Heron’s Formula Basics: Semiperimeter, Validity & Setup

The semiperimeter is defined as half the perimeter:

s = (a + b + c) / 2

This can be written as:

s = ½(a + b + c)

Hence, s = ½(a + b + c).

Heron’s Formula for the area of a triangle with semiperimeter s is:

A = √[s(s − a)(s − b)(s − c)]

Hence, the correct formula is A = √[s(s − a)(s − b)(s − c)].

Heron’s Formula naturally uses the expressions (s − a), (s − b), and (s − c), so using s simplifies the expression inside the square root.

Hence, the semiperimeter is used because it simplifies (s − a)(s − b)(s − c).

Compute the semiperimeter:

s = (10 + 10 + 12) / 2 = 32 / 2 = 16

Hence, s = 16 cm.

For a right triangle with legs a and b, area = ½ab.

Heron’s Formula also gives the same result.

Hence, it matches ½ab.

The expression s(s − a)(s − b)(s − c) is the product used before taking the square root. It gives the squared area of the triangle.

Hence, it represents the product used to find squared area.

Area is always measured in square units such as cm² or m². Hence, Heron’s Formula produces square units (cm²).

For an equilateral triangle, a = b = c, and semiperimeter:

s = 3a/2

Substitute into Heron's Formula:

A = √[s(s − a)³]

Hence, area simplifies to √[s(s − a)³].

The triangle 3–4–5 is a right triangle, and its area is:

A = ½·3·4 = 6

This is a simple integer area.

Hence, the triangle 3, 4, 5 gives the simplest integer area.

Check the triangle inequality:

4 + 9 = 13, which is NOT greater than 14 → invalid triangle.

Hence, Heron’s Formula cannot be applied because the triangle is invalid.

Compute (s − a):

s − a = 10 − 7 = 3

Hence, (s − a) = 3 cm.

For three lengths to form a triangle, they must satisfy the triangle inequality:

a + b > c

a + c > b

b + c > a

Hence, the sum of any two sides must be greater than the third side.

Given s = 12, the full perimeter is: perimeter = 2s = 24

Two sides are 7 and 9, so the third side is: third side = 24 − (7 + 9) = 24 − 16 = 8

Hence, the third side = 8 cm.

Heron's Formula requires the semiperimeter s and the values (s − a), (s − b), (s − c).

The most efficient first step is to compute s.

Hence, the first step is to find s.

Heron’s Formula is useful because it finds the area using only the side lengths, without needing the height.

Hence, it finds area without needing altitude.

Compute (s − a):

s − a = 8 − 7 = 1

Hence, (s − a) = 1 m.

Check if the sides form a right triangle:

5² + 12² = 25 + 144 = 169 = 13²

So it satisfies the Pythagorean theorem.

Hence, the triangle is right-angled.

Check triangle inequality.

For 7, 10, 2002:

7 + 10 = 17, which is NOT greater than 2002 → invalid.

Hence, 7, 10, 2002 is not a valid triangle.

Compute approximate areas:

Using Heron’s Formula, the triangle with sides 7, 8, 9 gives area ≈ 26.8, which is the largest among the choices.

Hence, the triangle 7, 8, 9 has the largest area.

If one side increases while the others stay fixed, area increases up to a maximum, then decreases toward zero as the triangle becomes nearly flat. Hence, area increases then decreases.