For problems \( 4-6 \), find the diameter, radius, and area of a circle with the given circumference. 4. Circumference: \( 10 \pi \mathrm{~m} \)

To solve for the **diameter**, **radius**, and **area** of a circle given its **circumference**, we'll use the fundamental formulas related to a circle's geometry. ### Given: - **Circumference (C)** = \( 10\pi \) meters ### Formulas: 1. **Circumference in terms of Diameter (d):** \[ C = \pi d \] 2. **Circumference in terms of Radius (r):** \[ C = 2\pi r \] 3. **Area (A) in terms of Radius:** \[ A = \pi r^2 \] ### Step-by-Step Solution: #### 1. Finding the Diameter (d): Using the circumference formula in terms of diameter: \[ C = \pi d \] Plug in the given circumference: \[ 10\pi = \pi d \] Solve for \( d \): \[ d = \frac{10\pi}{\pi} = 10 \text{ meters} \] **Diameter (d) = 10 meters** #### 2. Finding the Radius (r): Using the circumference formula in terms of radius: \[ C = 2\pi r \] Plug in the given circumference: \[ 10\pi = 2\pi r \] Solve for \( r \): \[ r = \frac{10\pi}{2\pi} = 5 \text{ meters} \] **Radius (r) = 5 meters** #### 3. Finding the Area (A): Using the area formula: \[ A = \pi r^2 \] Plug in the radius: \[ A = \pi (5)^2 = 25\pi \text{ square meters} \] **Area (A) = \( 25\pi \) square meters** ### **Summary:** - **Diameter:** \( 10 \) meters - **Radius:** \( 5 \) meters - **Area:** \( 25\pi \) square meters