Right triangle calculator - the result

You have entered cathetus b and angle α.

Right scalene triangle.

Sides: a = 4.28878300782 b = 15.8 c = 16.37114839517

Area: T = 33.87438576177
Perimeter: p = 36.45993140299
Semiperimeter: s = 18.23296570149

Angle ∠ A = α = 15.18333333333° = 15°11' = 0.26549991581 rad
Angle ∠ B = β = 74.81766666667° = 74°49' = 1.30657971687 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: h_a = 15.8
Height: h_b = 4.28878300782
Height: h_c = 4.1388153599

Median: m_a = 15.94547913657
Median: m_b = 8.98986309736
Median: m_c = 8.18657419758

Line segment c_a = 15.24884649979
Line segment c_b = 1.12330189538

Inradius: r = 1.85881730633
Circumradius: R = 8.18657419758

Vertex coordinates: A[16.37114839517; 0] B[0; 0] C[1.12330189538; 4.1388153599]
Centroid: CG[5.83215009685; 1.3799384533]
Coordinates of the circumscribed circle: U[8.18657419758; 0]
Coordinates of the inscribed circle: I[2.43296570149; 1.85881730633]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 164.81766666667° = 164°49' = 0.26549991581 rad
∠ B' = β' = 105.18333333333° = 105°11' = 1.30657971687 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other triangle characteristics, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Input data entered: cathetus b and angle α

b = 15.8 α = 15.183 °

2. From the angle α, we calculate angle β:

α + β + 90 ° = 180 ° β = 90 ° - α = 90 ° - 15.183 ° = 74.817 °

3. From the cathetus b and angle α, we calculate hypotenuse c:

cos α = b : c c = b / cos α = 15.8 / cos (15.183 °) = 16.371

4. From the hypotenuse c and angle α, we calculate cathetus a:

sin α = a : c a = c \cdot sin α = 16.371 \cdot sin (15.183 °) = 4.288

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides SSS.

a = 4.29 b = 15.8 c = 16.37

5. The triangle perimeter is the sum of the lengths of its three sides

p = a + b + c = 4.29 + 15.8 + 16.37 = 36.46

6. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles to be given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s = \frac{p}{2} = \frac{3 6 . 4 6}{2} = 18.23

7. The triangle area - from two legs

T = \frac{a b}{2} = \frac{4 . 2 9 \cdot 1 5 . 8}{2} = 33.87

8. Calculate the heights of the right triangle from its area.

h_{a} = b = 15.8 h_{b} = a = 4.29 T = \frac{c h _{c}}{2} h_{c} = \frac{2 T}{c} = \frac{2 \cdot 3 3 . 8 7}{1 6 . 3 7} = 4.14

9. Calculation of the inner angles of the triangle - basic use of sine function

sin α = \frac{a}{c} α = arcsin (\frac{a}{c}) = arcsin (\frac{4 . 2 9}{1 6 . 3 7}) = 15 ° 1 1^{'} sin β = \frac{b}{c} β = arcsin (\frac{b}{c}) = arcsin (\frac{1 5 . 8}{1 6 . 3 7}) = 74 ° 4 9^{'} γ = 90 °

10. Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of a triangle's inradius and semiperimeter (half the perimeter) is its area.

T = r s r = \frac{T}{s} = \frac{3 3 . 8 7}{1 8 . 2 3} = 1.86

11. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R = \frac{c}{2} = \frac{1 6 . 3 7}{2} = 8.19

12. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio of 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate a median's length from its side's lengths.

m_{a}^{2} = b^{2} + (a / 2)^{2} m_{a} = b^{2} + (a / 2)^{2} = 15 . 8^{2} + (4.29 / 2)^{2} = 15.945 m_{b}^{2} = a^{2} + (b / 2)^{2} m_{b} = a^{2} + (b / 2)^{2} = 4.2 9^{2} + (15.8 / 2)^{2} = 8.989 m_{c} = R = \frac{c}{2} = \frac{1 6 . 3 7}{2} = 8.186

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