The Pyramid Calculator focuses on a square-base pyramid with the apex above the center of the base. Enter base side length and perpendicular height to obtain volume, face slant height, lateral surface area and total surface area. The page redraws a labelled figure with the base and height marked to the entered values.
Volume uses the same one-third factor as a cone, but the base is a square rather than a circle. Related pages cover the Cone Calculator for a circular base, the Pyramid Frustum Calculator when the tip is cut off, and the Rectangular Prism Calculator for a box without an apex.
Calculate the volume of a pyramid
Volume of a pyramid is one-third times base area times perpendicular height. The Pyramid Calculator uses a square base of side a, so base area is a squared. Side length and height must be positive.
V = (1/3) * a^2 * hThe one-third factor matches cones and general pyramids with any polygonal base: replace a^2 with the actual base area when the base is not square. Forgetting the one-third triples the reported volume. Volume scales with the square of the side and linearly with height.
For square base a = 6 and h = 4:
V = (1/3) * 36 * 4 = 48No pi appears because the base is polygonal. When volume and base side are known, height recovers as 3 * V / a^2. When volume and height are known, side recovers as the positive square root of 3 * V / h.
Apex position matters for surface area but not for this volume formula when height is the perpendicular distance from the base plane to the apex. This page assumes a right square pyramid with the apex above the base center so slant heights on all four faces match.
Calculate the slant height
Slant height along a face runs from the midpoint of a base edge up the triangular face to the apex. Half the base side and the perpendicular height form a right triangle with slant height as the hypotenuse. The Pyramid Calculator applies that Pythagorean step before lateral area.
l = sqrt((a/2)^2 + h^2)This face slant is the height of each isosceles triangular face measured from the base edge to the apex along the face. Edge slant from a base corner to the apex is a longer length and is not used for the face-area formula on this page.
For a = 6 and h = 4:
l = sqrt(3^2 + 4^2) = sqrt(9 + 16) = 5The 3-4-5 triple appears because half of 6 is 3. That choice keeps face area and total surface area as whole numbers in the worked example.
Calculate lateral and total surface area
Lateral surface area is the sum of the four congruent triangular faces. Each face has base a and face height equal to slant height l, so one face area is (1/2) * a * l. Total surface area adds the square base. The Pyramid Calculator reports both.
Lateral SA = 4 * (1/2) * a * l = 2 * a * l
Total SA = Lateral SA + a^2Using perpendicular height in place of slant height under-counts each face. Net problems that unfold the four triangles need slant height on the face altitude.
For a = 6 and l = 5:
Lateral SA = 2 * 6 * 5 = 60
Base area = 36
Total SA = 96A pyramid frustum with two square faces needs a different volume formula that mixes both base areas. Use the Pyramid Frustum Calculator for truncated pyramids such as hoppers and tapered ducts.
Solve a square pyramid with base 6 and height 4
Take base side a = 6 and height h = 4 as the reference pyramid. Slant height is 5. Volume is 48. Surface areas are 60 lateral and 96 total. 1. Slant height
l = sqrt((6/2)^2 + 4^2) = 52. Volume
V = (1/3) * 6^2 * 4 = 483. Lateral surface area
Lateral SA = 2 * 6 * 5 = 604. Total surface area
Total SA = 60 + 36 = 96A cone with base radius 3 (so diameter matches the half-side geometry differently) and height 4 has volume 12 * pi, which is not comparable by side length alone. A square prism with base 6 and height 4 has volume 144, three times this pyramid volume, matching the one-third rule when bases and heights match.
Frequently asked questions
What is pyramid volume?
Pyramid volume is V = (1/3) * base area * height. For a square base, V = (1/3) * a^2 * h. With a = 6 and h = 4, V = 48.
How do I find slant height on a square pyramid?
Slant height along a face is l = sqrt((a/2)^2 + h^2). For a = 6 and h = 4, l = 5.
What is lateral surface area?
Lateral surface area is 2 * a * l for four triangular faces. For a = 6 and l = 5, lateral SA = 60.
Does the Pyramid Calculator require a square base?
This page models a square base. The volume formula still holds for other bases if base area replaces a^2.
How does a pyramid compare to a cone?
Both use (1/3) * base area * height. The cone uses pi * r^2 for base area; the pyramid uses the polygonal base area.
What is a pyramid frustum?
A pyramid frustum is a pyramid with the tip cut off parallel to the base. Use the Pyramid Frustum Calculator for two square sides and a height.
Can I start from volume?
Yes. With side a known, h = 3 * V / a^2. With height known, a = sqrt(3 * V / h).
What units does the Pyramid Calculator use?
Side and height share one length unit. Volume uses cube units; surface areas use square units.
What is total surface area for the reference pyramid?
Total surface area is 96: lateral 60 plus base 36, with a = 6 and l = 5.
How does edge slant differ from face slant?
Edge slant runs from a base corner to the apex and is longer than face slant l from an edge midpoint. Lateral area uses face slant.
Can the apex sit off-center?
Volume still uses (1/3)*base area*perpendicular height. Surface formulas on this page assume a right square pyramid with equal face slants.
Summary
Pyramid Calculator returns volume, face slant height, lateral surface area and total surface area for a square-base pyramid, with substituted arithmetic and a labelled figure. Worked numbers match base 6 and height 4: slant 5, volume 48 and total SA 96. Use the Cone Calculator for a circular base and the Pyramid Frustum Calculator when the apex is removed.