why is the equator hotter than the north and south poles?

Because the amount of light striking a unit area is greater at the equator than at either pole.

The following might help.

Take a ruled piece of paper and pretend the lines represent rays of light coming from the sun. Now take another piece of paper and lie is verticially on that one, such that the lines on the two pages line up. Note that each line-to-line interval in the second sheet receives exactly one line-to-line interval on the first sheet. That is analogous to the equator.

Now, hold the second sheet near its last line and start rotating it clockwise, until it has been rotated about 45 degrees. Now note that each line-to-line interval on the second is matched up with LESS than 1 line-to-line interval the first. Thus, less rays of sunlight are striking each unit area when the surface it tilted away from being perpendicular to the light source.

As you rotate it more and more, each unit on the rotating sheets receives less and less line-intervals (light rays). This models near the poles.

The sun appears overhead near the equator -- from 23.5 degrees North to 23.5 degrees South. The farther from the Equator the farther the sun's rays have to travel.

http://geography.about.com/library/misc/blequator.htm

The equator is hotter than the North and South Poles because of various factors, including the angle at which sunlight strikes the Earth, the distribution of solar energy, and the characteristics of the Earth's surface.

1. Angle of sunlight: The Earth is tilted at an angle of approximately 23.5 degrees, causing the Sun's rays to hit different parts of the Earth at different angles throughout the year. Near the equator, the sunlight hits almost directly overhead, resulting in more concentrated solar energy. In contrast, sunlight reaches the poles at a much lower angle, spreading the energy over a larger surface area.

2. Distribution of solar energy: The equator receives more direct and intense sunlight than the poles because the Sun's rays are more concentrated. As a result, the equator absorbs more solar energy, leading to higher temperatures. Near the poles, sunlight is spread out over a larger area, reducing the overall energy received.

3. Characteristics of the Earth's surface: The equator is surrounded by a large amount of water, which has a high heat capacity. Consequently, the ocean absorbs and stores heat, moderating the temperature and making the equatorial regions hotter. In contrast, the polar regions are covered by snow and ice, which have a low heat capacity and reflect sunlight back into space, contributing to lower temperatures.

To understand why the equator is hotter than the poles, one can study concepts such as Earth's axial tilt, solar radiation, heat transfer, and the properties of water and ice.