Surface Area of Polyhedra
How much wrapping paper does it take to cover a gift box? How much glass does it take to cover a pyramid-shaped skylight, like the giant glass pyramid of the Luxor hotel in Las Vegas? Both questions ask for the surface area of a polyhedron: the total area of all its faces. The trick is to unfold the solid and look at it flat.
Nets and Surface Area
Cut along some edges of a box and flatten it on a table. The flat shape you get is the net of the solid: all its faces joined along some edges. The surface area of the solid is just the area of its net.
Area is measured in square units: in.², ft², cm², m². Every formula on this page works with any unit, as long as all lengths use the same one. Many problems use centimeters or meters, and the formulas are the same.
A young designer is creating a smart speaker prototype in the shape of a cube. If the edge of the cube measures $9\text{ cm}$, calculate the total surface area of…
A young designer is creating a smart speaker prototype in the shape of a cube. If the edge of the cube measures $9\text{ cm}$, calculate the total surface area of plastic material needed to cover all its faces.
From the net of the figure, identify which polyhedron it is and calculate its total area knowing that each square has a side of $5\text{ cm}$.
From the net of the figure, identify which polyhedron it is and calculate its total area knowing that each square has a side of $5\text{ cm}$.
Prisms
When you unroll the lateral faces of a right prism, they form one big rectangle. Its width is the perimeter of the base and its height is the height of the prism.
Why $2B$? A closed prism has two identical bases: a top and a bottom. Here are the base areas you will need:
| Base | Area $B$ |
|---|---|
| Rectangle | $\ell w$ |
| Triangle | $\tfrac{1}{2}bh$ |
| Equilateral triangle, side $s$ | $\dfrac{\sqrt{3}}{4}s^2$ |
| Rhombus with diagonals $d_1, d_2$ | $\tfrac{1}{2}d_1 d_2$ |
| Trapezoid | $\tfrac{1}{2}(b_1 + b_2)h$ |
| Regular polygon, apothem $a$, perimeter $P$ | $\tfrac{1}{2}aP$ |
The apothem of a regular polygon is the distance from its center to the midpoint of a side.
For a rectangular prism you can also use the shortcut $SA = 2(\ell w + \ell h + wh)$. Check: $2(96 + 60 + 40) = 392$ ✓.
A shoe box has the shape of a rectangular prism with the following dimensions: $30\text{ cm}$ long, $18\text{ cm}$ wide and $12\text{ cm}$ high. Calculate the total area…
A shoe box has the shape of a rectangular prism with the following dimensions: $30\text{ cm}$ long, $18\text{ cm}$ wide and $12\text{ cm}$ high. Calculate the total area of cardboard used to manufacture the box (excluding flaps).
A musical instrument case has the shape of a right triangular prism. The base is an equilateral triangle of side $8\text{ cm}$ and the height of the prism is…
A musical instrument case has the shape of a right triangular prism. The base is an equilateral triangle of side $8\text{ cm}$ and the height of the prism is $40\text{ cm}$. If the area of the triangular base is approximately $27,7\text{ cm}^2$, calculate the total area of the case.
A metal bin without a lid is to be manufactured in the shape of a regular octagonal prism. The base has a perimeter of $64\text{ cm}$ and an apothem of $9,66\text{ cm}$.…
A metal bin without a lid is to be manufactured in the shape of a regular octagonal prism. The base has a perimeter of $64\text{ cm}$ and an apothem of $9,66\text{ cm}$. The height of the bin is $40\text{ cm}$. Calculate the total surface area of metal needed for its manufacture.
Finding a Missing Length in the Base
Often the problem does not give you the side of the base or the apothem. You can almost always find it with the Pythagorean Theorem, using a right triangle hidden inside the base.
Three right triangles show up again and again:
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Rhombus: the diagonals cross at right angles and cut each other in half. A rhombus with diagonals 10 and 24 has side $\sqrt{5^2 + 12^2} = 13$.
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Isosceles trapezoid: drop a perpendicular from each top corner. Each leg is the hypotenuse of a right triangle with one leg equal to the trapezoid's height and the other equal to $\frac{b_1 - b_2}{2}$. With bases 12 and 6 and height 4, the leg is $\sqrt{4^2 + 3^2} = 5$.
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Regular hexagon: it splits into 6 equilateral triangles. Its apothem is $a = \dfrac{\sqrt{3}}{2}s$ and its area is $B = \dfrac{3\sqrt{3}}{2}s^2$.
Calculate the lateral area of a right prism whose height is $12\text{ cm}$ and whose base is a rhombus with diagonals $16\text{ cm}$ and $12\text{ cm}$. (Hint: First you…
Calculate the lateral area of a right prism whose height is $12\text{ cm}$ and whose base is a rhombus with diagonals $16\text{ cm}$ and $12\text{ cm}$.
(Hint: First you must find the side of the rhombus using the Pythagorean theorem in one of the interior right-angled triangles of the base).
For a technology project, you have to build a toolbox in the shape of a regular hexagonal prism. The edge of the base measures $10\text{ in}$ and the height of the prism…
For a technology project, you have to build a toolbox in the shape of a regular hexagonal prism. The edge of the base measures $10\text{ in}$ and the height of the prism is $25\text{ in}$. If you want to cover all the outer faces (including the two bases) with a protective film, how many square inches of material will you need?
