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Find the perimeter and total area of the polygon

To find the perimeter and the total area of the polygon, we need to analyze the given shape, which is composed of a square and an isosceles triangle on top of it.

### Perimeter Calculation

1. **Square Side Length:**
   - Each side of the square is given as 12 units.

2. **Triangle Sides:**
   - The two equal sides of the isosceles triangle are each 10 units.
   - The base of the triangle (also the top side of the square) is 12 units.

To find the perimeter, we add up the lengths of all the sides:
- Perimeter of the square: \( 4 \times 12 = 48 \) units
- Perimeter of the triangle (excluding the base, as it is already included in the square’s perimeter): \( 2 \times 10 = 20 \) units

Thus, the total perimeter is:
\[ 48 + 20 = 68 \text{ units} \]

### Area Calculation

1. **Area of the Square:**
   - The area of the square is given by:
   \[ \text{Area of the square} = \text{side}^2 = 12^2 = 144 \text{ square units} \]

2. **Area of the Triangle:**
   - The height of the triangle is given as 8 units.
   - The base of the triangle is 12 units.

   The area of the triangle is:
   \[ \text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 8 = 48 \text{ square units} \]

Thus, the total area of the polygon is:
\[ 144 + 48 = 192 \text{ square units} \]

### Summary

- **Perimeter:** 68 units
- **Total Area:** 192 square units

Would you like more details or have any questions?

Here are 5 more questions you can ask:
1. How do you calculate the area of a different polygon?
2. Can you explain the Pythagorean theorem?
3. How do you find the area of a circle?
4. What is the formula for the perimeter of a rectangle?
5. How do you solve a quadratic equation?

**Tip:** When solving for the area of composite shapes, break the shape down into simpler components like squares, rectangles, triangles, and circles, and then sum their areas.

Learn how to find the perimeter and total area of a polygon composed of a square and an isosceles triangle. Includes detailed calculations and formulas for perimeter and area. Suitable for Grades 7-9.

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