📐 Slant Height Calculator
Calculate the slant height of cones and pyramids
How to Use This Calculator
Select Shape Type
Choose between Right Circular Cone or Square Pyramid from the dropdown menu.
Enter Dimensions
For a cone, enter the radius and height. For a square pyramid, enter half the base side length and height. Make sure all values are positive numbers.
Get Slant Height
Click "Calculate Slant Height" to get the length along the surface from the base edge to the apex. This is useful for calculating lateral surface areas.
Formulas
Right Circular Cone: s = √(r² + h²)
Slant height from base edge to apex
Square Pyramid: s = √((base/2)² + h²)
Slant height from base edge to apex
Where:
- s = slant height
- r = radius (for cone) or half base side (for pyramid)
- h = perpendicular height from base to apex
Note: Slant height is the length along the curved or slanted surface from the base edge to the apex. It's always greater than or equal to the perpendicular height.
Example 1: Right Circular Cone
Radius = 5 units, Height = 12 units
Slant Height = √(5² + 12²) = √(25 + 144) = √169 = 13 units
Example 2: Square Pyramid
Base side = 10 units (half = 5), Height = 8 units
Slant Height = √(5² + 8²) = √(25 + 64) = √89 ≈ 9.43 units
Example 3: Cone with radius 3, height 4
Slant Height = √(3² + 4²) = √(9 + 16) = √25 = 5 units
About Slant Height Calculator
Slant height is the length along the surface of a cone or pyramid from the base edge to the apex. Unlike the perpendicular height, slant height follows the slanted surface. This calculator finds the slant height for right circular cones and square pyramids using the Pythagorean theorem.
When to Use This Calculator
- Calculating Lateral Areas: Slant height is essential for finding lateral surface areas of cones and pyramids
- Architecture: Determine material lengths needed for slanted surfaces in cone or pyramid structures
- Engineering: Calculate dimensions for cone or pyramid components
- Mathematics Education: Teach students about the relationship between height and slant height
- Construction: Plan materials for slanted surfaces
- Design: Design cone or pyramid shapes with specific surface measurements
Why Use Our Calculator?
- ✅ Multiple Shapes: Handles both cones and square pyramids
- ✅ Instant Results: Calculate slant height immediately
- ✅ Step-by-Step Display: See the Pythagorean theorem calculation
- ✅ 100% Accurate: Uses precise mathematical formulas
- ✅ Educational: Helps understand geometric relationships
- ✅ Completely Free: No registration required
Understanding Slant Height
Key concepts about slant height:
- Definition: The length along the slanted or curved surface from the base edge to the apex
- Relationship to Height: Slant height ≥ perpendicular height (they're equal only if r = 0)
- Right Triangle: Forms the hypotenuse of a right triangle with base radius (or half base) and height as legs
- Lateral Area: Used in calculating lateral surface areas (πrs for cones, sum of triangular faces for pyramids)
- Visual: If you "unroll" a cone's lateral surface, it forms a sector of a circle with radius equal to the slant height
Real-World Applications
Cone Roof: A cone-shaped roof has radius 6 m and height 8 m. Slant height = √(36 + 64) = √100 = 10 m. This determines the length of roofing material needed along the slanted surface.
Pyramid Structure: A square pyramid has base 12 m and height 9 m. Half base = 6 m, so slant height = √(36 + 81) = √117 ≈ 10.82 m. This helps calculate lateral area for surface finishing.
Cone Packaging: A cone-shaped container with radius 5 cm and height 12 cm has slant height = 13 cm. Lateral area = π × 5 × 13 ≈ 204.20 cm² (wrapping material needed).
Frequently Asked Questions
What is slant height?
Slant height is the length along the slanted or curved surface from the base edge to the apex. For a cone, it's the length along the curved side. For a pyramid, it's the length along the triangular face from base edge to apex.
What's the difference between height and slant height?
Height is the perpendicular distance from the base to the apex (straight up). Slant height is the distance along the slanted surface from the base edge to the apex. Slant height is always ≥ height, and they're equal only if the shape has no base (point).
Why do I need slant height?
Slant height is essential for calculating lateral surface areas. For a cone, lateral area = πrs. For a pyramid, you need slant height to find the areas of the triangular faces. It's also useful for determining material lengths for slanted surfaces.
Can I calculate slant height if I know the base radius and height?
Yes! That's exactly what this calculator does. Using the Pythagorean theorem: slant height = √(radius² + height²). The radius and height form the legs of a right triangle, and the slant height is the hypotenuse.
Does this work for rectangular pyramids?
For rectangular pyramids, there are two different slant heights (one for each pair of triangular faces). This calculator is for square pyramids where all triangular faces have the same slant height. For rectangular pyramids, you'd need to calculate each pair separately.
How is slant height used in lateral area calculations?
For a cone: Lateral Area = π × radius × slant height. For a square pyramid: Each triangular face has area = (1/2) × base side × slant height, so total lateral area = 4 × (1/2) × base × slant height = 2 × base × slant height.