The Pyramid Frustum Calculator finds volume and surface measures of a truncated square pyramid from the side length a of the larger base, side length b of the smaller top, and vertical height h between parallel bases. The apex has been cut off by a plane parallel to the base. The page redraws a labelled figure with both sides and height marked.
Enter a, b and h with a greater than b. Related pages cover the full Pyramid Calculator before the cut, the Conical Frustum Calculator for circular truncated cones, and the Rectangular Prism Calculator when the sides do not taper.
Calculate the volume of a pyramid frustum
Volume of a pyramid frustum combines both base areas with a geometric-mean term under a one-third factor, matching the conical frustum pattern with square areas instead of pi r squared. The Pyramid Frustum Calculator uses side lengths for square bases, so areas are a squared and b squared.
V = (h / 3) * (a^2 + a*b + b^2)When b approaches 0 the formula reduces to (1/3) * a^2 * h, the full square pyramid. When a equals b it reduces to a^2 * h, the square prism. Those limits check the arithmetic.
For square bases a = 6, b = 3, h = 4:
V = (4 / 3) * (36 + 18 + 9) = (4 / 3) * 63 = 84Volume 84 is an integer fixture with no pi. Averaging the two prism volumes (a^2 * h + b^2 * h) / 2 gives (144 + 36) / 2 = 90, which is not 84. The middle term a*b is required; skipping it over-counts here.
Calculate the surface area
Each side face is an isosceles trapezoid when the frustum is right and the bases are centered. Slant height along a face depends on half the difference of base sides and the vertical height. Lateral area sums the four trapezoids. Total surface area adds both square bases.
slant m = sqrt(h^2 + ((a - b) / 2)^2)
One trapezoid area = (a + b) / 2 * m
Lateral SA = 2 * (a + b) * m
Total SA = Lateral SA + a^2 + b^2For a = 6, b = 3, h = 4:
m = sqrt(16 + (1.5)^2) = sqrt(16 + 2.25) = sqrt(18.25) approx 4.27
Lateral SA = 2 * (6 + 3) * m = 18 * m approx 76.87
Total SA = Lateral SA + 36 + 9 = Lateral SA + 45 approx 121.87Open hoppers that omit one base subtract that square. Net layouts need the trapezoid slant m, not only vertical h.
Solve a square frustum
Use larger base side 6, top side 3 and height 4 as the reference frustum. Volume is 84. Slant height along a face is sqrt(18.25). 1. Volume
V = (4 / 3) * (36 + 18 + 9) = 842. Face slant height
m = sqrt(4^2 + ((6 - 3) / 2)^2) = sqrt(18.25) approx 4.273. Base areas
Bottom = 36, top = 9A full pyramid with base 6 that continues to an apex beyond the top face would have height greater than 4; similar triangles can recover that apex height if needed, but frustum volume does not require it once a, b and h are known.
Compare with the Pyramid Calculator fixture base 6 height 4 volume 48: that solid still has its tip. Removing a smaller similar top changes the formula to the frustum form above.
Hoppers, tapered duct transitions and some architectural caps use this solid. Measure the outer side of each square opening and the clear vertical height between the parallel faces. If the openings are rectangles rather than squares, replace a^2 and b^2 with the actual face areas A1 and A2 and use V = (h/3)*(A1 + sqrt(A1*A2) + A2), which reduces to the side-length form when both faces are squares.
Similar-triangle checks help when a drawing gives the full pyramid height H to an imaginary apex and the cut height. If the small base is similar with scale factor k = b/a, then the distance from apex to the small face is k*H and the frustum height is H*(1-k). Volume can be computed as full-pyramid volume minus the small pyramid volume; that difference equals the frustum formula when the geometry is right and centered.
Work the face slant for a net
Lateral surface area needs face slant height m, not only vertical h. For a = 6, b = 3, h = 4, half the side difference is (6 - 3) / 2 = 1.5, so m = sqrt(4^2 + 1.5^2) = sqrt(18.25).
Each trapezoid area is ((a + b) / 2) * m = 4.5 * m, and four faces give lateral SA = 18 * m. Adding base squares 36 and 9 produces total SA.
A cardboard net for a closed frustum uses four congruent isosceles trapezoids plus two squares. Cutting with vertical height instead of m shortens the non-parallel sides and under-counts paper. Open hoppers that omit the large base subtract a^2 from total SA; omit the small base by subtracting b^2 instead.
Unit note: sides a and b and height h share one length unit. Volume uses cube units. Face slant m and every surface area use the same length unit for the square measure. Keep a > b for a tapering square frustum; a = b is the prism limit already noted above.
Frequently asked questions
What is pyramid frustum volume?
For square bases, V = (h/3) * (a^2 + a*b + b^2). With a = 6, b = 3, h = 4, V = 84.
What are a and b?
Side lengths of the larger and smaller square bases. Keep a > b for a tapering solid.
What is height h?
Vertical distance between the parallel square bases.
Does surface area include both bases?
Total surface area includes both squares plus four trapezoidal sides. Lateral area omits the bases.
Can bases be rectangles?
This page uses square bases. Rectangular frustums need general base areas in the same (h/3)*(A1 + sqrt(A1*A2) + A2) pattern.
How does this relate to a conical frustum?
Both use a one-third mix of the two ends with a middle product term. Circles use pi; squares use side squares. Use the Conical Frustum Calculator for circular ends.
What if a equals b?
The solid is a square prism (a rectangular prism with square bases). Volume becomes a^2 * h.
What units does the Pyramid Frustum Calculator use?
Sides and height share one length unit. Volume uses cube units; surface areas use square units.
Can volume be full pyramid minus small pyramid?
Yes, when both pyramids share the apex and the cut is parallel to the base. That difference matches V = (h/3)*(a^2 + a*b + b^2) for square bases.
What is face slant height used for?
Face slant height m is the altitude of each trapezoidal side face. Lateral surface area needs m; vertical height alone is not enough for the wall area.
Why is averaging the two prism volumes wrong?
(a^2*h + b^2*h)/2 drops the middle a*b term. For a = 6, b = 3, h = 4 that average is 90, but the true frustum volume is 84.
What is the reference frustum volume?
For a = 6, b = 3, h = 4, volume is 84 cubic units.
How do square and conical frustums differ?
Both mix two end measures with a middle product under h/3. Squares use side lengths; cones use radii and a pi factor.
Summary
Pyramid Frustum Calculator returns volume and surface measures for a truncated square pyramid from two base sides and height, with substituted arithmetic and a labelled figure. Worked numbers match a = 6, b = 3, h = 4: V = 84. Use the Pyramid Calculator for the uncut solid and the Conical Frustum Calculator for circular truncated cones.