Question

Explain the difference between the slant height and the altitude of a regular pyramid.

Short answer

The altitude is vertical and perpendicular to the base, while the slant height is an inclined line from apex to base edge midpoint.

Step-by-step solution

Define Regular Pyramid

A regular pyramid is a three-dimensional geometric figure with a polygonal base and triangular faces that converge at a single point called the apex. The base of the pyramid is regular, meaning all its sides are equal, and it's generally referred to as having a specific height.

Describe the Altitude of a Pyramid

The altitude of a regular pyramid is the perpendicular distance from the apex (top point) of the pyramid to the center of the base. It is a vertical line and is also known as the height of the pyramid.

Describe the Slant Height of a Pyramid

The slant height is the distance from the apex of the pyramid to the midpoint of one of the edges of the base face. In a regular pyramid, it is not vertical but inclined, forming a slant with respect to the base of the pyramid.

Compare Altitude and Slant Height

The key difference between the two is their orientation: the altitude is a vertical line from the apex perpendicular to the base, while the slant height is an inclined line from the apex to the midpoint of a base edge. They are used to calculate different properties of the pyramid, such as volume and surface area.

Key concepts

Regular Pyramid

A regular pyramid is a fascinating 3D shape. It has a base that's a regular polygon, which means all sides and angles are equal. Imagine a pyramid like the famous ones in Egypt but with a perfectly symmetrical base. From each vertex of the base, a triangular face rises to meet at a point above the base called the apex. This gives the pyramid its distinct shape.

• All lateral faces of a regular pyramid are congruent, meaning they have the same size and shape.
• The apex and the center of the base are directly aligned vertically in a regular pyramid.
• The symmetry in a regular pyramid makes calculation easier, especially when finding volume and surface area.

Understanding regular pyramids is crucial because it forms the basis for differentiating key measurements like the slant height and altitude.

Slant Height

The slant height in a regular pyramid is quite unique. It's not the same as the vertical height, and this can sometimes cause confusion. Picture the slant height as the side of the triangle that forms between the apex and the edge of the base.
It's called 'slant' because it's inclined rather than vertical. The slant height stretches from the apex down to the midpoint of any side of the base.

• In mathematical terms, if you imagine opening up a pyramid into its net, the slant height is the height of each triangular face of the pyramid.
• This measurement is essential when calculating the lateral surface area of the pyramid, as it directly affects the area of each triangular face.

Many problems involving pyramids will specify either the slant height or require it to be found to solve for other dimensions or attributes.

Altitude

Altitude in geometry, specifically for pyramids, is a critical concept to grasp. This is the height that measures the perpendicular distance from the very top of the pyramid, the apex, straight down to the center of the base.
It's an absolute vertical measure, unlike the slant height.

• You can think of the altitude as the spine of the pyramid, being purely upright and creating a perfect line down to the base.
• The altitude is fundamental when you calculate the volume of the pyramid, as it represents the true height needed in the volume formula: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Altitude} \).

The clarity of this concept helps in distinguishing it from other measures like the slant height and is therefore vital in understanding the true scale and proportion of the pyramid.