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?

1. Input data entered: side a, c and area S.

a = 80 ; ; c = 90 ; ; S = 3400 ; ;

2. From area T and side a we calculate h_a - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 80 } = 85 ; ;


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 ; ;

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

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

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 138.68**2+2 * 90**2 - 80**2 } }{ 2 } = 109.847 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 80**2 - 138.68**2 } }{ 2 } = 49.414 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 138.68**2+2 * 80**2 - 90**2 } }{ 2 } = 103.882 ; ;







#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?

1. Input data entered: side a, c and area S.

a = 80 ; ; c = 90 ; ; S = 3400 ; ; : Nr. 1

2. From area T and side a we calculate h_a - The area of the triangle is the product of the length of the base and the height divided by two:

T = fraction{ a h_a }{ 2 } ; ; ; ; h_a = fraction{ 2 T }{ a } = fraction{ 2 * 3400 }{ 80 } = 85 ; ; : Nr. 1


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 ; ;

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

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

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 98.83**2+2 * 90**2 - 80**2 } }{ 2 } = 85.636 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 90**2+2 * 80**2 - 98.83**2 } }{ 2 } = 69.341 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 98.83**2+2 * 80**2 - 90**2 } }{ 2 } = 77.837 ; ;
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