Triangle calculator

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

You have entered side a, c and height hc.

Triangle has two solutions: a=7.42; b=19.17114422912; c=12.5 and a=7.42; b=7.4210822089; c=12.5.

#1 Obtuse scalene triangle.

Sides: a = 7.42   b = 19.17114422912   c = 12.5

Area: T = 25
Perimeter: p = 39.09114422912
Semiperimeter: s = 19.54657211456

Angle ∠ A = α = 12.04328794408° = 12°2'34″ = 0.21101878977 rad
Angle ∠ B = β = 147.3798725566° = 147°22'43″ = 2.57222440085 rad
Angle ∠ C = γ = 20.57883949933° = 20°34'42″ = 0.35991607474 rad

Height: ha = 6.73985444744
Height: hb = 2.60880458236
Height: hc = 4

Median: ma = 15.75222379287
Median: mb = 3.71104110445
Median: mc = 13.12439399481

Inradius: r = 1.27990523212
Circumradius: R = 17.78215127251

Vertex coordinates: A[12.5; 0] B[0; 0] C[-6.25495119809; 4]
Centroid: CG[2.08334960064; 1.33333333333]
Coordinates of the circumscribed circle: U[6.25; 16.64769124702]
Coordinates of the inscribed circle: I[0.37442788544; 1.27990523212]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 167.9577120559° = 167°57'26″ = 0.21101878977 rad
∠ B' = β' = 32.62112744342° = 32°37'17″ = 2.57222440085 rad
∠ C' = γ' = 159.4221605007° = 159°25'18″ = 0.35991607474 rad




How did we calculate this triangle?

1. Input data entered: side a, c and height hc.

a = 7.42 ; ; c = 12.5 ; ; h_c = 4 ; ;

2. From side c we calculate T:

T = fraction{ c h_c }{ 2 } ; ; ; ; T = fraction{ 12.5 * 4 }{ 2 } = 25 ; ;


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 = 7.42 ; ; b = 19.17 ; ; c = 12.5 ; ;

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

p = a+b+c = 7.42+19.17+12.5 = 39.09 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 39.09 }{ 2 } = 19.55 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 19.55 * (19.55-7.42)(19.55-19.17)(19.55-12.5) } ; ; T = sqrt{ 625 } = 25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25 }{ 7.42 } = 6.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25 }{ 19.17 } = 2.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25 }{ 12.5 } = 4 ; ;

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{ 19.17**2+12.5**2-7.42**2 }{ 2 * 19.17 * 12.5 } ) = 12° 2'34" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.42**2+12.5**2-19.17**2 }{ 2 * 7.42 * 12.5 } ) = 147° 22'43" ; ; gamma = 180° - alpha - beta = 180° - 12° 2'34" - 147° 22'43" = 20° 34'42" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25 }{ 19.55 } = 1.28 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.42 }{ 2 * sin 12° 2'34" } = 17.78 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.17**2+2 * 12.5**2 - 7.42**2 } }{ 2 } = 15.752 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 7.42**2 - 19.17**2 } }{ 2 } = 3.71 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.17**2+2 * 7.42**2 - 12.5**2 } }{ 2 } = 13.124 ; ;







#2 Obtuse scalene triangle.

Sides: a = 7.42   b = 7.4210822089   c = 12.5

Area: T = 25
Perimeter: p = 27.3410822089
Semiperimeter: s = 13.67704110445

Angle ∠ A = α = 32.61772119332° = 32°37'2″ = 0.56992777411 rad
Angle ∠ B = β = 32.62112744342° = 32°37'17″ = 0.56993486451 rad
Angle ∠ C = γ = 114.7621513633° = 114°45'41″ = 2.00329662675 rad

Height: ha = 6.73985444744
Height: hb = 6.73877979691
Height: hc = 4

Median: ma = 9.58661984247
Median: mb = 9.58657211456
Median: mc = 44.0000000298

Inradius: r = 1.82987672491
Circumradius: R = 6.88328124875

Vertex coordinates: A[12.5; 0] B[0; 0] C[6.25495119809; 4]
Centroid: CG[6.2549837327; 1.33333333333]
Coordinates of the circumscribed circle: U[6.25; -2.88328124702]
Coordinates of the inscribed circle: I[6.25495889555; 1.82987672491]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.3832788067° = 147°22'58″ = 0.56992777411 rad
∠ B' = β' = 147.3798725566° = 147°22'43″ = 0.56993486451 rad
∠ C' = γ' = 65.23884863674° = 65°14'19″ = 2.00329662675 rad

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How did we calculate this triangle?

1. Input data entered: side a, c and height hc.

a = 7.42 ; ; c = 12.5 ; ; h_c = 4 ; ; : Nr. 1

2. From side c we calculate T:

T = fraction{ c h_c }{ 2 } ; ; ; ; T = fraction{ 12.5 * 4 }{ 2 } = 25 ; ; : 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 = 7.42 ; ; b = 7.42 ; ; c = 12.5 ; ;

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

p = a+b+c = 7.42+7.42+12.5 = 27.34 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 27.34 }{ 2 } = 13.67 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.67 * (13.67-7.42)(13.67-7.42)(13.67-12.5) } ; ; T = sqrt{ 625 } = 25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25 }{ 7.42 } = 6.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25 }{ 7.42 } = 6.74 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25 }{ 12.5 } = 4 ; ;

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{ 7.42**2+12.5**2-7.42**2 }{ 2 * 7.42 * 12.5 } ) = 32° 37'2" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7.42**2+12.5**2-7.42**2 }{ 2 * 7.42 * 12.5 } ) = 32° 37'17" ; ; gamma = 180° - alpha - beta = 180° - 32° 37'2" - 32° 37'17" = 114° 45'41" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25 }{ 13.67 } = 1.83 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7.42 }{ 2 * sin 32° 37'2" } = 6.88 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.42**2+2 * 12.5**2 - 7.42**2 } }{ 2 } = 9.586 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 12.5**2+2 * 7.42**2 - 7.42**2 } }{ 2 } = 9.586 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.42**2+2 * 7.42**2 - 12.5**2 } }{ 2 } = 4 ; ;
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