Short Answer 
The area of Figure 1 is calculated by finding the area of a triangle and subtracting the area of a quarter circle, resulting in A = 36(œÄ – 2) cm¬≤. For Figure 1’s perimeter, it combines the quarter circle’s circumference and a triangle’s side length to yield P = 6(œÄ + 2‚àö2) cm, while Figure 2’s area is 576 cm¬≤ with a perimeter of P = 24(2 + œÄ) cm.
Step 1: Calculate the Area of Figure 1
To determine the area of Figure 1, you need to calculate the area of a triangle and subtract the area of a quarter circle. Here’s how you do it:
- Calculate the area of the quarter circle using the formula A = (1/4)πr², where r = 12 cm, giving A = 36 cm².
- Now, find the area of the triangle with base (b) and height (h) both equal to 12 cm using A = (1/2)(b)(h), which results in A = 72 cm².
- Finally, subtract the quarter circle area from the triangle’s area to find the total area: A = 36(œÄ – 2) cm¬≤.
Step 2: Determine the Perimeter of Figure 1
The perimeter of Figure 1 comprises the perimeter of the quarter circle and the length of one side of the triangle. Here’s the process:
- Calculate the circumference of the quarter circle using C = (1/4)(2πr) which simplifies to C = 6 cm + 12√2 cm.
- Using the Pythagorean theorem, find the length of side AC, which comes out to be 12‚àö2 cm.
- Combine these measurements: P = 6(π + 2√2) cm for the total perimeter.
Step 3: Analyze Figure 2’s Area and Perimeter
Figure 2 consists of a semicircle and a square, and its area and perimeter can be calculated as follows:
- Calculate the area of the square, which is straight forward: A = b², hence A = 24² = 576 cm².
- Finding the perimeter involves summing up two sides of the square with the circumference of the semicircles: P = 2(AB) + πD.
- The final calculation gives you P = 24(2 + π) cm after factoring out the common term.