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Rechtwinkliges Dreieck: Hypotenuse, Seiten und Winkel

Hypotenuse, fehlende Kathete, spitze Winkel, Fläche und Umfang aus zwei Werten mit dem Satz des Pythagoras und Trigonometrie berechnen.

Aktualisiert Geprüfte Beispiele: 6

Ausprobieren
Fläche
cm²
Fläche: 6 cm²
Maximale Nachkommastellen: 4; Zum nächsten Wert, bei Gleichstand von null weg
Leg a
3cm
Leg b
4cm
Hypotenuse c
5cm
Angle α
36.8699°
Angle β
53.1301°
Umfang
12cm
Height onto the hypotenuse
2.4cm
Inradius
1cm
Circumradius
2.5cm

Legs of 3 and 4 cm meet at the right angle; the hypotenuse is 5 cm, the acute angles are 36.8699° and 53.1301°, and the triangle encloses 6 cm².

Right triangle, to scale

b = 4 cmc = 5 cma = 3 cmα = 36.87°β = 53.13°
So wird gerechnet S

  1. Hypotenuse (Pythagoras)

    c=a2+b2=32+42=5 cmc = \sqrt{a^2 + b^2} = \sqrt{3^2 + 4^2} = 5\,\text{cm}

  2. Angle α

    α=arctan⁡ab=arctan⁡34=36.869898∘\alpha = \arctan\frac{a}{b} = \arctan\frac{3}{4} = 36.869898^\circ

  3. Area and perimeter

    Area=12ab=12×3×4=6 cm2,P=a+b+c=12 cm\text{Area} = \tfrac{1}{2}ab = \tfrac{1}{2} \times 3 \times 4 = 6\,\text{cm}^{2},\qquad P = a + b + c = 12\,\text{cm}

  4. Height onto the hypotenuse, inradius, circumradius

    h=abc=2.4 cm,r=a+b−c2=1 cm,R=c2=2.5 cmh = \frac{ab}{c} = 2.4\,\text{cm},\quad r = \frac{a + b - c}{2} = 1\,\text{cm},\quad R = \frac{c}{2} = 2.5\,\text{cm}

    The circumcentre of a right triangle is the midpoint of its hypotenuse.

Über Rechtwinkliges Dreieck: Hypotenuse, Seiten und Winkel

A right triangle is fixed by any two of its parts as long as one of them is a side. With both legs known, Pythagoras' theorem gives the hypotenuse c = √(a² + b²). With a side and an acute angle, the rest follows from sin α = a/c, cos α = b/c and tan α = a/b, and the two acute angles always add up to 90°.

The default legs of 3 cm and 4 cm make the 3-4-5 triangle: hypotenuse 5 cm, area 6 cm², perimeter 12 cm and angles of 36.8699° and 53.1301°. Builders use that ratio to square a corner, since marks 3 m and 4 m from it along two walls should be exactly 5 m apart. The results also include the height onto the hypotenuse (ab/c), the inradius ((a + b − c)/2) and the circumradius (c/2).

Angle α is opposite leg a and β is opposite leg b. Angles can be typed in degrees, radians, gradians or turns and must lie strictly between 0° and 90°.

Durchgerechnete Beispiele

Legs 3 and 4

I know
Both legs (a and b)
Leg a (opposite α)
3 cm
Leg b (opposite β)
4 cm
Ergebniseinheit
Zentimeter (cm)
Hypotenuse c
5 cm
Fläche
6 cm²
Angle α
36.869898 °
Angle β
53.130102 °
Height onto the hypotenuse
2.4 cm
Inradius
1 cm
Circumradius
2.5 cm
Umfang
12 cm

Prüfquelle: 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4))

Leg 5 and hypotenuse 13

I know
Leg a and hypotenuse c
Leg a (opposite α)
5 cm
Hypotenuse c
13 cm
Ergebniseinheit
Zentimeter (cm)
Leg b
12 cm
Fläche
30 cm²
Umfang
30 cm
Angle α
22.619865 °

