# 삼각형 계산기: SSS, SAS, ASA, AAS, SSA

> 세 측정값으로 변, 각, 넓이, 둘레, 높이, 내접원과 외접원의 반지름을 구하고 SSA의 가능한 두 해를 확인하세요.

직접 계산할 수 있는 페이지: https://www.calcopenly.com/ko/geometry/triangle-solver
분야: 기하 계산기

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.

## 입력

- **I know** (선택 항목: SSS, SAS, ASA, AAS, SSA)
- **Side a (opposite A)**
- **Side b (opposite B)**
- **Side c (opposite C)**
- **Angle A**
- **Angle B**
- **Angle C (between a and b)**
- **결과 단위** (선택 항목: 밀리미터 (mm), 센티미터 (cm), 미터 (m), 킬로미터 (km), 인치 (in), 피트 (ft), 야드 (yd), 마일 (mi))

## 결과

- 넓이 — 주요 결과
- Side a
- Side b
- Side c
- Angle A (°)
- Angle B (°)
- Angle C (°)
- 둘레
- Inradius
- Circumradius
- Height onto a
- Height onto b
- Height onto c
- Type
- Number of triangles
- Second triangle: side c
- Second triangle: angle B (°)
- Second triangle: angle C (°)
- Second triangle: area

## 공식

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

## 계산 예제

### 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
- 결과 단위: 센티미터 (cm)
- **넓이: 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**
- 검증 출처: 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
- 결과 단위: 센티미터 (cm)
- **넓이: 10.825318 cm²**
- **Angle A: 60 °**
- **Angle B: 60 °**
- **Angle C: 60 °**
- **Type: Equilateral**
- **Circumradius: 2.886751 cm**
- 검증 출처: 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 °
- 결과 단위: 센티미터 (cm)
- **Side c: 5.298667 cm**
- **Angle A: 45.411694 °**
- **Angle B: 85.588306 °**
- **넓이: 13.207418 cm²**
- 검증 출처: 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 °
- 결과 단위: 센티미터 (cm)
- **Angle C: 80 °**
- **Side a: 6.527036 cm**
- **Side b: 8.793852 cm**
- **넓이: 28.262897 cm²**
- 검증 출처: Python 3.8 math: C = 80°, law of sines a = 10·sin40/sin80

### AAS A = 30°, B = 45°, a = 10

- I know: AAS
- Side a (opposite A): 10 cm
- Angle A: 30 °
- Angle B: 45 °
- 결과 단위: 센티미터 (cm)
- **Angle C: 105 °**
- **Side b: 14.142136 cm**
- **Side c: 19.318517 cm**
- **넓이: 68.30127 cm²**
- 검증 출처: Python 3.8 math: b = 10·sin45/sin30 = 10√2, c = 10·sin105/sin30

### SSA ambiguous case, two triangles

- I know: SSA
- Side a (opposite A): 6 cm
- Side b (opposite B): 8 cm
- Angle A: 35 °
- 결과 단위: 센티미터 (cm)
- **Number of triangles: 2**
- **Angle B: 49.886408 °**
- **Angle C: 95.113592 °**
- **Side c: 10.419047 cm**
- **넓이: 23.904479 cm²**
- **Second triangle: angle B: 130.113592 °**
- **Second triangle: angle C: 14.886408 °**
- **Second triangle: side c: 2.687386 cm**
- **Second triangle: area: 6.165685 cm²**
- 검증 출처: Python 3.8 math: asin(8·sin35°/6) and its supplement, c = 6·sin C/sin35°

## 자주 묻는 질문

### 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.

### “삼각형 계산기: SSS, SAS, ASA, AAS, SSA”의 정확도는 어느 정도인가요?

정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 8. 예를 들어 “SSS 3-4-5”은 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))와 대조해 확인합니다.

### 이 계산 방법의 출처는 무엇인가요?

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

## 출처

- [Weisstein, E. W. “Law of Sines”, “Law of Cosines”, “Triangle” — MathWorld](https://mathworld.wolfram.com/LawofSines.html)
- [OpenStax Precalculus 2e, §8.1 Non-right Triangles: Law of Sines (ambiguous case)](https://openstax.org/books/precalculus-2e/pages/8-1-non-right-triangles-law-of-sines)