Triangle calculator

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

You have entered side a, b and height ha.

Triangle has two solutions: a=8.55; b=8.56; c=14.72985480637 and a=8.55; b=8.56; c=8.70875870329.

#1 Obtuse scalene triangle.

Sides: a = 8.55   b = 8.56   c = 14.72985480637

Area: T = 32.06325
Perimeter: p = 31.83985480637
Semiperimeter: s = 15.91992740318

Angle ∠ A = α = 30.57219712649° = 30°34'19″ = 0.53435815574 rad
Angle ∠ B = β = 30.61215662609° = 30°36'42″ = 0.53442726204 rad
Angle ∠ C = γ = 118.8166462474° = 118°48'59″ = 2.07437384757 rad

Height: ha = 7.5
Height: hb = 7.49112383178
Height: hc = 4.35437896419

Median: ma = 11.26217156345
Median: mb = 11.25660167924
Median: mc = 4.35437935165

Inradius: r = 2.01440679742
Circumradius: R = 8.40550914283

Vertex coordinates: A[14.72985480637; 0] B[0; 0] C[7.35884655842; 4.35437896419]
Centroid: CG[7.36223378826; 1.4511263214]
Coordinates of the circumscribed circle: U[7.36442740318; -4.05112997794]
Coordinates of the inscribed circle: I[7.35992740318; 2.01440679742]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.4288028735° = 149°25'41″ = 0.53435815574 rad
∠ B' = β' = 149.3888433739° = 149°23'18″ = 0.53442726204 rad
∠ C' = γ' = 61.18435375258° = 61°11'1″ = 2.07437384757 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 = 8.55 ; ; b = 8.56 ; ; c = 14.73 ; ;

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

p = a+b+c = 8.55+8.56+14.73 = 31.84 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 31.84 }{ 2 } = 15.92 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.92 * (15.92-8.55)(15.92-8.56)(15.92-14.73) } ; ; T = sqrt{ 1028 } = 32.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 32.06 }{ 8.55 } = 7.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 32.06 }{ 8.56 } = 7.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 32.06 }{ 14.73 } = 4.35 ; ;

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{ 8.55**2-8.56**2-14.73**2 }{ 2 * 8.56 * 14.73 } ) = 30° 34'19" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.56**2-8.55**2-14.73**2 }{ 2 * 8.55 * 14.73 } ) = 30° 36'42" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.73**2-8.55**2-8.56**2 }{ 2 * 8.56 * 8.55 } ) = 118° 48'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 32.06 }{ 15.92 } = 2.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.55 }{ 2 * sin 30° 34'19" } = 8.41 ; ;





#2 Acute scalene triangle.

Sides: a = 8.55   b = 8.56   c = 8.70875870329

Area: T = 32.06325
Perimeter: p = 25.81875870329
Semiperimeter: s = 12.90987935165

Angle ∠ A = α = 59.35215897327° = 59°21'6″ = 1.03658806571 rad
Angle ∠ B = β = 59.46548727416° = 59°27'54″ = 1.03878578186 rad
Angle ∠ C = γ = 61.18435375258° = 61°11'1″ = 1.06878541778 rad

Height: ha = 7.5
Height: hb = 7.49112383178
Height: hc = 7.36442674782

Median: ma = 7.50114805851
Median: mb = 7.49329223917
Median: mc = 7.36442740318

Inradius: r = 2.48437720085
Circumradius: R = 4.96991296668

Vertex coordinates: A[8.70875870329; 0] B[0; 0] C[4.34439687511; 7.36442674782]
Centroid: CG[4.35105185947; 2.45547558261]
Coordinates of the circumscribed circle: U[4.35437935165; 2.39551475239]
Coordinates of the inscribed circle: I[4.34987935165; 2.48437720085]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120.6488410267° = 120°38'54″ = 1.03658806571 rad
∠ B' = β' = 120.5355127258° = 120°32'6″ = 1.03878578186 rad
∠ C' = γ' = 118.8166462474° = 118°48'59″ = 1.06878541778 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 = 8.55 ; ; b = 8.56 ; ; c = 8.71 ; ;

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

p = a+b+c = 8.55+8.56+8.71 = 25.82 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25.82 }{ 2 } = 12.91 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.91 * (12.91-8.55)(12.91-8.56)(12.91-8.71) } ; ; T = sqrt{ 1028 } = 32.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 32.06 }{ 8.55 } = 7.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 32.06 }{ 8.56 } = 7.49 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 32.06 }{ 8.71 } = 7.36 ; ;

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{ 8.55**2-8.56**2-8.71**2 }{ 2 * 8.56 * 8.71 } ) = 59° 21'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.56**2-8.55**2-8.71**2 }{ 2 * 8.55 * 8.71 } ) = 59° 27'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.71**2-8.55**2-8.56**2 }{ 2 * 8.56 * 8.55 } ) = 61° 11'1" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 32.06 }{ 12.91 } = 2.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.55 }{ 2 * sin 59° 21'6" } = 4.97 ; ;




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