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For a triangle with side lengths a, b, and c, we first compute the semiperimeter s. Then we apply Heron’s formula to get the area. This approach is useful when the height is not given and only side measurements are available.

Meaning of variables: a, b, and c are the three sides of the triangle. s is the semiperimeter, half of the perimeter. A is the area of the triangle in square units (for example, cm² or m²).

Before using the formula, check the triangle inequality: each side must be smaller than the sum of the other two sides. If any side is too long, the three lengths cannot form a triangle, so the area is not defined.

To compute the area using only three sides, follow these steps in order. Use brackets while subtracting to avoid sign errors during multiplication.

1. Write the three side lengths as a, b, and c in the same unit.
2. Check the triangle inequality: a < b + c, b < a + c, and c < a + b.
3. Find the semiperimeter: s = (a + b + c)/2.
4. Compute the product s(s-a)(s-b)(s-c).
5. Take the square root of the product to get the area A.

Find the area of a triangle with side lengths 9 cm, 6 cm, and 5 cm. We will apply Heron’s formula and simplify step by step. Keep units consistent, since the final area will be in cm².

Step 1: Compute the semiperimeter.

Step 2: Substitute into Heron’s formula.

Step 3: Simplify inside the square root.

Step 4: Convert to a decimal if needed.

Final answer: The area is ‎10√2 ≈ 14.14 cm².

One common mistake is skipping the triangle inequality check. If the side lengths do not satisfy it, Heron’s formula will lead to an invalid square root for real measurements.

Another error is calculating the perimeter instead of the semiperimeter. Heron’s formula uses s = (a + b + c)/2, so using (a + b + c) directly makes the product too large and gives the wrong area.

Students sometimes subtract in the wrong order, such as using (a - s) instead of (s - a). Each factor must be (s-a), (s-b), and (s-c) exactly, otherwise signs can flip and the multiplication becomes incorrect.

Rounding side lengths or intermediate products too early also changes the final answer. Keep exact values through the square root step and round only once at the end if a decimal is required.

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No, the triangle inequality must be satisfied. Each side must be less than the sum of the other two sides; otherwise, the triangle cannot be formed and an area cannot be computed.

It uses Heron’s formula: A = √(s(s-a)(s-b)(s-c)). The semiperimeter is s = (a + b + c)/2, and the area comes out in square units.

No, not when you use Heron’s formula. Knowing all three sides is enough to compute the area without finding the height or any angles.

Measure the three sides in feet and label them a, b, and c. Compute s = (a + b + c)/2 and then A = √(s(s-a)(s-b)(s-c)) to get the area in square feet.

Using Heron’s formula, the semiperimeter is 10 and the area is √(200) = 10√2. As a decimal, this is approximately 14.14 square units (square inches, cm², etc., depending on the input unit).

Use any consistent unit such as cm, m, or ft for all three sides. The output will be in square units of the same system, such as cm², m², or ft².

If the side lengths do not satisfy the triangle inequality, at least one of (s-a), (s-b), or (s-c) becomes zero or negative in a way that makes the product unsuitable for a real square root. This indicates the given lengths cannot form a real triangle.