Explanation of Slant Height
The slant height of a three-dimensional shape, such as a cone or pyramid, is the distance from the base to the apex (or top) along the slanted side. It's important to distinguish the slant height from the height of the shape, which is the distance from the base to the apex along a perpendicular line.For a cone, the slant height is the distance from the base to the tip along the curved surface. The slant height is a key component when calculating the surface area and volume of cones, as it is used in the Pythagorean theorem to find the height and in formulas for lateral area and total surface area.For a pyramid, the slant height is the distance from the base to the apex along one of the triangular faces. Just like with cones, the slant height is essential when calculating the surface area and volume of pyramids.Study Guide for Slant Height
1. Definitions
- Slant Height: The distance from the base to the apex along the slanted side of a cone or pyramid.
- Height: The distance from the base to the apex along a perpendicular line.
2. Formulas
- For a cone: Use the Pythagorean theorem to find the slant height (l) given the radius (r) and height (h): l = √(r^2 + h^2).
- For a pyramid: Use the Pythagorean theorem to find the slant height (l) given the base length (b) and height (h): l = √(b^2 + h^2).
3. Applications
- Surface Area: The slant height is used in the lateral area formula (L = πrl) for a cone and in the lateral faces area formula (A = (1/2)pl) for a pyramid.
- Volume: The slant height is used to calculate the volume of cones and pyramids, as it is needed to find the height in the volume formulas (V = (1/3)πr^2h for a cone and V = (1/3)Bh for a pyramid).
4. Practice Problems
Solve problems involving cones and pyramids to practice using the slant height in various calculations, such as finding surface area, volume, and other dimensions.
.◂Math Worksheets and Study Guides Eighth Grade. Solving equations and inequalities
Study Guide
Solving equations and inequalities 
Worksheet/Answer key
Solving equations and inequalities Worksheet/Answer key
Solving equations and inequalities Worksheet/Answer key
Solving equations and inequalities
Worksheet/Answer keyWorksheet/Answer keyWorksheet/Answer keyThe resources above cover the following skills:
Algebra (NCTM)
Represent and analyze mathematical situations and structures using algebraic symbols.
Develop an initial conceptual understanding of different uses of variables.