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?

1. Input data entered: side a, b and height ha.

a = 8.55 ; ; b = 8.56 ; ; h_a = 7.5 ; ;

2. From side a we calculate T:

T = fraction{ a h_a }{ 2 } ; ; ; ; T = fraction{ 8.55 * 7.5 }{ 2 } = 32.063 ; ;


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 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 8.56**2+14.73**2-8.55**2 }{ 2 * 8.56 * 14.73 } ) = 30° 34'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.55**2+14.73**2-8.56**2 }{ 2 * 8.55 * 14.73 } ) = 30° 36'42" ; ; gamma = 180° - alpha - beta = 180° - 30° 34'19" - 30° 36'42" = 118° 48'59" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.56**2+2 * 14.73**2 - 8.55**2 } }{ 2 } = 11.262 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.73**2+2 * 8.55**2 - 8.56**2 } }{ 2 } = 11.256 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.56**2+2 * 8.55**2 - 14.73**2 } }{ 2 } = 4.354 ; ;







#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

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How did we calculate this triangle?

1. Input data entered: side a, b and height ha.

a = 8.55 ; ; b = 8.56 ; ; h_a = 7.5 ; ; : Nr. 1

2. From side a we calculate T:

T = fraction{ a h_a }{ 2 } ; ; ; ; T = fraction{ 8.55 * 7.5 }{ 2 } = 32.063 ; ; : 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 = 8.55 ; ; b = 8.56 ; ; c = 8.71 ; ;

3. 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 ; ;

4. Semiperimeter of the triangle

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

5. 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 ; ;

6. 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 ; ;

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{ 8.56**2+8.71**2-8.55**2 }{ 2 * 8.56 * 8.71 } ) = 59° 21'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.55**2+8.71**2-8.56**2 }{ 2 * 8.55 * 8.71 } ) = 59° 27'54" ; ; gamma = 180° - alpha - beta = 180° - 59° 21'6" - 59° 27'54" = 61° 11'1" ; ;

8. Inradius

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

9. Circumradius

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

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.56**2+2 * 8.71**2 - 8.55**2 } }{ 2 } = 7.501 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.71**2+2 * 8.55**2 - 8.56**2 } }{ 2 } = 7.493 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.56**2+2 * 8.55**2 - 8.71**2 } }{ 2 } = 7.364 ; ;
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