When only a sloping edge is given, build a right triangle from the apex to the base centre.
Key formulas
Locate the foot
For a right square-based pyramid the foot of the perpendicular from the apex is the centre of the base. The distance from centre to a base vertex is (a√2)/2.
Numbers
Base side 6, lateral edge 5. Distance from centre to vertex = 3√2. Then h = √(25 − 18) = √7.
Refuse the shortcut
Never treat the lateral edge as h. That single swap is one of the most common volume errors.
Worked example
Worked example
Problem. A right square-based pyramid has base side 6 cm and lateral edge 5 cm from apex to a base vertex. Find the perpendicular height.
Solution. Distance from base centre to a vertex = (a√2)/2 = 3√2. Then h = √(ℓ² − d²) = √(25 − 18) = √7 cm.
Order
Find centre distance → Pythagoras → then volume if required.
Review quiz
Square base side 8, lateral edge 10 to a vertex. Height h is:
Practice questions
- Base side 8, lateral edge 10. Find h.
- Why is the centre-to-vertex length (a√2)/2?
Study notes
What if the pyramid is oblique?
The foot is not the centre; the question must give more information.
More in Spatial Geometry
- Pyramid volume: identify the base and perpendicular height
- Distance from a point to a plane using volume
- Finding the angle between a line and a plane
- Writing the equation of a plane in 3D
- Cylinder volume and curved surface area
- Cone volume when slant height is given
- Prism volume from a cross-section
- Sphere volume without the fear of π