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A pyramid with a square base has side lengths of 2 meters and a height of 3 meters. What is the volume?


​ ​This is the Cheops Great Pyramid of Giza located outside Cairo, Egypt. It was built four and a half thousand years ago. Back then, the Egyptians had already mastered mathematics and engineering with impressive precision. The Pyramid's base is almost a perfect square. Its side is approximately 230.4 meters long.

When the tip still existed, the Pyramid was 146 1/2 meters tall. So, we're going to calculate the volume of the Cheops Pyramid. We have the area of the base. It's 230.4 meters times itself. If we multiply the area of the base by the height, we get the volume of a cuboid.

The Pyramid fits completely inside the cuboid. The Pyramid's volume is actually exactly one-third of the volume of the cuboid. If you want to know why, follow this. We'll put the pyramid aside for a minute, and look at this shape instead - a cube. In the middle of the cube, there's a point.

Now, we draw the lines from each of the cube's vertices to this point, and then we cut the cube along the lines and take it apart like this. We get one, two, three, four, five, six pyramids. A cube can be divided into six equal pyramids. This is useful since the volume of a cube is easy to calculate and the volume of one of the small pyramids equals one-sixth of the cube's volume. ​ ​Now, let's take one of these small pyramids and make another cuboid like we did around the Cheops Pyramid. The area of the base of the cuboid equals the area of the face of the cube.

But the height, what is it? Look how the pyramids were attached. The height of every pyramid equals half the length of the cube's edge. Therefore, the volume of the cuboid has to be exactly half of the cube's volume, and if the volume of the whole cube equals the sum of the volumes of six pyramids, then half the cubes volume equals the sum of the volumes of three pyramids. And since the volume of the cuboid is three times the volume of the pyramid, the formula for calculating the volume of one pyramid is the area of the base times height divided by three.

If you lost the thread there, feel free to rewind the video and watch it again. Now, we can plug in the measurements from the Cheops Pyramid into the formula. The area of the base, times the height, divided by three. The area of the base is 230.4 meters squared. The height is 146 and 1/2 meters divided by three.

That's almost 2.6 million cubic meters. This is how much stone it took to build Egypt's largest pyramid. We calculated the volume of a pyramid with a square base. But pyramids can have other shapes. For instance, any polygon can be a base and pyramids can have different heights.

Regardless, a pyramid's volume is always one-third of the area of the base, times the height. They weren't stupid, those ancient Egyptians.