所有三角形公式一站式参考——面积、周长、角、定理
由 [email protected], Geometry Calculator Developer & Online Math Educator 审核 最后更新于 May 14, 2026
三角形是几何中公式最丰富的图形——仅面积就有六种形式，取决于已知条件。本页列出你在学校会遇到的所有三角形公式，按计算内容（面积、周长、角度、相似、全等）分类，并附有适用条件。需要数值答案时，请使用相关计算器。
| 名称 | 公式 | 说明 |
|---|---|---|
| 面积——底×高 | A = ½ × b × h |
b = 底；h = 对应底上的高。最简形式。 |
| 面积——海伦公式 | A = √[s(s−a)(s−b)(s−c)], s = (a+b+c)/2 |
已知三边，无需高。详见海伦公式参考。 |
| 面积——SAS（两边+夹角） | A = ½ × a × b × sin(C) |
a, b = 两边；C = 它们的夹角。 |
| 面积——坐标形式 | A = ½ × |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| |
已知三个顶点坐标时使用。 |
| 面积——内切圆半径 | A = r × s |
r = 内切圆半径；s = 半周长。逆用：r = A / s。 |
| 面积——外接圆半径 | A = (a × b × c) / (4R) |
R = 外接圆半径。逆用：R = abc / (4A)。 |
| 周长 | P = a + b + c |
三边之和。等边三角形：P = 3a。 |
| 内角和 | ∠A + ∠B + ∠C = 180° |
任意三角形的内角和为180°。 |
| 外角定理 | ext = sum of 2 opposite interior angles |
外角等于不相邻的两个内角之和。 |
| 勾股定理 | a² + b² = c² |
仅适用于直角三角形。c 是斜边（直角所对边）。详见勾股定理参考。 |
| 30-60-90 三角形 | sides ratio 1 : √3 : 2 |
短直角边 : 长直角边 : 斜边，分别对应 30°、60°、90°。推导与示例见特殊直角三角形页面。 |
| 45-45-90 三角形 | sides ratio 1 : 1 : √2 |
等腰直角三角形。两直角边相等，斜边 = 直角边 × √2。推导与示例见特殊直角三角形页面。 |
| 正弦定理 | a/sin(A) = b/sin(B) = c/sin(C) = 2R |
适用于任意三角形。最适合已知ASA、AAS或SSA的情况。 |
| 余弦定理 | c² = a² + b² − 2ab × cos(C) |
适用于任意三角形。最适合SSS（求角）或SAS（求第三边）。是勾股定理的推广（当C = 90°时，cos C = 0）。 |
| 相似——AA | two pairs of equal angles → similar |
如果两对对应角相等，则第三对角也相等（内角和），三角形相似。 |
| 相似比 | k = corresponding sides ratio |
对于相似三角形：边长缩放k倍，面积缩放k²倍，体积（若为立体）缩放k³倍。 |
| 全等公理 | SSS, SAS, ASA, AAS, HL |
满足以下5个条件之一，三角形全等（形状和大小完全相同）。HL仅适用于直角三角形。 |
| 等边三角形——所有公式 | A = (√3/4)·a², h = (√3/2)·a, P = 3a |
当所有边都等于 a 时。高平分底边，同时也是中线、角平分线和垂线。详见等边三角形参考。 |
When the height of a triangle is unknown, calculating area becomes difficult if relying solely on the base-height relationship. Heron's formula provides a robust alternative that requires only the lengths of the three sides. This method is particularly useful in surveying, engineering, and geometry problems where altitude measurement is impractical or impossible.
The process begins by determining the semi-perimeter, denoted as s. This value is half the sum of all three side lengths: a, b, and c. The formula for semi-perimeter is:
s = (a + b + c) / 2
Once s is established, the area A is calculated using the product of the differences between the semi-perimeter and each side length:
A = √(s(s - a)(s - b)(s - c))
Consider a triangle with side lengths of 5, 12, and 13 units. First, calculate the semi-perimeter: s = (5 + 12 + 13) / 2 = 15. Next, substitute these values into the area equation: A = √(15(15 - 5)(15 - 12)(15 - 13)). Simplifying the terms inside the square root yields A = √(15 × 10 × 3 × 2), which equals √900. The final area is 30 square units.
