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. Common figures include triangles, squares, rectangles, circles, etc.
2. insert step 2: Recall the general formula for the area of the identified figure. For example, the area of a square is given by \(A = s^2\), where \(s\) is the side length.
3. insert step 3: If the figure is a triangle, use the formula \(A = \frac{1}{2} \times \text{base} \times \text{height}\). If the triangle is equilateral, the area can also be expressed in terms of \(s\) as \(A = \frac{\sqrt{3}}{4} s^2\).
4. insert step 4: For a circle, the area is given by \(A = \pi s^2\), where \(s\) is the radius.
5. insert step 5: Substitute the given or known values of \(s\) into the appropriate formula to express the area in terms of \(s\).

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 a crucial component in Heron's formula for calculating the area of a triangle.

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 for triangles where height is not easily determined.