Question

use the equation to solve for the area of the inner square. start by finding the area of the outer square. area of outer square - sum of the areas of the 4 triangles = area of inner square

Explanation:

Step1: Determine outer square side length

From the grid, the outer square has a side length of 8 units (counting the grid squares). The area of a square is \( \text{side}^2 \), so the area of the outer square is \( 8 \times 8 = 64 \) square units.

Step2: Analyze the 4 triangles

Each of the 4 triangles is a right triangle with legs of length 4 units (since the outer square side is 8, and the triangles are formed by dividing the sides into two equal parts). The area of one triangle is \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8 \) square units.

Step3: Calculate sum of 4 triangles' areas

There are 4 such triangles, so the total area of the 4 triangles is \( 4 \times 8 = 32 \) square units.

Step4: Find inner square area

Using the given equation: \( \text{Area of outer square} - \text{Sum of areas of 4 triangles} = \text{Area of inner square} \). Substituting the values: \( 64 - 32 = 32 \) square units.

Answer:

The area of the inner square is 32 square units. The area of the outer square is 64 square units, the sum of the areas of the 4 triangles is 32 square units, and the area of the inner square is 32 square units.