# Right triangle calculator (A,b)

Please enter two properties of the right triangle

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

You have entered cathetus b, angle α and angle γ.

### Right scalene triangle.

Sides: a = 1.45497474683   b = 3.5   c = 3.7888372701

Area: T = 2.53770580695
Perimeter: p = 8.73881201693
Semiperimeter: s = 4.36990600847

Angle ∠ A = α = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 3.5
Height: hb = 1.45497474683
Height: hc = 1.33993920133

Median: ma = 3.57442750217
Median: mb = 2.27325025241
Median: mc = 1.89441863505

Vertex coordinates: A[3.7888372701; 0] B[0; 0] C[0.55547943372; 1.33993920133]
Centroid: CG[1.44877223461; 0.44664640044]
Coordinates of the circumscribed circle: U[1.89441863505; -0]
Coordinates of the inscribed circle: I[0.86990600847; 0.58106873836]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 157.5° = 157°30' = 0.39326990817 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 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: cathetus b, angle α and angle γ

$b = 3.5 \ \\ α = 22.5^\circ \ \\ γ = 90^\circ$

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

$α + β + 90^\circ = 180^\circ \ \\ β = 90^\circ - α = 90^\circ - 22.5 ^\circ = 67.5 ^\circ$

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

$\cos α = b:c \ \\ c = b/\cos α = 3.5/\cos(22.5 ^\circ ) = 3.788$

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

$\sin α = a:c \ \\ a = c \cdot \ \sin α = 3.788 \cdot \ \sin(22.5 ^\circ ) = 1.45$

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.

$a = 1.45 \ \\ b = 3.5 \ \\ c = 3.79$

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

$p = a+b+c = 1.45+3.5+3.79 = 8.74$

### 6. 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.

$s = \dfrac{ p }{ 2 } = \dfrac{ 8.74 }{ 2 } = 4.37$

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

$T = \dfrac{ ab }{ 2 } = \dfrac{ 1.45 \cdot \ 3.5 }{ 2 } = 2.54$

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

$h _a = b = 3.5 \ \\ \ \\ h _b = a = 1.45 \ \\ \ \\ T = \dfrac{ c h _c }{ 2 } \ \\ \ \\ \ \\ h _c = \dfrac{ 2 \ T }{ c } = \dfrac{ 2 \cdot \ 2.54 }{ 3.79 } = 1.34$

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

$\sin α = \dfrac{ a }{ c } \ \\ α = \arcsin(\dfrac{ a }{ c } ) = \arcsin(\dfrac{ 1.45 }{ 3.79 } ) = 22^\circ 30' \ \\ \sin β = \dfrac{ b }{ c } \ \\ β = \arcsin(\dfrac{ b }{ c } ) = \arcsin(\dfrac{ 3.5 }{ 3.79 } ) = 67^\circ 30' \ \\ γ = 90^\circ$

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.

$T = rs \ \\ r = \dfrac{ T }{ s } = \dfrac{ 2.54 }{ 4.37 } = 0.58$

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.

$R = \dfrac{ c }{ 2 } = \dfrac{ 3.79 }{ 2 } = 1.89$

### 12. 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.

$m_a^2 = b^2 + (a/2)^2 \ \\ m_a = \sqrt{ b^2 + (a/2)^2 } = \sqrt{ 3.5^2 + (1.45/2)^2 } = 3.574 \ \\ \ \\ m_b^2 = a^2 + (b/2)^2 \ \\ m_b = \sqrt{ a^2 + (b/2)^2 } = \sqrt{ 1.45^2 + (3.5/2)^2 } = 2.273 \ \\ \ \\ m_c = R = \dfrac{ c }{ 2 } = \dfrac{ 3.79 }{ 2 } = 1.894$

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.

Examples for right triangle calculation:

## A right triangle in word problems in mathematics:

• Triangle P2
Can triangle have two right angles?
• Vector 7
Given vector OA(12,16) and vector OB(4,1). Find vector AB and vector |A|.
• Height 2
Calculate the height of the equilateral triangle with side 38.
• The ditch
Ditch with cross section of an isosceles trapezoid with bases 2m 6m are deep 1.5m. How long is the slope of the ditch?
• Cableway
Cableway has a length of 1800 m. The horizontal distance between the upper and lower cable car station is 1600 m. Calculate how much meters altitude is higher upper station than the base station.
• Euclid2
In right triangle ABC with right angle at C is given side a=27 and height v=12. Calculate the perimeter of the triangle.
• Calculate
Calculate the length of a side of the equilateral triangle with an area of 50cm2.
• Trapezoid - RR
Find the area of the right angled trapezoid ABCD with the right angle at the A vertex; a = 3 dm b = 5 dm c = 6 dm d = 4 dm
• Cable car
Cable car rises at an angle 45° and connects the upper and lower station with an altitude difference of 744 m. How long is "endless" tow rope?
• Area of RT 2
Calculate the area of right triangle whose legs have a length 5.8 cm and 5.8 cm.
• RT triangle and height
Calculate the remaining sides of the right triangle if we know side b = 4 cm long and height to side c h = 2.4 cm.