Pyramid nets and triangular faces
Distinguish a face's slant height from the vertical height of a solid.
Before you begin
- Triangle area
- Square area and nets
Separate the base and side faces
A square-based pyramid has one square base and four triangular side faces. In a symmetric pyramid, the side triangles are equal, so calculate one triangle and multiply by four. For a pyramid with unequal faces, calculate each face separately. A net spreads the triangles around the base so their areas can be counted without trying to measure a three-dimensional drawing.
Use the perpendicular height within each face
The triangle-area formula needs the height measured along the face from its apex perpendicular to its base edge. This is the slant height, not the vertical height inside the pyramid. These lengths lie in different planes. A question about total surface area includes the base unless it explicitly asks for only the side faces. Check that every given length belongs to the shape you are calculating.
Worked example
A square pyramid has base edge 6 cm and four triangular faces each with perpendicular face height 5 cm. Find its total surface area.
Show the worked solution
- The square base has area 6 × 6 = 36 cm².
- Each triangle has area ½ × 6 × 5 = 15 cm², so all four total 60 cm².
- Add the base and sides: 36 + 60 = 96 cm².
96 cm².
Try it yourself
For a triangular side face, which height belongs in ½ × base × height?
Common mistakes
- Do not replace slant height with vertical height.
- State whether the base is included.
What can you explain now?
A square pyramid has base edge 4 m and four faces each of perpendicular height 3 m. Find its side area and total surface area.
Compare with the explanation
Side area 24 m²; total surface area 40 m².
Each triangle is ½ × 4 × 3 = 6 m². Four give 24 m². Adding the 4 × 4 = 16 m² base gives 40 m².
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Common Core mathematics standards ↗