Find a formula for the area of each figure in terms of s.

Verified step by step guidance

1. <insert step 1: Identify the type of figure you are dealing with. For example, it could be a square, triangle, or circle.>
2. <insert step 2: Recall the general formula for the area of the identified figure. For instance, for a square, the area is \( s^2 \), for a triangle, it is \( \frac{1}{2} \times \text{base} \times \text{height} \), and for a circle, it is \( \pi s^2 \) where \( s \) is the radius.>
3. <insert step 3: Substitute the given variable \( s \) into the general formula. For example, if the figure is a square, the area formula becomes \( s^2 \).>
4. <insert step 4: Simplify the expression if necessary. For example, if the figure is a triangle and \( s \) represents both the base and the height, the area formula becomes \( \frac{1}{2} s^2 \).>
5. <insert step 5: Verify that the formula is dimensionally consistent and makes sense for the given figure.>

Key Concepts

Area of a Triangle

The area of a triangle can be calculated using the formula A = 1/2 * base * height. However, when the lengths of the sides are known, Heron's formula can be used, which states that the area A is given by A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter and a, b, c are the lengths of the sides.

Semi-Perimeter

The semi-perimeter of a triangle is defined as half of the perimeter. It is calculated as s = (a + b + c) / 2, where a, b, and c are the lengths of the triangle's sides. The semi-perimeter is crucial in Heron's formula for calculating the area of a triangle when only the side lengths are known.

Heron's Formula

Heron's formula provides a way to calculate the area of a triangle when the lengths of all three sides are known. It utilizes the semi-perimeter and is expressed as A = √(s(s-a)(s-b)(s-c)). This formula is particularly useful in trigonometry for finding areas without needing to know the height or angles of the triangle.