 86.97441340859

Vertex coordinates: A[165; 0] B[0; 0] C[54.9643660374; 54.9643660374]
Centroid: CG[73.32112201247; 18.32112201247]
Coordinates of the circumscribed circle: U[82.5; -27.5366339626]
Coordinates of the inscribed circle: I[59.86551769693; 24.79769682145]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.4587659717° = 153°27'28″ = 0.46332512291 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 71.54223402826° = 71°32'32″ = 1.89329432611 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 = 77.73 ; ; b = 123 ; ; c = 165 ; ;

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

p = a+b+c = 77.73+123+165 = 365.73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 365.73 }{ 2 } = 182.87 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 182.87 * (182.87-77.73)(182.87-123)(182.87-165) } ; ; T = sqrt{ 20561708.21 } = 4534.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4534.5 }{ 77.73 } = 116.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4534.5 }{ 123 } = 73.73 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4534.5 }{ 165 } = 54.96 ; ;

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{ 77.73**2-123**2-165**2 }{ 2 * 123 * 165 } ) = 26° 32'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 123**2-77.73**2-165**2 }{ 2 * 77.73 * 165 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 165**2-77.73**2-123**2 }{ 2 * 123 * 77.73 } ) = 108° 27'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4534.5 }{ 182.87 } = 24.8 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 77.73 }{ 2 * sin 26° 32'32" } = 86.97 ; ;





#2 Acute scalene triangle.

Sides: a = 155.6154883853   b = 123   c = 165

Area: T = 9077.998801915
Perimeter: p = 443.6154883853
Semiperimeter: s = 221.8077441926

Angle ∠ A = α = 63.45876597174° = 63°27'28″ = 1.10875450977 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 71.54223402826° = 71°32'32″ = 1.24986493925 rad

Height: ha = 116.6732618896
Height: hb = 147.6109723889
Height: hc = 110.0366339626

Median: ma = 122.9765615391
Median: mb = 148.1165650889
Median: mc = 113.4299476056

Inradius: r = 40.92773825094
Circumradius: R = 86.97441340859

Vertex coordinates: A[165; 0] B[0; 0] C[110.0366339626; 110.0366339626]
Centroid: CG[91.67987798753; 36.67987798753]
Coordinates of the circumscribed circle: U[82.5; 27.5366339626]
Coordinates of the inscribed circle: I[98.80774419265; 40.92773825094]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116.5422340283° = 116°32'32″ = 1.10875450977 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 108.4587659717° = 108°27'28″ = 1.24986493925 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 = 155.61 ; ; b = 123 ; ; c = 165 ; ;

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

p = a+b+c = 155.61+123+165 = 443.61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 443.61 }{ 2 } = 221.81 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 221.81 * (221.81-155.61)(221.81-123)(221.81-165) } ; ; T = sqrt{ 82410048.04 } = 9078 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9078 }{ 155.61 } = 116.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9078 }{ 123 } = 147.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9078 }{ 165 } = 110.04 ; ;

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{ 155.61**2-123**2-165**2 }{ 2 * 123 * 165 } ) = 63° 27'28" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 123**2-155.61**2-165**2 }{ 2 * 155.61 * 165 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 165**2-155.61**2-123**2 }{ 2 * 123 * 155.61 } ) = 71° 32'32" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9078 }{ 221.81 } = 40.93 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 155.61 }{ 2 * sin 63° 27'28" } = 86.97 ; ;




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