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
- Identify the relationship between diameter and radius
The radius ($r$) of a sphere is half of its diameter ($d$). This can be expressed as: $$ r = \frac{d}{2} $$
- Write the formula for the surface area of a sphere
The surface area ($A$) of a sphere is calculated using the formula: $$ A = 4\pi r^2 $$
- Substitute the radius in terms of the diameter
Now, replace $r$ with $\frac{d}{2}$ in the surface area formula: $$ A = 4\pi \left(\frac{d}{2}\right)^2 $$
- Simplify the equation
Calculating the squared term gives us: $$ A = 4\pi \cdot \frac{d^2}{4} $$
- 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.
