Sphere Diameter Calculation from Surface Area

Let's find the diameter of a sphere when we know its surface area. The question gives us the surface area of the sphere as 1386 cm² and asks for the diameter in cm.

Understanding Sphere Formulas

To solve this problem, we need to use the formula for the surface area of a sphere and the relationship between the radius and diameter.

• The surface area (A) of a sphere with radius (r) is given by the formula: \[A = 4 \pi r^2\]
• The diameter (d) of a sphere is twice its radius (r): \[d = 2r\]

Step-by-Step Calculation

We are given the surface area \(A = 1386\) cm². We can use the surface area formula to first find the radius (r), and then use the radius to find the diameter (d).

Step 1: Find the Radius (r)

Substitute the given surface area into the formula \(A = 4 \pi r^2\). We will use the value of \(\pi \approx \frac{22}{7}\) for this calculation.

\[1386 = 4 \times \frac{22}{7} \times r^2\]

\[1386 = \frac{88}{7} \times r^2\]

Now, we need to isolate \(r^2\). Multiply both sides by \(\frac{7}{88}\).

\[r^2 = 1386 \times \frac{7}{88}\]

Let's simplify the fraction \(\frac{1386}{88}\) first. Both numbers are divisible by 2:

\[r^2 = \frac{1386 \div 2}{88 \div 2} \times 7 = \frac{693}{44} \times 7\]

Now, let's see if 693 is divisible by 11 (since 44 is \(4 \times 11\)). The sum of alternating digits in 693 is \((6+3) - 9 = 9 - 9 = 0\). Since the result is 0, 693 is divisible by 11.

\[693 \div 11 = 63\]

\[44 \div 11 = 4\]

So the expression becomes:

\[r^2 = \frac{63}{4} \times 7\]

\[r^2 = \frac{63 \times 7}{4}\]

\[r^2 = \frac{441}{4}\]

To find r, take the square root of both sides:

\[r = \sqrt{\frac{441}{4}}\]

We know that \(\sqrt{441} = 21\) and \(\sqrt{4} = 2\).

\[r = \frac{21}{2} \text{ cm}\]

So, the radius of the sphere is 10.5 cm.

Step 2: Find the Diameter (d)

The diameter is twice the radius.

\[d = 2r\]

Substitute the value of r we found:

\[d = 2 \times \frac{21}{2}\]

\[d = 21 \text{ cm}\]

Conclusion

The diameter of the sphere with a surface area of 1386 cm² is 21 cm.

Revision Table: Sphere Properties

Additional Information: Sphere Calculations

Spheres are fundamental 3D shapes, and understanding their properties like surface area and volume is crucial in geometry. The value of \(\pi\) is an irrational number, often approximated as 3.14 or \(\frac{22}{7}\) for calculations. The choice of approximation can slightly affect the final answer, but \(\frac{22}{7}\) is commonly used when it helps simplify calculations involving multiples of 7, as seen in this problem with 1386 (which is divisible by 7: \(1386 = 7 \times 198\)). Always pay attention to the units given in the problem (cm² for area, cm for diameter) and ensure your final answer has the correct units.