Prisms, cylinders and exposed surfaces

A prism repeats one cross-section

A prism with constant cross-sectional area A and perpendicular length l has volume Al. A triangular prism whose cross-section has base 6 cm and height 4 cm has area 12 cm². If the prism is 10 cm long, its volume is 120 cm³. Use the perpendicular length between matching end faces, not an unrelated sloping edge.

For its total surface area, include both triangular ends and the rectangular side faces. If the triangle’s three side lengths are 5, 5 and 6 cm, the lateral rectangles total (5 + 5 + 6) × 10 = 160 cm². Add 2 × 12 = 24 cm² for the ends: total 184 cm².

A cylinder is a circular prism model

A cylinder of radius r and perpendicular height h has volume π r²h. Its curved surface unwraps to a rectangle of width 2π r and height h, so curved area is 2π rh.

A closed cylinder includes two circular ends: total area 2π rh + 2π r². For r = 3 cm, h = 8 cm, volume is 72π cm³ and total area 66π cm². An open-top container includes only one circular end, giving 57π cm². Read which surfaces are requested.

Volume units need three factors

One cube of edge 1 m contains 100³ = 1 000 000 cubes of edge 1 cm. Thus 1 m³ = 1,000,000 cm³. For example 350,000 cm³ = 0.35 m³. The square-unit conversion factor is not enough for volume.

Joining solids hides some faces

Two cuboids joined face-to-face have the sum of their volumes, provided they do not overlap. Their exposed surface area is the sum of both surface areas minus twice the shared face area, because each original solid counted that now-hidden face.

Check your understanding 1 Count the end faces

A cylinder has radius 2 cm and height 5 cm. Find its volume, curved surface area and total surface area when both ends are closed.

Your answer or drawing on paper

Show answer

Volume is 20π cm³. Curved area is 20π cm². Two end circles add 8π cm², so total area is 28π cm². Matching numerical coefficients do not make volume and area the same quantity.

A capacity model uses internal dimensions

For a container, internal dimensions describe capacity; external dimensions include its material. A stated 1 mm wall thickness cannot be ignored if the task asks for the material volume rather than internal capacity. Model the solid and the empty region separately.