Surface Area & Nets

Unfold a solid and its surface becomes a flat shape whose area you already know how to find.

Cut a cardboard box along its edges and lay it flat.

What you get is six rectangles, joined at the folds.

unfold the box and lay it flat a box six flat rectangles the wrapping paper is the surface area, and now you can measure it

Nothing has been added or lost. The cardboard is exactly what was wrapped around the box.

And now every piece is flat, so every piece has an area you can already calculate.

That flat shape is called a net, and its total area is the surface area of the solid. There is no new formula to learn — only a solid to unfold.

A box unfolded into its net, showing six faces 7×53×5 7×53×5 7×37×3 surface area = the area of the whole net 2(7×5) + 2(3×5) + 2(7×3) = 142 volume is different: 7 × 5 × 3 = 105

Unfolding a solid

Cut along enough edges and a solid opens out flat without stretching anything.

The net shows every face at its true size, which is what makes measuring possible.

One solid can have several different nets, depending on where you cut.

Count the faces first

A rectangular box has six, a triangular prism five, a square pyramid five. If your net has the wrong number, something has been missed.

Boxes

A rectangular box has three pairs of identical faces: front and back, top and bottom, two ends.

So find the area of three different faces, add them, and double.

For 5 × 3 × 4: 15 + 20 + 12 = 47, doubled to 94 square units.

A 5 by 3 by 4 box and its net: two faces of 20, two of 15 and two of 12 square units, 94 in all 20 15 12 543 3 × 412 5 × 420 3 × 412 5 × 420 5 × 315 5 × 315 (15 + 20 + 12) × 2 = 47 × 2 = 94 square units three different faces, each one twice — same shade, same area

Three areas, then double

Working out all six separately is fine but slower, and it is easier to lose one. Pairing them halves the work.

Prisms

A prism has two identical ends and a set of rectangles wrapped around them.

Surface area is twice the end plus the area of the wrapping.

The wrapping is one long rectangle, as tall as the prism and as wide as the perimeter of the end.

Perimeter times length

The side faces unroll into a single rectangle. Its width is the perimeter of the end face, which saves computing each side separately.

Pyramids

A square pyramid has a square base and four identical triangles.

Each triangle needs the slant height, measured up the face, not the vertical height of the pyramid.

Total is the base plus four triangles.

A square pyramid with base 6, vertical height 4 and slant height 5, and its net of one square and four triangles, 96 square units h ℓ 6 ℓ = 5 runs up the face h = 4 is inside — too short 6 × 636 151515 ℓ = 5 15 36 + 4 × (½ × 6 × 5) = 36 + 60 = 96 square units

Slant height, not height

The triangle’s height runs up the sloping face. Using the pyramid’s vertical height makes every triangle too small.

Why surface area matters

Surface area is how much material wraps a solid — paint, wrapping paper, cardboard, skin.

Volume is how much fits inside. The two answer different questions and use different units.

Surface area is in square units; volume is in cubic units.

Square or cubic

Wrapping a present needs surface area; filling it needs volume. An answer in cm³ to a wrapping question is answering the wrong question.

Worked examples

A box

Find the surface area of a box 5 by 3 by 4.

  1. Three different faces: 5 × 3 = 15, 5 × 4 = 20, 3 × 4 = 12. One of each pair.
  2. Add them: 15 + 20 + 12 = 47. Half the total.
  3. Double: 94 square units. Every face has a twin.

94

A cube

Find the surface area of a cube with edge 7 cm.

  1. Every face is a 7 by 7 square, so each is 49 cm². All six are identical.
  2. There are six faces. A cube unfolds into six squares.
  3. 6 × 49 = 294 cm². Not 73 = 343 — that would be the volume.

294 cm²

A pyramid

A square pyramid has base edge 6 and slant height 5. Find its surface area.

  1. Base: 6 × 6 = 36. A square.
  2. Each triangle: 12 × 6 × 5 = 15, and there are four. Slant height is the triangle’s height.
  3. 36 + 4(15) = 96 square units. Base plus the four sloping faces.

96

Words to know

Net
A flat pattern that folds up into a solid.
Surface area
The total area of all the faces.
Face
One flat surface of a solid.
Prism
A solid with two identical parallel ends joined by rectangles.
Pyramid
A solid with a polygon base and triangular faces meeting at a point.

Mistakes to watch for

Forgetting a pair of faces. A box has six faces in three matching pairs; computing three areas and forgetting to double halves the answer.

Using the pyramid’s vertical height for its triangles. The triangles need the slant height, measured up the sloping face, which is always the longer of the two.

Giving surface area in cubic units. It is an area, so the units are squared: 294 cm², not 294 cm³.

Confusing surface area with volume. Wrapping a box needs surface area; filling it needs volume. For a 7 cm cube those are 294 and 343 — close enough to hide a mistake.

Drawing a net with the wrong number of faces. Count them against the solid before calculating: six for a box, five for a triangular prism, five for a square pyramid.

Packaging designers work in nets. A cereal box begins life as a single flat piece of card, printed and then folded.

A cube has eleven genuinely different nets — a fact that takes some patience to verify.

Surface area decides how fast something cools, which is why radiators have fins and why a crushed ice cube melts faster than a whole one.

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