ত্রিভুজ ক্যালকুলেটর: SSS, SAS, ASA, AAS ও SSA

Q: How do you find the angles of a triangle from its three sides?

Rearrange the law of cosines: cos A = (b² + c² − a²)/(2bc), and the same pattern for B and C. For sides 7, 8 and 9, cos A = (64 + 81 − 49)/144 = 0.6667, so A = 48.19°; B = 58.41° and C = 73.40° follow the same way, and the three add up to 180°. The largest angle is always opposite the longest side.

Q: How do you find the area of a triangle without the height?

Take half the product of two sides and the sine of the angle between them: Area = ½ab sin C. Sides 7 and 8 with C = 73.40° give ½ × 7 × 8 × 0.9583 ≈ 26.83. From three sides alone, Heron's formula gives the same answer: the semi-perimeter is 12, and √(12 × 5 × 4 × 3) = √720 ≈ 26.83.

Q: How can you tell if a triangle is acute, right or obtuse from its sides?

Compare the square of the longest side c with the sum of the squares of the other two. If c² 41, obtuse.

তিনটি মাপ থেকে ত্রিভুজের বাহু, কোণ, ক্ষেত্রফল, পরিসীমা, উচ্চতা, অন্তর্বৃত্ত ও পরিবৃত্তের ব্যাসার্ধ বের করুন। SSA-এর দুটি সম্ভাব্য সমাধান দেখুন।

হালনাগাদ ২৮ সেপ্টেম্বর, ২০২৬যাচাই করা উদাহরণ: ৮

ক্ষেত্রফল

cm²

সর্বাধিক দশমিক স্থান: ৪; নিকটতম; সমান দূরত্বে শূন্য থেকে দূরে

Side a: 7cm
Side b: 8cm
Side c: 9cm
Angle A: 48.1897°
Angle B: 58.4119°
Angle C: 73.3985°
পরিসীমা: 24cm
Inradius: 2.2361cm
Circumradius: 4.6957cm
Height onto a: 7.6665cm
Height onto b: 6.7082cm
Height onto c: 5.9628cm
Type: Scalene, acute
Number of triangles: 1

The triangle has sides 7, 8 and 9 cm with angles 48.1897°, 58.4119° and 73.3985°; it is scalene, acute and encloses 26.8328 cm².

The triangle, to scale

যেভাবে হিসাব করা হয় S

Law of cosines for each angle
$\begin{gathered} \cos A = \frac{b^2 + c^2 - a^2}{2bc} = 0.666667 \Rightarrow A = 48.189685^\circ \\[6pt] \cos B = \frac{a^2 + c^2 - b^2}{2ac} = 0.52381 \Rightarrow B = 58.411864^\circ \\[6pt] \cos C = \frac{a^2 + b^2 - c^2}{2ab} = 0.285714 \Rightarrow C = 73.39845^\circ \end{gathered}$
ক্ষেত্রফল
$\text{Area} = \tfrac{1}{2}ab\sin C = \tfrac{1}{2} \times 7 \times 8 \times \sin 73.39845^\circ = 26.832816\,\text{cm}^{2}$
Inradius and circumradius
$\begin{gathered} r = \frac{\text{Area}}{s} = \frac{26.832816}{12} = 2.236068\,\text{cm} \\[6pt] R = \frac{a}{2\sin A} = 4.695743\,\text{cm} \end{gathered}$
s is the semi-perimeter.
Heights
$\begin{gathered} h_a = \frac{2\,\text{Area}}{a} = 7.666519\,\text{cm} \\[6pt] h_b = \frac{2\,\text{Area}}{b} = 6.708204\,\text{cm},\quad h_c = \frac{2\,\text{Area}}{c} = 5.962848\,\text{cm} \end{gathered}$

ত্রিভুজ ক্যালকুলেটর: SSS, SAS, ASA, AAS ও SSA সম্পর্কে

Three measurements fix a triangle as long as at least one of them is a side. The law of cosines, c² = a² + b² − 2ab cos C, solves three sides (SSS) and two sides with the angle between them (SAS). The law of sines, a/sin A = b/sin B = c/sin C, solves two angles and a side (ASA, AAS). The area comes from ½ab sin C, and the results add the perimeter, all three heights, the inradius and the circumradius.

The default sides of 7, 8 and 9 cm give angles of 48.19°, 58.41° and 73.40° and an area of 26.8328 cm². Surveying and roof framing use the same two laws to find a length that cannot be measured directly from ones that can.

Two sides and an angle that is not between them (SSA) can fit two triangles, one or none. With a = 6, b = 8 and A = 35°, both B = 49.89° and B = 130.11° work, so both triangles are drawn and compared in a table.