Prüfquelle: 5-12-13 triple; Python 3.8 math: degrees(asin(5/13))

30-60-90 from hypotenuse 10 and α = 30°

I know
Hypotenuse c and angle α
Hypotenuse c
10 cm
Angle α
30 °
Ergebniseinheit
Zentimeter (cm)
Leg a
5 cm
Leg b
8.660254 cm
Angle β
60 °
Fläche
21.650635 cm²

Prüfquelle: Side opposite 30° is half the hypotenuse; Python 3.8 math: 10*cos(radians(30)), 25*sqrt(3)/2

45° angle gives equal legs (edge case)

I know
Leg a and angle α
Leg a (opposite α)
7 cm
Angle α
45 °
Ergebniseinheit
Zentimeter (cm)
Leg b
7 cm
Hypotenuse c
9.899495 cm
Fläche
24.5 cm²
Angle β
45 °

Prüfquelle: tan 45° = 1 so b = a; Python 3.8 math: 7*sqrt(2)

Fragen

How do you find the hypotenuse of a right triangle?

Square both legs, add them and take the square root: c = √(a² + b²). Legs of 3 and 4 give √25 = 5, and legs of 6 in and 8 in give 10 in. If you know one leg and the angle opposite it instead, divide by the sine: c = a / sin α, so a 5 cm leg opposite 30° means a 10 cm hypotenuse.

How do you find a missing side of a right triangle?

For a missing leg, subtract the squares: b = √(c² − a²). A hypotenuse of 13 and a leg of 5 give √(169 − 25) = 12. The hypotenuse must be longer than either leg, or no right triangle exists. With one side and an angle, use SOHCAHTOA: opposite = hypotenuse × sin α and adjacent = hypotenuse × cos α.

What are the side ratios of a 30-60-90 and a 45-45-90 triangle?

A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the side opposite 30° is half the hypotenuse; a hypotenuse of 10 gives legs of 5 and 8.6603. A 45-45-90 triangle has equal legs and a hypotenuse √2 ≈ 1.4142 times a leg, so legs of 7 give a hypotenuse of 9.8995.

What does SOHCAHTOA mean?

It is a memory aid for the three trigonometric ratios in a right triangle: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. In the 3-4-5 triangle, the angle opposite the side of 3 has sin = 0.6, cos = 0.8 and tan = 0.75, and each inverse function returns the same 36.87°.

How do you check a corner is square with the 3-4-5 rule?

Mark 3 units along one side and 4 along the other, measured from the corner; the diagonal between the marks is exactly 5 when the angle is 90°. A longer diagonal means the angle is too wide: 5.05 m on 3 m and 4 m legs is 91.2°, and 4.95 m is 88.8°. Multiples such as 6-8-10 or 9-12-15 give a more precise check on large slabs.

Wie genau arbeitet „Rechtwinkliges Dreieck: Hypotenuse, Seiten und Winkel“?

Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 6. Beispielsweise wird „Legs 3 and 4“ anhand von 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4)) geprüft.

Woher stammt die Methode?

Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem); OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry.

Über diesen Rechner

a2+b2=c2,sin⁡α=ac,cos⁡α=bc,tan⁡α=ab,α+β=90∘a^2 + b^2 = c^2,\quad \sin\alpha = \frac{a}{c},\quad \cos\alpha = \frac{b}{c},\quad \tan\alpha = \frac{a}{b},\quad \alpha + \beta = 90^\circ

Quellen

  1. Euclid, Elements, Book I, Proposition 47 (Pythagorean theorem)
  2. OpenStax Precalculus 2e, §5.4 Right Triangle Trigonometry

Anhand von Quellen geprüft

Dieser Rechner enthält 6 Rechenbeispiele mit Ergebnissen aus unabhängigen Quellen. Sie laufen in der Testsuite und können auch hier ausgeführt werden.

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