The Basic Formula: A = 1/2 × Base × Height
The most fundamental formula for the area of a triangle is A = 1/2 × b × h, where b is the length of the base and h is the perpendicular height (the shortest distance from the base to the opposite vertex). This formula works for every triangle — right, acute, or obtuse — as long as you use the perpendicular height, not the length of a slanted side.
Why does this formula work? A triangle is exactly half of a parallelogram. If you duplicate a triangle and flip it, the two triangles together form a parallelogram with area base × height. Since the triangle is half the parallelogram, its area is 1/2 × base × height. This is not just a formula to memorize — understanding why it works helps you remember it and apply it correctly.
Any side of a triangle can serve as the base. The height is then the perpendicular distance from that base to the opposite vertex. For right triangles, this is particularly easy: the two legs are already perpendicular, so you can use one as the base and the other as the height. For a right triangle with legs 6 cm and 8 cm: A = 1/2 × 6 × 8 = 24 cm².
Finding the Height When It Is Not Given
In many geometry problems, the height is not given directly. For obtuse triangles, the height often falls outside the triangle when drawn from the base, which can be confusing. You may need to extend the base line and draw the perpendicular from the opposite vertex to this extended line.
If you know two sides and the included angle, you can calculate the height using trigonometry. If side b is the base and angle A is at one end of the base, then h = c × sin(A), where c is the other side adjacent to angle A. This leads directly to the trigonometric area formula discussed below.
In coordinate geometry, if you know the coordinates of all three vertices, you can find the height using the distance from a point to a line. Alternatively, you can use the coordinate formula: A = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|. For example, a triangle with vertices (0,0), (4,0), and (2,3) has area A = 1/2 |0(0-3) + 4(3-0) + 2(0-0)| = 1/2 |0 + 12 + 0| = 6 square units.
Heron's Formula: Area from Three Sides
Heron's formula (also called Hero's formula) calculates the area of a triangle when you know all three side lengths but not the height. Named after Hero of Alexandria (c. 10-70 CE), the formula uses the semi-perimeter s = (a + b + c) / 2, where a, b, and c are the three side lengths.
The formula is: A = √(s(s-a)(s-b)(s-c)). Let's find the area of a triangle with sides 7, 8, and 9. First, calculate the semi-perimeter: s = (7 + 8 + 9) / 2 = 12. Then: A = √(12 × 5 × 4 × 3) = √(720) = √(144 × 5) = 12√5 ≈ 26.83 square units.
Heron's formula is especially useful in real-world problems where you can measure distances but not angles — for instance, surveying a triangular plot of land. It also works perfectly for all types of triangles without needing to identify the height. A quick check: if any of the terms (s-a), (s-b), or (s-c) is zero, the triangle is degenerate (the three points are collinear). If any term is negative, the three lengths cannot form a valid triangle.
The Trigonometric Area Formula
When you know two sides and the included angle (SAS), use the formula: A = 1/2 × a × b × sin(C), where a and b are two sides and C is the angle between them. This formula is derived directly from A = 1/2 × base × height, substituting h = b × sin(C).
Example: a triangle has sides of 10 cm and 14 cm with an included angle of 30°. A = 1/2 × 10 × 14 × sin(30°) = 1/2 × 10 × 14 × 0.5 = 35 cm². Since sin(30°) = 0.5, this calculation is clean, but the formula works for any angle.
This formula reveals an important geometric fact: for two fixed side lengths, the area is maximized when the included angle is 90° (since sin(90°) = 1). As the angle approaches 0° or 180°, the area approaches zero because the triangle collapses into a line segment. This insight is useful in optimization problems.
Special Triangle Areas
Equilateral triangle: Since all sides are equal (length s), the area formula simplifies to A = (s² × √3) / 4. For an equilateral triangle with side 6: A = (36 × √3) / 4 = 9√3 ≈ 15.59 square units. This formula comes from using A = 1/2 × s × h where h = s × sin(60°) = s√3/2.
Isosceles triangle: If the two equal sides have length a and the base has length b, the height is h = √(a² - (b/2)²). Then A = 1/2 × b × √(a² - b²/4). For an isosceles triangle with equal sides 5 and base 6: h = √(25 - 9) = √16 = 4, so A = 1/2 × 6 × 4 = 12 square units.
Right triangle: The simplest case. The two legs are the base and height, so A = 1/2 × leg₁ × leg₂. If you only know the hypotenuse and one leg, use the Pythagorean theorem to find the other leg first. For a right triangle with hypotenuse 13 and one leg 5: other leg = √(169 - 25) = √144 = 12, so A = 1/2 × 5 × 12 = 30 square units.
Common Mistakes and How to Avoid Them
The most frequent mistake is using a slanted side instead of the perpendicular height in the base-height formula. The height must be perpendicular to the base. If you are given a triangle with sides 5, 12, and 13, and you use A = 1/2 × 5 × 12 = 30, that only works because this is a right triangle (5² + 12² = 13²) and the legs are perpendicular. For a non-right triangle with sides 5, 12, and 14, using 1/2 × 5 × 12 would give the wrong answer.
Another common error in Heron's formula is forgetting to take the square root at the end, or making arithmetic errors when computing s(s-a)(s-b)(s-c). Work carefully through each subtraction and multiplication. A good practice is to verify that a + b > c, a + c > b, and b + c > a (triangle inequality) before proceeding.
When using the trigonometric formula, make sure your calculator is set to the correct angle mode (degrees vs. radians). Using sin(30) in radian mode gives a completely different answer than sin(30°). On standardized tests, angles are almost always in degrees unless stated otherwise. If your area calculation gives you a very strange number, check your angle mode first. Need help with a triangle problem? Snap a photo and ScanSolve will identify the right formula and solve it step by step.
