Question

What is the area of a circle of radius \(24.87 \mathrm{~m}\) ?

Short answer

The area is approximately 1943.22 m².

Step-by-step solution

Formula for Area of a Circle

To find the area of a circle, we use the formula: \[ A = \pi r^2 \]where \( A \) is the area and \( r \) is the radius of the circle.

Substitute the Radius

Given that the radius \( r \) is \( 24.87 \) meters, we substitute it into the formula:\[ A = \pi (24.87)^2 \]

Calculate the Area

First, compute \( (24.87)^2 \):\[ (24.87)^2 = 618.5569 \]Now, substitute back to find \( A \):\[ A = \pi \times 618.5569 \]

Calculate Final Area Value

To calculate \( A \), multiply \( 618.5569 \) by \( \pi \) (approximately \( 3.14159 \)):\[ A \approx 3.14159 \times 618.5569 \approx 1943.2229 \]Therefore, the area is approximately \( 1943.22 \mathrm{~m}^2 \).

Key concepts

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space. It includes numerous concepts, such as points, lines, surfaces, and solids. Geometry is essential for understanding the world around us, as it is all about the physical space we live in. In this particular problem, we are dealing with the geometry of circles. Circles are fascinating shapes that have equal distances from a central point to all points on their outline, known as the boundary. This equal distance is called the radius. Understanding geometry helps you not only solve mathematical problems but also to grasp more complex concepts in a visual and practical way. Circles, in geometric terms, are two-dimensional shapes with a flat surface. The calculation of the area of a circle is a fundamental exercise in geometry, allowing us to apply the concepts of space and measurement in practical scenarios.

Radius of a Circle

The radius of a circle is one of its most important features. It is the straight-line distance from the center of the circle to any point on its boundary. Understanding the concept of a circle's radius is crucial when calculating its area.

• The radius helps us know the size of the circle, as larger radii mean larger circles.
• When you know the radius, you can also find other related measurements, like the circumference and diameter, using simple formulas.
• For instance, the diameter is twice the radius.

In our exercise, the radius is given as 24.87 meters. This is a fairly large circle in practical terms, possibly the size of a large sports field or a small performance stage. Keep in mind that the radius is always measured in a straight line, which makes it quite simple to understand and use in further calculations.

Mathematical Formulas

Mathematical formulas are tools that help us solve numerical problems by providing a set structure to follow. They turn complex problems into manageable steps.

• The formula to find the area of a circle is \( A = \pi r^2 \).
• In this formula, \( A \) represents the area, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the circle.

In the exercise example, once you substitute the given radius (24.87 meters) into the formula, you can find the area by squaring the radius and multiplying by \( \pi \). Mathematical formulas like this one not only aid in quick calculations but also ensure precision. They are universally applicable, meaning once you understand the formula, you can apply it to any circle, regardless of its size. Thus, mathematical formulas are indispensable when it comes to solving everyday problems related to shapes and measurements.