Calculate slant height for cones and pyramids
Step 1: Identify the 3D Shape
Determine if you're working with a cone or pyramid (and pyramid type if needed).
Why: The slant height calculation differs because base geometry differs.
Step 2: Measure the Vertical Height
Find h, the perpendicular distance from the apex to the base center.
Why: This vertical component is one leg of the right triangle formed by the slant height.
Step 3: Determine the Base Dimension
For cones: measure radius r. For square pyramids: measure side s (and divide by 2).
Why: The horizontal leg connects the apex to the base edge (radius for cones, s/2 for pyramids).
Step 4: Apply the Pythagorean Theorem
The slant height l is the hypotenuse: l = √(h² + r²) or l = √(h² + (s/2)²).
Why: The apex, base edge, and center form a right triangle where the slant height is the hypotenuse.
Step 5: Confirm the Result is Valid
Verify that l > h (slant height always exceeds vertical height for non-degenerate shapes).
Why: This confirms the geometry is valid; l = h would mean a degenerate flat shape.
Cone: l = √(h² + r²)
Square Pyramid: l = √(h² + (s/2)²)
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