Why, in a pyramid, is the slant height different from the altitude?
Think about this: why is running down a starwell a longer distance that the elevator?
Because it takes more time to complete?
Nope: how does exactly slant height compare to vertical height? That is the question in your original q.
In a pyramid, the slant height refers to the length of the line segment joining the apex (top) of the pyramid to any point on the edge of the base. On the other hand, the altitude refers to the perpendicular distance from the apex to the base of the pyramid.
The reason why the slant height is different from the altitude is due to the shape of the pyramid. Unlike a cone, which has a circular base, the base of a pyramid is typically a polygon (e.g., square, triangle) with straight edges. Since the slant height measures the distance along the surface of the pyramid, it follows the shape of the edges. In contrast, the altitude measures the perpendicular distance from the apex to the base, which is shorter than the slant height in most cases.
To understand the slant height and altitude of a pyramid, you can use trigonometry and geometry. Here's a step-by-step explanation of how to find the slant height and altitude:
1. Identify the pyramid's base shape: Determine the shape of the base (e.g., square, triangle) and note the lengths of its sides.
2. Find the apothem (for regular pyramids): If the base is a regular polygon (all sides and angles are equal), you can find the apothem, which is the perpendicular distance from the center of the base to any of its sides. The apothem can be used to find the altitude.
3. Calculate the slant height: If you know the height of the pyramid (the perpendicular distance from the base to the apex), you can find the slant height using the Pythagorean theorem. The slant height, height, and half of the base length form a right triangle, allowing you to calculate the slant height.
4. Determine the altitude: For regular pyramids, you can use the concept of similar triangles to find the altitude. The altitude is proportional to the apothem, so you can use the known relationship between the slant height and the apothem to find the altitude.
Keep in mind that for irregular pyramids, you may need to use various geometric principles and formulas to calculate the slant height and altitude.