Method 1: base times height, divided by 2
The most familiar triangle-area formula is area = 1/2 x base x height, where height is the perpendicular distance from the chosen base to the opposite vertex -- not the length of a slanted side. This method applies whenever a base and its perpendicular height are both known or can be measured directly, which is common for right triangles and for triangles drawn on a grid or diagram with a marked altitude.
Worked example: a triangle has a base of 8 units and a perpendicular height of 5 units. Applying the formula: area = 1/2 x 8 x 5 = 1/2 x 40 = 20 square units. This method requires no trigonometry and no knowledge of the triangle's angles or its other two side lengths -- only the one side chosen as the base and the straight-line height measured perpendicular to it.
Method 2: Heron's formula, from three side lengths alone
When only the three side lengths of a triangle are known -- with no height and no angle given directly -- Heron's formula calculates the area from those three lengths alone. It first computes the semi-perimeter, s = (a + b + c) / 2, then finds the area as the square root of s(s - a)(s - b)(s - c).
Worked example: a triangle has sides a = 9, b = 10 and c = 11. Step 1: the semi-perimeter is s = (9 + 10 + 11) / 2 = 30 / 2 = 15. Step 2: area = the square root of 15 x (15-9) x (15-10) x (15-11) = the square root of 15 x 6 x 5 x 4 = the square root of 1800, which is approximately 42.4264 square units. Before applying Heron's formula, it is worth confirming the three lengths actually form a triangle: each side must be shorter than the sum of the other two (the triangle inequality), which 9, 10 and 11 satisfy.
Method 3: 1/2 x a x b x sin(C), from two sides and the included angle
When two sides of a triangle and the angle between them (the included angle) are known -- but not a perpendicular height -- the trigonometric area formula applies: area = 1/2 x a x b x sin(C), where a and b are the two known sides and C is the angle between them.
Worked example: a triangle has sides a = 7 and b = 10, with an included angle of C = 40 degrees. Applying the formula: area = 1/2 x 7 x 10 x sin(40 degrees) = 35 x sin(40 degrees). Since sin(40 degrees) is approximately 0.6428, the area is approximately 35 x 0.6428 = 22.4976 square units. This method is especially useful in surveying and navigation problems, where two measured distances and the angle between them are often easier to obtain directly than a perpendicular height.
Which method should you use?
The three methods are not competing approaches to the same problem -- each is suited to a different combination of known measurements, and choosing the one that matches the available data avoids unnecessary extra steps such as first calculating a height or an angle that was not directly given.
| Known values | Best method | Formula |
|---|---|---|
| Base and perpendicular height | Base x height / 2 | area = 1/2 x base x height |
| All three side lengths (SSS), no angle known | Heron's formula | area = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2 |
| Two sides and the included angle (SAS) | Trigonometric formula | area = 1/2 x a x b x sin(C) |
Common mistakes when finding a triangle's area
The most frequent error in the base-height method is substituting a slanted side length for the perpendicular height -- the two are only equal when the triangle happens to be right-angled at the base, so using a slanted side elsewhere inflates the calculated area. The height must always be measured along a line perpendicular to the chosen base, even if that line does not correspond to any actual drawn side of the triangle.
In the trigonometric method, a common error is using an angle that is not actually included between the two known sides -- the formula area = 1/2 x a x b x sin(C) requires C to be the angle directly between sides a and b, not one of the triangle's other two angles. Using Heron's formula on three lengths that violate the triangle inequality (where one side is longer than the sum of the other two) will attempt to take the square root of a negative number, signaling that no such triangle exists.
よくある質問
What is the formula for the area of a triangle?
The most common formula is area = 1/2 x base x height, where height is the perpendicular distance from the base to the opposite vertex. When only the three side lengths are known, Heron's formula gives the area instead; when two sides and the included angle are known, area = 1/2 x a x b x sin(C) applies. All three formulas produce the same area for the same triangle.
How do you find the area of a triangle with only the three side lengths?
Use Heron's formula: calculate the semi-perimeter s = (a + b + c) / 2, then area = the square root of s(s-a)(s-b)(s-c). For a triangle with sides 9, 10 and 11, s = 15 and area = the square root of (15 x 6 x 5 x 4) = the square root of 1800, approximately 42.4264 square units.
How do you find the area of a triangle with two sides and an angle?
Use the trigonometric formula area = 1/2 x a x b x sin(C), where a and b are the two known sides and C is the angle included between them. For sides 7 and 10 with an included angle of 40 degrees, area = 1/2 x 7 x 10 x sin(40 degrees), approximately 22.4976 square units.
Can you find a triangle's area without knowing its height?
Yes. If only the three side lengths are known, Heron's formula finds the area without any height measurement. If two sides and the angle between them are known, the trigonometric formula 1/2 x a x b x sin(C) also avoids needing a height, since the sine of the included angle effectively substitutes for it.
Why must the height be perpendicular in the base-height formula?
The base-height formula area = 1/2 x base x height is derived from the area of a rectangle with the same base and height, cut diagonally -- and that derivation only holds when the height is measured along a line perpendicular to the base. Substituting a slanted side length in place of the true perpendicular height produces an incorrect, inflated area.
参考文献
- Weisstein, Eric W. "Heron's Formula" and "Triangle Area." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Larson R. Geometry. Cengage Learning (triangle area formulas and the triangle inequality).
- Larson R. Trigonometry. Cengage Learning (the trigonometric area formula, 1/2 ab sin C).