Question

What is the area of a circle with a radius of \(12.77 \mathrm{~m}\) ? Recall that the area of a circle is given by area \(=\pi\) (radius) \(^{2}\).

Short answer

The area is approximately 512.85 m².

Step-by-step solution

Recall the Formula

The formula for finding the area of a circle is \[A = \pi r^{2}\]where \(A\) is the area and \(r\) is the radius of the circle.

Substitute the Radius Value

Given the radius \(r = 12.77\, \text{m}\), substitute this value into the formula:\[A = \pi (12.77)^{2}\]

Calculate the Square of the Radius

Compute \(12.77^{2}\):\[12.77 \times 12.77 = 163.0729\]

Compute the Area

Substitute \(163.0729\) back into the formula and multiply by \(\pi\). Use \(\pi \approx 3.14159\):\[A = 3.14159 \times 163.0729 \approx 512.85 \, \text{m}^2\]

Final Answer

The calculated area for the circle with a radius of \(12.77\, \text{m}\) is approximately:\[512.85 \, \text{m}^2\]

Key concepts

Understanding Geometry

Geometry is all around us and is a branch of mathematics that deals with shapes and their properties. It involves understanding and calculating how different forms relate to one another. Circles are particularly important in geometry because of their unique properties: every point on the edge is equidistant from the center, called the radius.

Using geometry, we can calculate various circle properties, such as circumference, diameter, and area. The area is the amount of space a shape covers on the plane. In everyday life, understanding geometry helps us when designing roads, buildings, or even artworks. It makes it easier to visualize how different shapes fit together and interact.

Applying Mathematics in Problem Solving

Mathematics is a powerful tool that lets us solve many types of problems, like our exercise on finding the area of a circle. The key to mastering mathematics is practice and understanding how to apply different formulas. These mathematical concepts form a foundation used across other subjects.

Let's break it down:

• First, find the known values— in our exercise, it's the radius.
• Next, recall and utilize the appropriate formula, here it's the area formula for a circle.
• Finally, substitute and compute the values.

By following structured steps, we can systematically approach and solve mathematical problems with confidence. Each formula and equation serves to simplify complex problems into manageable calculations. This structured approach not only helps in geometry but also trains our mind to think critically and make reasoned decisions.

Exploring the Circle Formula

The circle formula is an essential part of geometry. It helps us find critical measurements of a circle easily. Let’s unpack the formula used to calculate a circle’s area:
The area of a circle is given by the formula \(A = \pi r^2\), where \(A\) is the area, \(\pi\) is a constant approximately equal to 3.14159, and \(r\) is the radius.

• The \(\pi\) (pi) element is a mathematical constant representing the ratio of a circle's circumference to its diameter.
• The \(r^2\) indicates we need to square the radius, meaning multiplying the radius by itself.

Using the formula is straightforward. Input the radius, square it, multiply by \(\pi\), and you have your answer. This is a classic example of how formulas simplify calculations, enabling you to focus on understanding the conceptual elements rather than getting bogged down by manual computation. Once you grasp how the formula works, applying it to similar problems becomes intuitive and efficient.