طرح‌ها

حل مثلث

هر مثلثی را حل کنید — نه فقط مثلث قائم‌الزاویه — از روی سه ضلع، دو ضلع و زاویهٔ بینشان یا دو زاویه و ضلع بینشان: همهٔ ضلع‌ها، همهٔ زاویه‌ها، محیط، مساحت و نوع مثلث.

ضلع a روبه‌روی زاویهٔ A است

نتیجه

مقادیر را وارد کنید و برای دیدن نتیجه «محاسبه» را بزنید.

فرمول

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

The law of cosines links three sides and one angle.

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

The law of sines: each side over the sine of its opposite angle is the same.

Area=s(s−a)(s−b)(s−c),s=a+b+c2\text{Area} = \sqrt{s(s-a)(s-b)(s-c)},\quad s = \frac{a+b+c}{2}

Heron’s formula gives the area from the three sides.

نحوه استفاده از این ماشین‌حساب

  1. Choose what you know: three sides, two sides and the angle between them, or two angles and the side between them.
  2. Side a is opposite angle A, side b opposite angle B and side c opposite angle C.
  3. Press Calculate to get every side, every angle, the perimeter, the area and the triangle type.

مثال حل‌شده

A triangle with sides 3, 4 and 5

Check the triangle inequality: 3 + 4 > 5, so the triangle exists.

Law of cosines: cos A = (16 + 25 − 9)/40 = 4/5, so A ≈ 36.869898°; cos B = 3/5, so B ≈ 53.130102°; cos C = 0, so C = 90°.

Heron: s = 6 and the area is √(6 × 3 × 2 × 1) = 6.

پرسش‌های متداول

Why does it say no triangle exists?

Any two sides must add up to more than the third side, and the angles must add up to exactly 180°.

Why is SSA (two sides and an angle not between them) not offered?

SSA can give two different triangles or none at all — the “ambiguous case”. Sumzo AI can walk through it.

بیشتر بدانید