Problem 12 - Elementary Algebra

Question

Find the area of a circle with a radius of \(10 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The area of the circle is \100\pi \mathrm{~cm}^2.

Step 1 - Identify the Formula

The formula to calculate the area of a circle is given by \[ A = \pi r^2 \], where \(A\) is the area and \(r\) is the radius of the circle.

Step 2 - Substitute the Radius

Substitute the given radius into the formula. In this case, \(r = 10 \mathrm{~cm}\). So, the formula becomes \[ A = \pi (10 \mathrm{~cm})^2 \].

Step 3 - Calculate the Area

First, square the radius: \[ 10 \mathrm{~cm} \times 10 \mathrm{~cm} = 100 \mathrm{~cm}^2 \]. Then, multiply by \(\pi\): \[ A = \pi \times 100 \mathrm{~cm}^2 \].

Step 4 - Write the Final Answer

The area of the circle is \[ A = 100\pi \mathrm{~cm}^2 \].

Key Concepts

circle area formula

The area of a circle can be calculated using a specific formula. The formula is \( A = \pi r^2 \) where \( A \) represents the area of the circle, \( \pi \) is a constant value approximately equal to 3.14159, and \( r \) is the radius of the circle. This formula tells you how much space is inside the circle.

It is quite straightforward: just square the radius and then multiply by \( \pi \). Remember, squaring a number means multiplying the number by itself. For example, if the radius is 4 cm, you would calculate 4 cm * 4 cm. The result is then multiplied by \( \pi \) (which is often approximated as 3.14 for easier calculations). So, the area formula helps you figure out the size of the flat surface enclosed by the circle.

radius

The radius is a key concept when dealing with circles. It refers to the distance from the center of the circle to any point on its edge. In simplest terms, it's half the diameter of the circle. For instance, if you know the diameter of a circle is 20 cm, then the radius would be half of that, which is 10 cm.

Named \( r \) in mathematical formulas, the radius is essential for calculating the circle's area. The formula \( A = \pi r^2 \) directly uses the radius. Without knowing the radius, you cannot compute the area. If ever you're given the diameter and need to find the area, simply divide the diameter by 2 to get the radius first.

pi

Pi, denoted by the Greek letter \( \pi \), is an important mathematical constant used in circle calculations. Numerically, \( \pi \) is approximately equal to 3.14159, but it is an irrational number, meaning it has an infinite number of decimal places without repeating.

In the formula for the area of a circle, pi helps relate the diameter and radius to the circle’s area. When using \( \pi \), you essentially scale the radius squared to match the circle’s true area. For simple or quick calculations, \( \pi \) is often rounded to 3.14, making it much easier to work with. This small factor greatly simplifies complex computations involving circles.