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High School Trigonometry: Homework Help Resource30 chapters | 203 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

A sphere is a perfectly round three-dimensional object. This lesson will give a mathematical definition of a sphere, discuss the formulas associated with spheres and finish with a quiz.

A **sphere** is a geometrical figure that is perfectly round, 3-dimensional and circular - like a ball. Geometrically, a sphere is defined as the set of all points equidistant from a single point in space. The distance from an outer point to the center of the sphere is the **radius**(*r*) and the maximum straight distance from one side of the sphere to another is the **diameter** (*d*).

A **hemisphere** is what you would call half a sphere, if you were to split a sphere down the middle.

A **great circle** of the sphere is a circle that has the same radius and center as the sphere itself.

A sphere is the geometrical figure that occupies the biggest space, but has the smallest surface area. In other words, when something needs to be as small as possible but still have a large volume, it takes the shape of a sphere. That is why a balloon is round when you blow it up. It wants to hold as much air as possible with the smallest amount of surface. This occurs quite often in nature - common examples include bubbles and water drops.

The planet Earth is called a **spheroid** because it is extremely close to being a sphere, but is not perfectly round. It is elongated a bit at the North and South poles.

Here are the most common formulas associated with a sphere:

The **volume** (*V*) of a sphere is the amount of 'stuff' that could fit inside the sphere. Usually, when discussing volume, it is in terms of a liquid or gas. An example would be, 'What is the volume of air that can fit inside a basketball?' The formula for volume is:

*V* = (4/3?*r*^3

**Surface area** is just what its name implies: it's the area of the surface of an object. So, if you could cut open the basketball from the previous example and lay its surface out flat, you could see the surface area? This is useful for determining how much material would be needed to cover the sphere in question. The formula for surface area is:

*SA* = 4?*r*^2

Using our formulas, let's take a look at some example problems using spheres:

1.) What is the volume of a ball with a radius of 6 inches? (We'll use 3.14 for pi and round our answer to the nearest .10.)

Since we are determining the volume, we use the equation:

*V* = (4/3) ?*r*^3

We know the radius - the rest is just simple math.

*V* = (4/3)(3.14)(6^3)

*V* = 904.3 in^3

2.) What is the surface area of a sphere whose radius is 3 ft? (Use 3.14 for pi and round your answer to the nearest .10.)

We know our formula for surface area is:

*SA* = 4?*r*^2

*SA* = (4)(3.14)(3^2)

*SA* = 113.04 ft^2

3.) What is the volume of a water drop if the diameter is 1.2 cm? (Use 3.14 for pi and round your answer to the nearest .10.)

This problem is a bit trickier, since we are given the diameter and not the radius. To find the radius, just divide the diameter in half. Then, you can use the equation for volume.

1.2/2 = 0.6cm

*V* = (4/3) ?*r*^3

*V* = (4/3)(3.14)(0.6^3)

*V* = 81.0 cm^3

A **sphere** is a geometrical figure that is perfectly round with no corners or edges. It is a 3-dimensional shape, and, using the given equations, it is not difficult to find the volume or surface area. The **volume**(*V*) of a sphere is the amount of 'stuff' that could fit inside the sphere. The formula for volume is:

*V* = (4/3) ?*r*^3

**Surface area** is just what its name implies. It's the area of the surface of an object. The formula for surface area is:

*SA* = 4?*r*^2

Spheres are found quite often in nature, since they are the perfect shape to fit a large volume in a small area. A **hemisphere** is what you would call half a sphere if you were to split a sphere down the middle. A **great circle** of the sphere is a circle that has the same radius and center as the sphere itself.

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High School Trigonometry: Homework Help Resource30 chapters | 203 lessons

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