Triangle calculator

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

You have entered side a, c and area S.

Triangle has two solutions: a=80; b=138.6832601023; c=90 and a=80; b=98.82988225849; c=90.

#1 Obtuse scalene triangle.

Sides: a = 80   b = 138.6832601023   c = 90

Area: T = 3400
Perimeter: p = 308.6832601023
Semiperimeter: s = 154.3411300512

Angle ∠ A = α = 33.01216253219° = 33°42″ = 0.57661615533 rad
Angle ∠ B = β = 109.1888136454° = 109°11'17″ = 1.90656924852 rad
Angle ∠ C = γ = 37.88002382244° = 37°48'1″ = 0.66597386151 rad

Height: ha = 85
Height: hb = 49.03328271163
Height: hc = 75.55655555556

Median: ma = 109.8477311816
Median: mb = 49.41444112925
Median: mc = 103.8821817048

Inradius: r = 22.02991003687
Circumradius: R = 73.42202005415

Vertex coordinates: A[90; 0] B[0; 0] C[-26.29436879249; 75.55655555556]
Centroid: CG[21.23554373584; 25.18551851852]
Coordinates of the circumscribed circle: U[45; 58.0133152367]
Coordinates of the inscribed circle: I[15.65986994885; 22.02991003687]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.9888374678° = 146°59'18″ = 0.57661615533 rad
∠ B' = β' = 70.81218635463° = 70°48'43″ = 1.90656924852 rad
∠ C' = γ' = 142.2199761776° = 142°11'59″ = 0.66597386151 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 = 80 ; ; b = 138.68 ; ; c = 90 ; ;

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

p = a+b+c = 80+138.68+90 = 308.68 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 308.68 }{ 2 } = 154.34 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 154.34 * (154.34-80)(154.34-138.68)(154.34-90) } ; ; T = sqrt{ 11560000 } = 3400 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 80 } = 85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3400 }{ 138.68 } = 49.03 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3400 }{ 90 } = 75.56 ; ;

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{ 80**2-138.68**2-90**2 }{ 2 * 138.68 * 90 } ) = 33° 42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 138.68**2-80**2-90**2 }{ 2 * 80 * 90 } ) = 109° 11'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 90**2-80**2-138.68**2 }{ 2 * 138.68 * 80 } ) = 37° 48'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3400 }{ 154.34 } = 22.03 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 80 }{ 2 * sin 33° 42" } = 73.42 ; ;





#2 Acute scalene triangle.

Sides: a = 80   b = 98.82988225849   c = 90

Area: T = 3400
Perimeter: p = 268.8298822585
Semiperimeter: s = 134.4144411293

Angle ∠ A = α = 49.86333583292° = 49°51'48″ = 0.87702797789 rad
Angle ∠ B = β = 70.81218635463° = 70°48'43″ = 1.23659001684 rad
Angle ∠ C = γ = 59.32547781245° = 59°19'29″ = 1.03554127063 rad

Height: ha = 85
Height: hb = 68.80658384401
Height: hc = 75.55655555556

Median: ma = 85.63662545115
Median: mb = 69.34113005115
Median: mc = 77.83768041916

Inradius: r = 25.29549067537
Circumradius: R = 52.32111413685

Vertex coordinates: A[90; 0] B[0; 0] C[26.29436879249; 75.55655555556]
Centroid: CG[38.76545626416; 25.18551851852]
Coordinates of the circumscribed circle: U[45; 26.69327299859]
Coordinates of the inscribed circle: I[35.58655887075; 25.29549067537]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.1376641671° = 130°8'12″ = 0.87702797789 rad
∠ B' = β' = 109.1888136454° = 109°11'17″ = 1.23659001684 rad
∠ C' = γ' = 120.6755221875° = 120°40'31″ = 1.03554127063 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 = 80 ; ; b = 98.83 ; ; c = 90 ; ;

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

p = a+b+c = 80+98.83+90 = 268.83 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 268.83 }{ 2 } = 134.41 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 134.41 * (134.41-80)(134.41-98.83)(134.41-90) } ; ; T = sqrt{ 11560000 } = 3400 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 80 } = 85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3400 }{ 98.83 } = 68.81 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3400 }{ 90 } = 75.56 ; ;

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{ 80**2-98.83**2-90**2 }{ 2 * 98.83 * 90 } ) = 49° 51'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 98.83**2-80**2-90**2 }{ 2 * 80 * 90 } ) = 70° 48'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 90**2-80**2-98.83**2 }{ 2 * 98.83 * 80 } ) = 59° 19'29" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3400 }{ 134.41 } = 25.29 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 80 }{ 2 * sin 49° 51'48" } = 52.32 ; ;




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