Half Circle Triangle 05Eb75

1. **State the problem:** We have an equilateral triangle with side length $12$ units joined to a half circle on one side. We need to find the perimeter of the combined shape. 2. **Identify the shape and perimeter components:** - The equilateral triangle has three equal sides of length $12$ units. - The half circle is joined along one side of the triangle, so the diameter of the half circle equals the side length of the triangle, which is $12$ units. 3. **Formula for perimeter:** The perimeter of the combined shape is the sum of: - The two remaining sides of the triangle (since one side is shared with the half circle and not part of the perimeter), and - The curved half circle perimeter (half the circumference of a full circle). 4. **Calculate the half circle circumference:** The circumference of a full circle is $$C = 2\pi r$$ where $r$ is the radius. Since the diameter $d = 12$, the radius is $$r = \frac{12}{2} = 6$$. Half circle circumference is half of full circumference: $$\frac{1}{2} \times 2\pi r = \pi r = \pi \times 6 = 6\pi$$. 5. **Calculate the triangle sides contributing to perimeter:** Two sides of the equilateral triangle remain, each $12$ units: $$2 \times 12 = 24$$. 6. **Sum the perimeter:** $$P = 24 + 6\pi$$. 7. **Final answer:** The perimeter of the resulting shape is $$\boxed{24 + 6\pi}$$ units.