This approach ensures accuracy regardless of whether the triangle is right-angled, acute, or obtuse. It eliminates the need to derive height through trigonometric functions first, streamlining calculations for scalene triangles where no obvious geometric properties simplify the problem. Understanding this formula expands the toolkit available for solving complex spatial problems efficiently.
For non-right triangles, standard trigonometric ratios like SOH CAH TOA do not apply directly. Instead, the law of sines and the law of cosines allow for the solution of any triangle given specific combinations of sides and angles. These laws are fundamental in navigation, physics, and advanced geometry.
The law of sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides. Mathematically, this is expressed as:
a / sin(A) = b / sin(B) = c / sin(C)
This law is most effective when two angles and one side are known (ASA or AAS cases), or two sides and a non-included angle (SSA case). For example, if side a is 10, angle A is 30°, and angle B is 45°, you can solve for side b by rearranging the equation: b = (a × sin(B)) / sin(A).
The law of cosines generalizes the Pythagorean theorem for any triangle. It relates the lengths of the sides to the cosine of one of its angles:
c² = a² + b² - 2ab cos(C)
This formula is essential when two sides and the included angle are known (SAS), or when all three sides are known and an angle needs to be found (SSS). Unlike the law of sines, the law of cosines avoids ambiguity in the SSA case, providing a unique solution for the third side. Mastery of both laws enables complete resolution of any triangular configuration encountered in technical applications.
Certain triangles possess fixed angle measures that create predictable side ratios, simplifying calculations without requiring complex trigonometric functions. Recognizing these patterns allows for rapid mental math and efficient problem-solving in standardized tests and practical design tasks.
The 45-45-90 triangle is an isosceles right triangle. Its angles are 45°, 45°, and 90°. The sides follow a ratio of 1 : 1 : √2. If the legs (the sides opposite the 45° angles) have length x, the hypotenuse is always x√2. For instance, if a leg is 7 units, the hypotenuse is exactly 7√2 units.
The 30-60-90 triangle arises from bisecting an equilateral triangle. Its angles are 30°, 60°, and 90°. The side ratios are consistently 1 : √3 : 2. The shortest side (opposite the 30° angle) is x. The longer leg (opposite the 60° angle) is x√3, and the hypotenuse (opposite the 90° angle) is 2x. If the shortest side is 4, the hypotenuse is 8, and the remaining side is 4√3.
These special triangles are frequently used in architecture for roof trusses and in computer graphics for rendering angles. Memorizing these ratios saves time and reduces computational errors compared to using a calculator for every step. They serve as building blocks for understanding more complex geometric relationships involving regular polygons and circular sectors.
Not every set of three lengths can form a triangle. The triangle inequality theorem establishes the necessary conditions for three segments to enclose a space. It states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.
If the sides are labeled a, b, and c, three inequalities must hold true simultaneously:
a + b > ca + c > bb + c > aFailure to meet any one of these conditions means the segments cannot connect to form a closed figure. For example, lengths of 1, 2, and 5 cannot form a triangle because 1 + 2 = 3, which is less than 5. The shorter sides would never reach each other to meet the longest side.
Conversely, lengths of 3, 4, and 5 work because 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3. This theorem is critical for validating geometric constructions and ensuring that calculated dimensions are physically possible. It also helps identify degenerate triangles, where the sum of two sides equals the third, resulting in a flat line rather than a two-dimensional shape.
Even experienced students and professionals make errors when applying geometric principles. Avoiding these pitfalls ensures accuracy in calculations and interpretations of spatial data.
A frequent mistake is using the wrong formula for area. Students often default to (base × height) / 2 even when height is unknown, leading to unnecessary complexity or incorrect assumptions. In such cases, switching to Heron's formula is the correct approach. Another error involves misidentifying the hypotenuse in right triangles, which leads to incorrect applications of the Pythagorean theorem. Always verify that the longest side is opposite the largest angle before assigning variables.
Another common issue is ignoring the triangle inequality theorem during problem setup. When given three potential side lengths, it is crucial to check validity before proceeding with area or angle calculations. Attempting to solve for a triangle that cannot exist wastes time and produces nonsensical results.
To mitigate these errors, always sketch the triangle and label known values. Verify that the sum of any two sides exceeds the third. Choose the simplest applicable method: use special triangle ratios when angles match 30-60-90 or 45-45-90 patterns, resort to the law of sines or law of cosines for general cases, and use Heron's formula when only side lengths are provided. Double-checking units and significant figures at the end of the calculation further enhances reliability.