Question

1. a circle’s circumference is approximately 76 cm. estimate the radius, diameter, and area of the circle.

Explanation:

Step1: Recall circumference formula

The formula for the circumference \( C \) of a circle is \( C = \pi d \) (or \( C = 2\pi r \)), where \( d \) is the diameter and \( r \) is the radius. We know \( C \approx 76 \) cm and use \( \pi \approx 3.14 \) (or \( \frac{22}{7} \), but 3.14 is common for estimation). First, find the diameter: \( d=\frac{C}{\pi} \).
Substituting \( C = 76 \) and \( \pi \approx 3.14 \), we get \( d\approx\frac{76}{3.14}\approx24.2 \) cm (or using \( \pi\approx\frac{22}{7} \), \( d\approx\frac{76\times7}{22}=\frac{532}{22}\approx24.18 \) cm, so approximately 24 cm for a simpler estimate).

Step2: Find the radius

The radius \( r \) is half the diameter, so \( r=\frac{d}{2} \). Using the estimated diameter \( d\approx24 \) cm, \( r\approx\frac{24}{2}=12 \) cm (or using the more precise \( d\approx24.2 \), \( r\approx\frac{24.2}{2}=12.1 \) cm, so approximately 12 cm).

Step3: Find the area

The formula for the area \( A \) of a circle is \( A=\pi r^{2} \). Using \( r\approx12 \) cm and \( \pi\approx3.14 \), we get \( A\approx3.14\times12^{2}=3.14\times144 = 452.16 \) cm² (or with \( r\approx12.1 \), \( A\approx3.14\times(12.1)^{2}\approx3.14\times146.41\approx459.7 \) cm², so approximately 452 cm² or 460 cm² depending on precision).

Answer:

• Radius: Approximately \( \boldsymbol{12} \) cm (or more precisely ~12.1 cm)
• Diameter: Approximately \( \boldsymbol{24} \) cm (or more precisely ~24.2 cm)
• Area: Approximately \( \boldsymbol{452} \) cm² (or ~460 cm²)

(Note: If using \( \pi\approx\frac{22}{7} \) throughout:
Diameter: \( d=\frac{76}{\frac{22}{7}}=\frac{76\times7}{22}=\frac{532}{22}\approx24.18 \) cm ≈24 cm
Radius: \( r=\frac{24.18}{2}\approx12.09 \) cm ≈12 cm
Area: \( A=\frac{22}{7}\times(12.09)^{2}\approx\frac{22}{7}\times146.17\approx460 \) cm²)