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Segitiga siku-siku: hipotenusa, sisi dan sudut

Hitung hipotenusa, sisi yang belum diketahui, sudut lancip, luas dan keliling dari dua nilai dengan teorema Pythagoras dan trigonometri.

Diperbarui Contoh terverifikasi: 6

Coba
Luas
cm²
Luas: 6 cm²
Tempat desimal maksimum: 4; Terdekat, jika berjarak sama menjauhi nol
Leg a
3cm
Leg b
4cm
Hypotenuse c
5cm
Angle α
36.8699°
Angle β
53.1301°
Keliling
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°
Cara menghitung 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.

Tentang Segitiga siku-siku: hipotenusa, sisi dan sudut

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°.

Contoh penyelesaian

Legs 3 and 4

I know
Both legs (a and b)
Leg a (opposite α)
3 cm
Leg b (opposite β)
4 cm
Satuan hasil
Sentimeter (cm)
Hypotenuse c
5 cm
Luas
6 cm²
Angle α
36.869898 °
Angle β
53.130102 °
Height onto the hypotenuse
2.4 cm
Inradius
1 cm
Circumradius
2.5 cm
Keliling
12 cm

Sumber pemeriksaan: 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
Satuan hasil
Sentimeter (cm)
Leg b
12 cm
Luas
30 cm²
Keliling
30 cm
Angle α
22.619865 °

Sumber pemeriksaan: 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 °
Satuan hasil
Sentimeter (cm)
Leg a
5 cm
Leg b
8.660254 cm
Angle β
60 °
Luas
21.650635 cm²

Sumber pemeriksaan: 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 °
Satuan hasil
Sentimeter (cm)
Leg b
7 cm
Hypotenuse c
9.899495 cm
Luas
24.5 cm²
Angle β
45 °

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

Pertanyaan

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.

Seberapa akurat “Segitiga siku-siku: hipotenusa, sisi dan sudut”?

Akurasi bergantung pada masukan dan asumsi metode. Aritmetika desimal memakai 50 digit signifikan, tetapi perkiraan, metode numerik dan data sumber bisa kurang presisi; pembulatan yang ditampilkan tidak menghilangkan batasan itu. Contoh penyelesaian yang diperiksa dengan sumber independen: 6. Misalnya, “Legs 3 and 4” diperiksa dengan 3-4-5 triple; h = 3·4/5; r = (3 + 4 − 5)/2; Python 3.8 math: degrees(atan(3/4)).

Dari mana metode ini berasal?

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

Tentang kalkulator ini

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

Sumber

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

Diperiksa dengan referensi

Kalkulator ini mencakup 6 contoh perhitungan dengan jawaban dari sumber independen. Contoh tersebut dijalankan dalam rangkaian pengujian dan dapat Anda jalankan di sini juga.

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