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Square frustum: volume and surface area calculator

Sizes

    How to calculate the volume and surface area of a square frustum

    A square pyramid with its top sliced off parallel to the base, like a bucket turned square. The volume formula h(a² + ab + b²)/3 was already known to the Egyptians (Moscow papyrus, about 1850 BC). The four sides are trapezoids with height h_s.

    Formulas

    Bottom area
    S_1 = a²
    Top area
    S_2 = b²
    Slant height
    h_s = √(h² + ((a - b)/2)²)
    Lateral edge
    e = √(h² + ((a - b)²)/2)
    Lateral area
    S_l = 2(a + b) h_s
    Surface area
    S = S_1 + S_2 + S_l
    Volume
    V = h/3 (a² + ab + b²)
    Vertices
    N_V = 8
    Edges
    N_E = 12
    Faces
    N_F = 6

    Worked example

    Given: bottom edge a = 5 cm, top edge b = 3 cm, height h = 4 cm.

    1. Bottom areaS_1 = 5² = 25 cm²
    2. Top areaS_2 = 3² = 9 cm²
    3. Slant heighth_s = √(4² + ((5 - 3)/2)²) = 4.1231 cm
    4. Lateral edgee = √(4² + ((5 - 3)²)/2) = 4.2426 cm
    5. Lateral areaS_l = 2 · (5 + 3) · 4.1231 = 65.9697 cm²
    6. Surface areaS = 25 + 9 + 65.9697 = 99.9697 cm²
    7. VolumeV = 4/3 · (5² + 5 · 3 + 3²) = 65.3333 cm³= 0.0653 l
    8. VerticesN_V = 8
    9. EdgesN_E = 12
    10. FacesN_F = 6

    Common questions about the square frustum

    How do you calculate the volume of a square frustum?

    Use V = h/3 (a² + ab + b²). For example, with a = 5 cm, b = 3 cm and h = 4 cm, V = 65.3333 cm³.

    How do you calculate the surface area of a square frustum?

    Use S = S_1 + S_2 + S_l. For example, with a = 5 cm, b = 3 cm and h = 4 cm, S = 99.9697 cm².

    Can I find the bottom edge from the volume?

    Yes. In Solve, choose the volume under “I know”, type its value and choose the bottom edge under “Find”. The other sizes stay as they are. For example, with b = 3 cm and h = 4 cm and V = 160 cm³, a = 9.1419 cm.

    Related shapes

    All 37 shapes and their main formulas

    Each one has its own model, sizes you can change and the formula behind every number. Pick one to open it in the lab.

    How the lab works

    Change a size, watch everything follow

    Drag a slider or type an exact value. The model, every result and every worked formula update at once, and the grid under the model counts one unit per square.

    See what a number means

    Tap a result and the model shows it: water fills a volume, paint sweeps a surface or an area, a dot walks the perimeter, vertices and edges are counted one by one.

    Work backwards

    Know the area and need the radius? Solve finds the size that gives the value you typed, then shows every other measurement for it.

    One set of formulas everywhere

    Every formula is written once and evaluated by the same engine in this page and on the API, with a test that checks both give the same numbers.

    Questions

    Can I work backwards from an area or a volume?

    Yes. Open Solve, choose the measurement you know, type its value and pick the size to find. The lab searches for the size that gives exactly that value and shows every other measurement for it, for example a circle with an area of 200 cm² has a radius of 7.9788 cm and a circumference of 50.1326 cm.

    How accurate are the results?

    Every formula is exact and computed in double precision, then rounded to the number of decimals you choose. Two values have no exact elementary formula and are labelled as approximations: the perimeter of an ellipse (Ramanujan) and the surface of an ellipsoid (Knud Thomsen, within 1.07 %).

    Which units can I use?

    Millimetres, centimetres, decimetres, metres, kilometres, inches, feet and yards. Areas and volumes follow the length unit (cm², cm³), and every volume is also given in litres.

    Is there an API?

    Yes. The same engine answers at https://techtools.cz/geometry-lab-api/ without a key: list the shapes, compute any of them, solve backwards, or evaluate an expression. The documentation is at https://techtools.cz/geometry-lab-api/docs.

    For developers and AI agents

    Public JSON, no key, no sign-in. Every response carries the formula, the worked step and the value.

    curl 'https://techtools.cz/geometry-lab-api/compute?shape=cone&r=3&h=4&unit=cm'
curl 'https://techtools.cz/geometry-lab-api/solve?shape=circle&given=A&value=200&unknown=r'
curl 'https://techtools.cz/geometry-lab-api/calc?expr=2pi*r^2&vars%5Br%5D=3'
curl 'https://techtools.cz/geometry-lab-api/shapes'