Question

Find the surface area of a sphere with a diameter of 24 meters. Round to the nearest hundredth.

Short answer

The surface area of the sphere is approximately 1809.56 square meters.

Step-by-step solution

Understand the Given Information

We are given the diameter of the sphere, which is 24 meters. To find the surface area, we will need to first find the radius of the sphere.

Calculate the Radius

The formula to find the radius of a sphere when the diameter is given is: \[ r = \frac{d}{2} \] where \( d \) is the diameter. Substituting the given diameter of 24 meters: \( r = \frac{24}{2} = 12 \) meters.

Use the Surface Area Formula for a Sphere

The formula for the surface area of a sphere is: \[ A = 4\pi r^2 \] where \( r \) is the radius. We've already calculated \( r = 12 \) meters. Substitute \( r \) into the formula: \( A = 4\pi (12)^2 \).

Calculate the Surface Area

First, calculate \( 12^2 = 144 \). Then multiply by 4 and \( \pi \): \[ A = 4 \pi \times 144 = 576 \pi \].

Simplify and Round the Answer

Using the approximate value \( \pi \approx 3.14159 \), calculate \( 576 \times 3.14159 \approx 1809.56 \). Round to the nearest hundredth: the surface area is approximately 1809.56 square meters.

Key concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It helps us to understand spatial relationships and solve practical problems. One common shape found in geometry is the sphere.

A sphere is a perfectly round three-dimensional shape, like a ball. Made up of all points that are the same distance from a central point. Geometrically, it has interesting properties, as each part looks the same from whichever angle you view it.

Understanding spheres is important in many fields, including physics, engineering, and architecture. They often occur in nature and can be seen in things like planets or bubbles.

Diameter

The diameter is an important concept in geometry, especially when dealing with circles and spheres.

The diameter is a straight line stretching across the sphere and passing through the center. It is the longest distance you can measure across a sphere. In simple terms, it connects two opposite points on the sphere's surface.

Knowing the diameter helps in calculations involving the size of the sphere. This includes surface area and volume calculations. For example, if you know the diameter, you can quickly find the radius, which is half of the diameter. In our exercise, the diameter given is 24 meters.

Radius Calculation

To calculate the radius of a sphere from its diameter, you use a simple yet crucial formula. The radius is half the diameter of the sphere. This relationship is fundamental when working with round shapes.

The formula to find the radius is:

• \( r = \frac{d}{2} \)

where \( d \) is the diameter.

For example, if the diameter is 24 meters, the radius would be

• \( r = \frac{24}{2} = 12 \text{ meters} \)

The radius is useful since it is directly used in calculating the surface area and volume for spheres.

Mathematical Formula

Mathematical formulas are tools that help us perform calculations. They break down complex concepts into easy-to-follow steps. When dealing with the sphere, the surface area is often what we want to find.

The mathematical formula for the surface area of a sphere is:

• \( A = 4\pi r^2 \)

where \( r \) is the radius.
If we have a radius of 12 meters, from our previous calculation, we can substitute it into the formula:

• \( A = 4\pi (12)^2 \)

Calculate \( 12^2 \) which is 144. Then multiply this by 4 and \( \pi \):

• \( A = 4 \times 144 \times \pi = 576\pi \)

When using the approximate value of \( \pi \) (3.14159), the result becomes approximately 1809.56 square meters. This process shows how formulas make it easy to compute the surface area efficiently.