A square pyramid has a square base and four triangular faces meeting at a point. Enter the base side and vertical height to get its volume, slant height and surface area.
Enter the base side and vertical height.
Read the volume, slant height and surface area.
Volume: V = ⅓ × a² × h
Slant height: l = √(h² + (a/2)²)
A square pyramid has a square base and four triangular faces that meet at a single point, the apex. Given the side of the base and the vertical height, this calculator returns the volume, the slant height of each triangular face and the total surface area including the base.
Pyramids are familiar from Egypt, but closer to home they appear as roof tops on gazebos and kiosks, the spires on temple towers, tent tops, packaging for sweets, and glass paperweights. In school mathematics the square pyramid is the standard example for two ideas: why a pointed solid holds one third of its matching prism, and how the vertical height differs from the slant height used for surface area.
1. Measure the base side a and the vertical height h, from the centre of the base straight up to the apex.
2. Find the base area: a².
3. Volume: V = ⅓ × a² × h.
4. Slant height of each face: l = √(h² + (a/2)²). This runs from the apex down the middle of a face to the midpoint of a base edge.
5. Area of the four triangular faces: 4 × ½ × a × l = 2al.
6. Total surface area: TSA = a² + 2al. Leave out a² if the pyramid has no base, such as a tent or roof.
Take a cube of side a and pick one corner. Join that corner to every vertex of the three faces that do not touch it. The cube splits into three identical square pyramids, each with one face of the cube as its base and height a. Each must hold a³ ÷ 3. The general rule, V = ⅓ × base area × height, holds for any pyramid or cone, because shearing or stretching a solid while keeping its base area and height fixed does not change its volume (Cavalieri's principle).
A pyramid has three different lengths that people confuse. The vertical height h runs from the base centre up to the apex. The slant height l runs from the apex down the centre of a face to the midpoint of a base edge; the triangle formed by h, a/2 and l is right-angled, so l = √(h² + (a/2)²). The lateral edge runs from the apex to a base corner and is longer again, √(h² + a²/2), because the corner is further from the centre than the edge midpoint. Face areas need the slant height, not either of the others.
The volume formula works for any base shape if you replace a² with the base area, so a rectangular pyramid uses l × b × h ÷ 3. The surface-area formula here, though, assumes a square base, because only then are all four faces the same triangle. A rectangular base gives two pairs of faces with different slant heights. As the height shrinks towards zero the slant height approaches a/2 and the faces lie flat, so the surface area approaches twice the base area, as expected.
Imran is designing a glass paperweight shaped like a square pyramid, with a base side of 6 cm and a vertical height of 4 cm, and needs the glass volume and the surface to be polished.
Volume = ⅓ × base area × height: = ⅓ × 6² × 4 = 48
Slant height = √(h² + (a/2)²): = √(4² + 3²) = 5
Total surface area = a² + 2 × a × slant height: = 36 + 2 × 6 × 5 = 96
Answer: Volume 48; Slant height 5; Total surface area 96
Using the vertical height in the triangle area formula instead of the slant height.
Using the lateral edge, measured from apex to corner, as if it were the slant height.
Forgetting the one-third factor and reporting the volume of the matching prism.
Including the base area for a tent or roof that has no floor.
Applying the square-base surface formula to a rectangular-base pyramid.
Estimating roofing sheets or tiles for pyramid-shaped gazebo and kiosk roofs.
Calculating canvas for a pyramid tent, excluding the floor.
Finding material and weight of pyramid-shaped castings, paperweights and packaging.
Class 9 to Class 12 mensuration and solid geometry problems.
Estimating the volume of stockpiles of sand or grain that settle into a pointed heap on a square base.
Is the height the same as the slant height?
No. Height is vertical from base to apex; slant height runs along a face.
Does this work for a rectangular base?
The volume formula does (use length × breadth), but the surface-area formula here assumes a square base.