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Surface area of a sphere in terms of diameter

Understand the Problem

The question is asking for the formula to calculate the surface area of a sphere using its diameter. The surface area of a sphere is typically expressed in terms of its radius, but this question seeks the expression in terms of the diameter instead.

Answer

The formula is $A = \pi d^2$.
Answer for screen readers

The formula for the surface area of a sphere in terms of its diameter is: $$ A = \pi d^2 $$

Steps to Solve

  1. Identify the relationship between diameter and radius

The radius ($r$) of a sphere is half of its diameter ($d$). This can be expressed as: $$ r = \frac{d}{2} $$

  1. Write the formula for the surface area of a sphere

The surface area ($A$) of a sphere is calculated using the formula: $$ A = 4\pi r^2 $$

  1. Substitute the radius in terms of the diameter

Now, replace $r$ with $\frac{d}{2}$ in the surface area formula: $$ A = 4\pi \left(\frac{d}{2}\right)^2 $$

  1. Simplify the equation

Calculating the squared term gives us: $$ A = 4\pi \cdot \frac{d^2}{4} $$

  1. Finalize the surface area formula

The $4$ in the numerator and denominator cancel out, resulting in: $$ A = \pi d^2 $$

The formula for the surface area of a sphere in terms of its diameter is: $$ A = \pi d^2 $$

More Information

This formula shows that the surface area of a sphere increases with the square of its diameter, meaning that even a small increase in diameter can lead to a larger increase in surface area. This concept is important in geometry and has applications in various fields including physics and engineering.

Tips

  • Confusing diameter and radius: Remember that the diameter is twice the radius. Always verify the relationships between the measurements.
  • Forgetting to square the radius when applying the surface area formula. Ensure that the entire term for the radius is squared, including any fractions.
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