Surface area of solids
Surface area is the total area of all the faces or curved surfaces of a solid. Imagining the solid unfolded into a flat net makes it easy to calculate.
Solids and nets
A polyhedron is a solid with flat polygon faces, such as a cube, prism, or pyramid; cylinders, cones, and spheres have curved surfaces. A net is a flat pattern that folds into a solid. Surface area is the area of the net, measured in square units.
Euler's formula
For any convex polyhedron, the number of vertices minus edges plus faces equals 2: V - E + F = 2, a result published by Leonhard Euler in the 1750s. A cube has 8 vertices, 12 edges, and 6 faces: 8 - 12 + 6 = 2.
Prisms
A prism has two congruent parallel bases joined by rectangular sides. Its surface area is twice the base area plus the lateral area, which is the base perimeter times the height. A rectangular box 5 by 4 by 3 has surface area 2(20 + 15 + 12) = 94 square units.
Cylinders
A cylinder's net is two circles and a rectangle whose width is the circumference. Its surface area is \(2\pi r^2 + 2\pi rh\). A can of radius 4 cm and height 10 cm has surface area \(32\pi + 80\pi = 112\pi \approx 352\) square centimetres.
Pyramids
A pyramid's surface area is its base area plus the area of its triangular faces. For a regular pyramid, the lateral area is half the base perimeter times the slant height, the height of each triangular face. A square pyramid with base 6 and slant height 5 has surface area 36 + (1/2)(24)(5) = 96.
Cones
A cone's surface area is \(\pi r^2 + \pi r\ell\), where \(\ell\) is the slant height from the tip to the edge of the base. A cone with radius 3 and height 4 has slant height 5 by Pythagoras, so its surface area is \(9\pi + 15\pi = 24\pi \approx 75.4\).
Spheres
A sphere's surface area is \(4\pi r^2\), exactly four times the area of its largest cross-section. Archimedes proved this and was so proud of the related sphere-cylinder results that he reportedly asked for a sphere inside a cylinder to be carved on his tomb. Earth, with radius about 6,371 km, has a surface area of about 510 million square kilometres.
Worked example
A company designs a closed cylindrical tin holding food, with radius 5 cm and height 12 cm. The metal needed is 2 pi (25) + 2 pi (5)(12) = 50 pi + 120 pi = 170 pi, about 534 square centimetres. Making the tin wider and shorter with the same volume could reduce the metal used.
Key takeaways
Surface area is the total area of a solid's net. Convex polyhedra satisfy Euler's formula V - E + F = 2. A prism's surface area is two bases plus perimeter times height; a cylinder's is 2 pi r squared + 2 pi r h. A regular pyramid's is the base plus half the perimeter times the slant height; a cone's is pi r squared + pi r l. A sphere's surface area is 4 pi r squared, as Archimedes proved. Slant heights often come from the Pythagorean theorem.
Flashcard
Solid with flat polygon faces
Check your understanding
What is the surface area of a rectangular box 5 by 4 by 3?