Surface Area of Prisms and Pyramids

Surface area is the total area of every flat face on the outside of a 3D solid. Imagine unfolding a cardboard box flat onto a table: every panel you see is a face, and the sum of those face areas is the surface area. Surface area answers questions like how much wrapping paper, how much paint, or how much sheet metal you need to cover a solid. The formulas are identical whether the measurements are US customary (sq inches, sq feet) or metric (cm^2, m^2) — only the unit labels change. For prisms and pyramids, every face is a rectangle, square, or triangle, so once you can find those three areas you can find any surface area.

Rectangular Prism

• A rectangular prism has 6 rectangular faces in 3 matching pairs: top and bottom, front and back, left and right.
• Top and bottom each have area l x w. Front and back each have area l x h. Left and right each have area w x h.
• Add all six: SA = 2lw + 2lh + 2wh.
• Customary example: a box with l = 5 in, w = 3 in, h = 4 in has SA = 2(5)(3) + 2(5)(4) + 2(3)(4) = 30 + 40 + 24 = 94 sq in.
• Metric example: a box with l = 5 cm, w = 3 cm, h = 4 cm has SA = 94 cm^2 — same arithmetic, different unit label.
• Tip: compute the area of each unique face, then double each one before adding.

Cube

• A cube is a special rectangular prism where length, width, and height are all equal: l = w = h = s.
• It has 6 identical square faces, each with area s x s = s^2.
• Add all six: SA = 6 x s^2.
• Customary example: a cube with side 4 in has SA = 6 x 4^2 = 6 x 16 = 96 sq in.
• Metric example: a cube with side 4 cm has SA = 96 cm^2 — same number, different unit.
• Tip: square the side first, then multiply by 6.

Triangular Prism

• A triangular prism has 5 faces: 2 triangular ends and 3 rectangular sides.
• Each triangular end has area (1/2) x b x h, where b and h are the base and height of the triangle.
• Each rectangular side has area = (one edge of the triangle) x (length of the prism).
• SA = 2 x (area of triangle) + (sum of the three rectangle areas).
• Example (in cm): triangle with base 6 cm and height 4 cm gives 2 x (1/2 x 6 x 4) = 24 cm^2. If the prism is 10 cm long and the triangle has sides 5 cm, 5 cm, 6 cm, the rectangles are 5 x 10 = 50, 5 x 10 = 50, and 6 x 10 = 60 — that is 160 cm^2. Total SA = 24 + 160 = 184 cm^2. (Same problem with inches gives 184 sq in.)

Pyramids

• A square pyramid has 1 square base and 4 identical triangular faces.
• SA = (area of square base) + 4 x (area of one triangular face) = b^2 + 4 x (1/2) x b x l, where l is the slant height.
• Customary example: square base side 6 in, slant height 5 in gives SA = 6^2 + 4 x (1/2) x 6 x 5 = 36 + 60 = 96 sq in.
• Metric example: square base side 6 cm, slant height 5 cm gives SA = 96 cm^2.
• A triangular pyramid (tetrahedron) has 4 triangular faces. If all four are identical, SA = 4 x (1/2) x b x h.
• Tip: always add the base AND every slanted face — do not forget the bottom.