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Regular pyramids – triangular and square base

Mathematics3D GeometryAges 13–14

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Use with my class

A regular pyramid is drawn in 3D; drag it to rotate in any direction, or switch on auto-rotate. Change the base edge, the height, and the base shape, and watch the height SO, the slant height SM, and the right triangle SOM appear on the solid. The lateral surface area Sxq = p·d, the total surface area, and the volume V = ⅓·(base area)·h update instantly as the numbers change. The 2D | 3D button switches to a true 3D pyramid (WebGL) with apex S, height SO, slant height SM and right triangle SOM; drag to rotate, scroll or pinch to zoom.

Lesson: Regular triangular pyramid and regular square pyramid: lateral surface area and volume

What it shows

A regular pyramid has a regular polygon as its base and its apex S directly above the centre O of the base. Two lengths are easily confused: the height SO, perpendicular to the base, and the slant height SM, the height of a triangular side face. They form the right triangle SOM with OM = r, the inradius of the base, so by Pythagoras SM = √(h² + r²) and the slant height is always longer than the height. The lateral surface area equals half the base perimeter times the slant height, and the volume is one third of the base area times the height.

How to use

Choose Square Pyramid or Triangular Pyramid, then change Base edge a and Height h. Turn on Height and Slant Height to show SO, SM and the right triangle SOM, and rotate the solid by dragging or with Auto-Rotate. Ask students to calculate the slant height, lateral surface area and volume before reading the values below the figure. The 2D | 3D button opens a WebGL model that can be zoomed.

Parameters you can change

  • Base edge a 2–10 cm
  • Height h 2–12 cm
  • Pyramid type Square pyramid (square base), Regular triangular pyramid (equilateral triangle base)
  • Show slant height SM and triangle SOM
  • Show height SO
  • Auto-rotate

Questions to explore

  1. Why is the slant height SM always longer than the height SO?
  2. If the base edge doubles and the height stays the same, how many times larger does the volume become?
  3. When the height doubles, does the lateral surface area also double, and how does triangle SOM explain it?

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