Slant Height

Definition

The slant height of a three-dimensional geometric shape, such as a cone or a pyramid, is the distance measured along the slanted side from the base to the apex or vertex. It represents the length of the inclined or angled surface of the shape, as opposed to the height which is the perpendicular distance from the base to the apex.

5 Must Know Facts For Your Next Test

1. The slant height is an important measurement in calculating the surface area of cones and pyramids, as it is used to determine the lateral surface area.
2. For a cone, the slant height is the distance from the vertex to the circumference of the base, along the side of the cone.
3. For a pyramid, the slant height is the distance from the vertex to the edges of the base, along the triangular sides.
4. The slant height, along with the base dimensions, can be used to calculate the total surface area of a cone or pyramid.
5. Knowing the slant height is also necessary for finding the volume of a cone or pyramid using the formula: $V = \frac{1}{3}Bh$, where $B$ is the base area and $h$ is the height.

Review Questions

• Explain how the slant height of a cone or pyramid is defined and how it is used to calculate the lateral surface area.
  • The slant height of a cone or pyramid is the distance measured along the inclined or angled side from the base to the apex or vertex. For a cone, the slant height is the distance from the vertex to the circumference of the base, along the side of the cone. For a pyramid, the slant height is the distance from the vertex to the edges of the base, along the triangular sides. The slant height is a crucial measurement used to calculate the lateral surface area of these three-dimensional shapes, which excludes the base. By knowing the slant height and the base dimensions, you can determine the total surface area of a cone or pyramid.
• Describe how the slant height of a cone or pyramid is related to the volume formula for these shapes.
  • The slant height of a cone or pyramid is an important factor in the volume formula for these three-dimensional shapes. The formula for the volume of a cone or pyramid is $V = \frac{1}{3}Bh$, where $B$ is the base area and $h$ is the height. While the height is the perpendicular distance from the base to the apex, the slant height represents the inclined or angled distance along the side. Knowing the slant height is necessary to determine the base area, which is then used in conjunction with the height to calculate the overall volume of the cone or pyramid.
• Analyze how the slant height of a cone or pyramid differs from the height of the shape and explain the significance of this distinction.
  • The slant height of a cone or pyramid is distinct from the height of the shape, which is the perpendicular distance from the base to the apex. The slant height represents the inclined or angled distance along the side of the shape, rather than the straight, vertical distance. This distinction is significant because the slant height is the key measurement used to calculate the lateral surface area of the cone or pyramid, excluding the base. Additionally, the slant height, along with the base dimensions, is necessary to determine the total surface area of the shape. Furthermore, the slant height is an important factor in the volume formula for cones and pyramids, as it is used to calculate the base area, which is then combined with the height to find the overall volume.

Related terms

Cone:

A three-dimensional geometric shape with a circular base and a single vertex or apex, where the sides taper inward to form a point.

Pyramid: A three-dimensional geometric shape with a polygonal base and triangular sides that meet at a single vertex or apex.

Lateral Surface Area: The surface area of the slanted or inclined sides of a three-dimensional shape, excluding the base.