# 11.8.4. Heron's Formula

Determine to the nearest 3 decimal places the area of an isosceles triangle whose side length is cm and base angle is .

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### Example

Determine to the nearest 3 decimal places the area of an isosceles triangle whose side length is cm and base angle is .

### Solution

The isosceles triangle is composed of two right-angled triangles. Each has interior angles measuring , , and , and the side opposite to the angle has a length of 49 cm. The sine rule states that the lengths of the sides of a triangle are proportional to the sines of the opposite angles. First, use the sine rule to find the approximate length, , of the side opposite the angle.

Now use the sine rule to find the approximate length, , of the side opposite the angle.

According to Heron's formula, the area of a triangle whose side lengths are , , and is equal to , where is half the triangle's perimeter. Each right-angled triangle has a perimeter of approximately , so is equal to approximately . Find the approximate area of each right-angled triangle using Heron's formula.

Thus, the approximate area of the isosceles triangle is .

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