# Right triangle calculator (A,S)

Please enter two properties of the right triangle

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

You have entered area S and angle α.

### Right scalene triangle.

Sides: a = 5.09766953733   b = 7.84882226365   c = 9.35879325858

Area: T = 20
Perimeter: p = 22.30328505956
Semiperimeter: s = 11.15114252978

Angle ∠ A = α = 33° = 0.57659586532 rad
Angle ∠ B = β = 57° = 0.99548376736 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 7.84882226365
Height: hb = 5.09766953733
Height: hc = 4.27444484033

Median: ma = 8.25215861799
Median: mb = 6.43223365402
Median: mc = 4.67989662929

Inradius: r = 1.7933492712
Circumradius: R = 4.67989662929

Vertex coordinates: A[9.35879325858; 0] B[0; 0] C[2.77658592499; 4.27444484033]
Centroid: CG[4.04545972786; 1.42548161344]
Coordinates of the circumscribed circle: U[4.67989662929; 0]
Coordinates of the inscribed circle: I[3.30332026613; 1.7933492712]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 147° = 0.57659586532 rad
∠ B' = β' = 123° = 0.99548376736 rad
∠ C' = γ' = 90° = 1.57107963268 rad

# How did we calculate this triangle?

The calculation of the triangle progress in two phases. The first phase is such that we try to calculate all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase is the calculation of other characteristics of the triangle, 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: angle α and area S

$α = 33^\circ \ \\ S = 20$

### 2. From angle α we calculate angle β:

$α + β + 90^\circ = 180^\circ \ \\ β = 90^\circ - α = 90^\circ - 33 ^\circ = 57 ^\circ$

### 3. From area S, angle α and angle β we calculate hypotenuse c:

$c^2 \ \sin α \sin β = 2 \ T \ \\ c = \sqrt{ \dfrac{ 2 \ T }{ \sin α \sin β } } \ \\ c = \sqrt{ \dfrac{ 2 \cdot \ 20 }{ \sin 33^\circ \cdot \ \sin 57^\circ } } = 9.358$

### 4. From area S and hypotenuse c we calculate height h:

$T = \dfrac{ c \cdot \ h }{ 2 } \ \\ h = 2 \cdot \ T / c = 2 \cdot \ 20 / 9.358 = 4.274$

### 5. From hypotenuse c and angle α we calculate cathetus a:

$\sin α = a:c \ \\ a = c \cdot \ \sin α = 9.358 \cdot \ \sin(33 ^\circ ) = 5.097$

### 6. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

$c^2 = a^2+b^2 \ \\ b = \sqrt{ c^2 - a^2 } = \sqrt{ 9.358^2 - 5.097^2 } = 7.848$

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.

$a = 5.1 \ \\ b = 7.85 \ \\ c = 9.36$

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

$p = a+b+c = 5.1+7.85+9.36 = 22.3$

### 8. Semiperimeter of the triangle

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

$s = \dfrac{ p }{ 2 } = \dfrac{ 22.3 }{ 2 } = 11.15$

### 9. The triangle area - from two legs

$T = \dfrac{ ab }{ 2 } = \dfrac{ 5.1 \cdot \ 7.85 }{ 2 } = 20$

### 10. Calculate the heights of the right triangle from its area.

$h _a = b = 7.85 \ \\ \ \\ h _b = a = 5.1 \ \\ \ \\ T = \dfrac{ c h _c }{ 2 } \ \\ \ \\ \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 20 }{ 9.36 } = 4.27$

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

$\sin α = \dfrac{ a }{ c } \ \\ α = \arcsin(\dfrac{ a }{ c } ) = \arcsin(\dfrac{ 5.1 }{ 9.36 } ) = 33^\circ \ \\ \sin β = \dfrac{ b }{ c } \ \\ β = \arcsin(\dfrac{ b }{ c } ) = \arcsin(\dfrac{ 7.85 }{ 9.36 } ) = 57^\circ \ \\ γ = 90^\circ$

### 12. Inradius

An incircle of a triangle is a circle which is tangent to each side. An incircle center is called 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 the inradius and semiperimeter (half the perimeter) of a triangle is its area.

$T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 20 }{ 11.15 } = 1.79$

### 13. 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. Circumcenter (center of circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

$R = \dfrac{ c }{ 2 } = \dfrac{ 9.36 }{ 2 } = 4.68$

### 14. Calculation of medians

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. 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 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 the length of a median from the lengths of its side.

$m_a^2 = b^2 + (a/2)^2 \ \\ m_a = \sqrt{ b^2 + (a/2)^2 } = \sqrt{ 7.85^2 + (5.1/2)^2 } = 8.252 \ \\ \ \\ m_b^2 = a^2 + (b/2)^2 \ \\ m_b = \sqrt{ a^2 + (b/2)^2 } = \sqrt{ 5.1^2 + (7.85/2)^2 } = 6.432 \ \\ \ \\ m_c = R = \dfrac{ c }{ 2 } = \dfrac{ 9.36 }{ 2 } = 4.679$

The right triangle calculators compute angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in the real world. Two independent properties entirely determine any right-angled triangle. The calculator provides a step-by-step explanation for each calculation.

A right triangle is a kind of triangle that has one angle that measures C=90°. In a Right triangle, the side c that is opposite of the C=90° angle, is the longest side of the triangle and is called the hypotenuse. The variables a, b are the lengths of the shorter sides, also called legs or arms. Variables for angles are A, B, or α (alpha) and β (beta). Variable h refers to the altitude(height) of the triangle, which is the length from the vertex C to the hypotenuse of the triangle.

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