সমাধান করা উদাহরণ

SSS 3-4-5

I know: SSS
Side a (opposite A): 3 cm
Side b (opposite B): 4 cm
Side c (opposite C): 5 cm
ফলাফলের একক: সেন্টিমিটার (cm)

ক্ষেত্রফল: 6 cm²
Angle C: 90 °
Angle A: 36.869898 °
Angle B: 53.130102 °
Inradius: 1 cm
Circumradius: 2.5 cm
Type: Scalene, right
Number of triangles: 1

যাচাইয়ের উৎস: Right triangle: area ½·3·4, r = (3 + 4 − 5)/2, R = hypotenuse/2; Python 3.8 math: degrees(acos(0.8)), degrees(acos(0.6))

Equilateral, side 5

I know: SSS
Side a (opposite A): 5 cm
Side b (opposite B): 5 cm
Side c (opposite C): 5 cm
ফলাফলের একক: সেন্টিমিটার (cm)

ক্ষেত্রফল: 10.825318 cm²
Angle A: 60 °
Angle B: 60 °
Angle C: 60 °
Type: Equilateral
Circumradius: 2.886751 cm

যাচাইয়ের উৎস: Python 3.8 math: sqrt(3)/4*25, 5/sqrt(3)

SAS a = 5, b = 7, C = 49°

I know: SAS
Side a (opposite A): 5 cm
Side b (opposite B): 7 cm
Angle C (between a and b): 49 °
ফলাফলের একক: সেন্টিমিটার (cm)

Side c: 5.298667 cm
Angle A: 45.411694 °
Angle B: 85.588306 °
ক্ষেত্রফল: 13.207418 cm²

যাচাইয়ের উৎস: Python 3.8 math: law of cosines and ½ab·sin C

ASA A = 40°, c = 10, B = 60°

I know: ASA
Side c (opposite C): 10 cm
Angle A: 40 °
Angle B: 60 °
ফলাফলের একক: সেন্টিমিটার (cm)

Angle C: 80 °
Side a: 6.527036 cm
Side b: 8.793852 cm
ক্ষেত্রফল: 28.262897 cm²

যাচাইয়ের উৎস: Python 3.8 math: C = 80°, law of sines a = 10·sin40/sin80

প্রশ্ন

How do you find a missing side of a triangle that is not right-angled?

With two sides and the angle between them, use the law of cosines: c² = a² + b² − 2ab cos C. For a = 5, b = 7 and C = 49°, c = √(74 − 70 cos 49°) ≈ 5.2987. With two angles and any side, use the law of sines, a/sin A = b/sin B. When C = 90° the cosine term is zero and the law of cosines becomes Pythagoras' theorem.

What is the ambiguous case of the law of sines?

It is the SSA case: two sides and an angle opposite one of them can fit two different triangles. For an acute angle A, find the height h = b sin A. If a < h there is no triangle, if a = h one right triangle, if h < a < b two triangles, and if a ≥ b one. For a = 6, b = 8 and A = 35°, h = 4.589, so B is either 49.89° or 130.11°.

How do you find the angles of a triangle from its three sides?

Rearrange the law of cosines: cos A = (b² + c² − a²)/(2bc), and the same pattern for B and C. For sides 7, 8 and 9, cos A = (64 + 81 − 49)/144 = 0.6667, so A = 48.19°; B = 58.41° and C = 73.40° follow the same way, and the three add up to 180°. The largest angle is always opposite the longest side.

How do you find the area of a triangle without the height?

Take half the product of two sides and the sine of the angle between them: Area = ½ab sin C. Sides 7 and 8 with C = 73.40° give ½ × 7 × 8 × 0.9583 ≈ 26.83. From three sides alone, Heron's formula gives the same answer: the semi-perimeter is 12, and √(12 × 5 × 4 × 3) = √720 ≈ 26.83.

How can you tell if a triangle is acute, right or obtuse from its sides?

Compare the square of the longest side c with the sum of the squares of the other two. If c² < a² + b² the triangle is acute, if they are equal it has a right angle, and if c² is larger it is obtuse. Sides 7, 8 and 9 give 81 < 113, so acute; 3, 4 and 5 give 25 = 25, right; 4, 5 and 8 give 64 > 41, obtuse.

“ত্রিভুজ ক্যালকুলেটর: SSS, SAS, ASA, AAS ও SSA” কতটা নির্ভুল?

নির্ভুলতা আপনার ইনপুট ও পদ্ধতির অনুমানের ওপর নির্ভর করে। দশমিক গণনায় 50টি সার্থক অঙ্ক ব্যবহৃত হয়, কিন্তু আনুমানিক হিসাব, সংখ্যাগত পদ্ধতি ও উৎসের তথ্য কম নির্ভুল হতে পারে; প্রদর্শিত মান রাউন্ড করলে এই সীমাবদ্ধতাগুলি দূর হয় না। স্বতন্ত্র উৎসের সমাধানের সঙ্গে যাচাই করা উদাহরণ: ৮। যেমন, “SSS 3-4-5” উদাহরণটি Right triangle: area ½·3·4, r = (3 + 4 − 5)/2, R = hypotenuse/2; Python 3.8 math: degrees(acos(0.8)), degrees(acos(0.6))-এর সঙ্গে যাচাই করা হয়।

এই পদ্ধতির উৎস কী?

Weisstein, E. W. “Law of Sines”, “Law of Cosines”, “Triangle” — MathWorld; OpenStax Precalculus 2e, §8.1 Non-right Triangles: Law of Sines (ambiguous case).

এই ক্যালকুলেটর সম্পর্কে

\begin{gathered} \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,\qquad c^2 = a^2 + b^2 - 2ab\cos C \\ \text{Area} = \tfrac{1}{2}ab\sin C,\qquad r = \frac{\text{Area}}{s} \end{gathered}

উৎস

তথ্যসূত্রের সঙ্গে যাচাই করা

এই ক্যালকুলেটরে স্বাধীন উৎস থেকে উত্তর নেওয়া ৮টি সমাধান করা উদাহরণ রয়েছে। এগুলি পরীক্ষার সমষ্টিতে চালানো হয়, আর আপনি এখানেও চালাতে পারেন।