Right triangle calculator (A,a) - the result

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R

You have entered cathetus a and angle α.

Right scalene triangle.

The lengths of the sides of the triangle:
a = 145
b = 261.58769245144
c = 299.0866474246

Area: T = 18965.0522027291
Perimeter: p = 705.67333987603
Semiperimeter: s = 352.83766993802

Angle ∠ A = α = 29° = 0.50661454831 rad
Angle ∠ B = β = 61° = 1.06546508437 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Altitude (height) to the side a: ha = 261.58769245144
Altitude (height) to the side b: hb = 145
Altitude (height) to the side c: hc = 126.82198575352

Median: ma = 271.44879122721
Median: mb = 195.2743986412
Median: mc = 149.5433237123

Line segment ca = 228.78990793102
Line segment cb = 70.29773949357

Inradius: r = 53.75502251342
Circumradius: R = 149.5433237123

Vertex coordinates: A[299.0866474246; 0] B[0; 0] C[70.29773949357; 126.82198575352]
Centroid: CG[123.12879563939; 42.27332858451]
Coordinates of the circumscribed circle: U[149.5433237123; -0]
Coordinates of the inscribed circle: I[91.25497748658; 53.75502251342]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 151° = 0.50661454831 rad
∠ B' = β' = 119° = 1.06546508437 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 a and angle α

a=145 α=29°

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

α+β+90°=180° β=90°α=90°29°=61°

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

sinα=a:c c=a/sinα=145/sin(29°)=299.086

4. From the cathetus a and hypotenuse c, we calculate cathetus b - Pythagorean theorem:

c2=a2+b2 b=c2a2=299.08621452=261.587

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=145 b=261.59 c=299.09

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


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.


7. The triangle area - from two legs

T=2ab=2145 261.59=18965.05

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

ha=b=261.59  hb=a=145  T=2chc   hc=c2 T=299.092 18965.05=126.82

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

sinα=ca α=arcsin(ca)=arcsin(299.09145)=29° sinβ=cb β=arcsin(cb)=arcsin(299.09261.59)=61° 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=rs r=sT=352.8418965.05=53.75

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.


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.

ma2=b2+(a/2)2 ma=b2+(a/2)2=261.592+(145/2)2=271.448  mb2=a2+(b/2)2 mb=a2+(b/2)2=1452+(261.59/2)2=195.274  mc=R=2c=2299.09=149.543

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See triangle basics on Wikipedia or more details on solving triangles.