Solve for the area of shaded regions in geometric figures, including a triangle with an inscribed circle and a rectangle within a rectangle.

Let’s solve each problem step by step.

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Problem 1: Circle inside a triangle

We are given:
- A right triangle with base = 12 cm, height = 14 cm.
- A circle inside it with radius = 3 cm.
- We need to find the area of the shaded region — that means the part of the triangle NOT covered by the circle.
- Use π = 3.14

Step 1: Find area of the triangle
Area of triangle = (base × height) ÷ 2
= (12 × 14) ÷ 2
= 168 ÷ 2
= 84 cm²

Step 2: Find area of the circle
Area of circle = π × r²
= 3.14 × 3²
= 3.14 × 9
= 28.26 cm²

Step 3: Subtract circle area from triangle area to get shaded region
Shaded area = Triangle area – Circle area
= 84 – 28.26
= 55.74 cm²

✔ Double-check:
Triangle: 12×14=168 → ÷2 = 84 ✔️
Circle: 3²=9 → 9×3.14=28.26 ✔️
Subtract: 84 - 28.26 = 55.74 ✔️

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Problem 2: Rectangle inside a rectangle

We are given:
- Outer rectangle: width = 14 in, height = 12 in
- Inner rectangle: width = 10 in, height = 8 in
- Shaded region is the border between them — so we subtract inner area from outer area.

Step 1: Area of outer rectangle
= length × width
= 14 × 12
= 168 in²

Step 2: Area of inner rectangle
= 10 × 8
= 80 in²

Step 3: Shaded area = Outer – Inner
= 168 – 80
= 88 in²

✔ Double-check:
Outer: 14×12 = 168 ✔️
Inner: 10×8 = 80 ✔️
Difference: 168 - 80 = 88 ✔️

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Final Answer:
1. 55.74 cm²
2. 88 in²