(Note: Remember that to calculate the area of the base you will first need to find the apothem of the hexagon using the Pythagorean theorem).
A right prism of height $20\text{ cm}$ has an isosceles trapezoid as its base. The parallel sides of the trapezoid measure $14\text{ cm}$ and $6\text{ cm}$, and its…
A right prism of height $20\text{ cm}$ has an isosceles trapezoid as its base. The parallel sides of the trapezoid measure $14\text{ cm}$ and $6\text{ cm}$, and its height is $3\text{ cm}$. Calculate the total area of the prism. Hint: You will first need to calculate the length of the slanted sides of the base trapezoid using the Pythagorean theorem.
Working From Other Information
Sometimes the problem hides what you need. Three tools help.
1. Volume gives you the height. The volume of a prism is $V = B \cdot h$. If $V = 240$ in.³ and $B = 40$ in.², then $h = 240 \div 40 = 6$ in.
2. The space diagonal. In a rectangular prism, the segment joining two opposite corners passes through the inside of the box. Apply the Pythagorean Theorem twice (once across the base, once up the height) and you get
$$d = \sqrt{\ell^2 + w^2 + h^2}$$When the diagonal is given and a side is missing, write $d^2 = \ell^2 + w^2 + h^2$, substitute what you know, and solve for the unknown. Relationships such as "the length is twice the width" turn into $\ell = 2w$.
3. Convert units first. If the lengths come in different units, convert them before you compute. Remember $1\text{ ft} = 12\text{ in.}$ and $1\text{ m} \approx 39.37\text{ in.}$
Calculate the lateral area of a regular pentagonal prism knowing that its volume is $450\text{ cm}^3$, its base area is $30\text{ cm}^2$ and the side of the base…
Calculate the lateral area of a regular pentagonal prism knowing that its volume is $450\text{ cm}^3$, its base area is $30\text{ cm}^2$ and the side of the base measures $4\text{ cm}$.
(Hint: First use the volume formula to find the height of the prism).
An architecture studio is designing a loft in the shape of a rectangular prism. It is known that the length of the room is twice its width, and that the height is…
An architecture studio is designing a loft in the shape of a rectangular prism. It is known that the length of the room is twice its width, and that the height is exactly $3\text{ m}$. If the space diagonal (the maximum distance between two opposite corners) measures exactly $7\text{ m}$, calculate the total area of the walls and the ceiling (excluding the floor) that must be covered with insulating paint.
An industrial waste container has been manufactured in the shape of a right prism whose base is a rhombus. The diagonals of the base rhombus measure $160\text{ in}$ and…
An industrial waste container has been manufactured in the shape of a right prism whose base is a rhombus. The diagonals of the base rhombus measure $160\text{ in}$ and $120\text{ in}$. If the height of the container is $2\text{ meters}$, calculate the total outer surface area of the container in square feet ($m^2$).
Pyramids
The lateral faces of a regular pyramid are identical triangles. Each triangle has a base equal to a side $s$ of the base polygon and a height equal to the slant height $\ell$ of the pyramid. (Some problems call the slant height the apothem of the pyramid.)
Why $\tfrac{1}{2}P\ell$? Each face has area $\tfrac{1}{2}s\ell$. Adding $n$ faces gives $\tfrac{1}{2}(ns)\ell$, and $ns$ is the perimeter.
The slant height is not the height of the pyramid. If you are given the height $h$, build a right triangle inside the pyramid: one leg is $h$, the other is the distance from the center of the base to the midpoint of a side, and the hypotenuse is $\ell$.
Calculate the lateral area of a regular square pyramid if we know that the side of the base measures $10\text{ cm}$ and the apothem of the pyramid (the height of its…
Calculate the lateral area of a regular square pyramid if we know that the side of the base measures $10\text{ cm}$ and the apothem of the pyramid (the height of its lateral faces) measures $15\text{ cm}$.
An architecture company has designed a glass skylight in the shape of a regular square pyramid. The edge of the base measures $4\text{ m}$ and the height of the pyramid…
An architecture company has designed a glass skylight in the shape of a regular square pyramid. The edge of the base measures $4\text{ m}$ and the height of the pyramid is $3\text{ m}$. Calculate the total surface area of glass needed to cover the four lateral faces (the base is left open to let light into the building).
Frustums
A frustum of a pyramid is what is left when you slice off the top of a pyramid with a plane parallel to the base. It has two parallel bases (a bigger one $B$ and a smaller one $b$), and its lateral faces are trapezoids.
Each lateral face of a regular square frustum is a trapezoid with parallel sides $s_1$ and $s_2$ and height equal to the slant height $\ell$. Adding all the faces:
$$L = \frac{1}{2}(P_1 + P_2)\,\ell \qquad\qquad SA = L + B_1 + B_2$$where $P_1, P_2$ are the perimeters of the two bases and $B_1, B_2$ their areas.
If you are given the height $h$ of the frustum instead of $\ell$, slice the frustum through the midpoints of two opposite base edges and look at the cut from the side. You see an isosceles trapezoid whose legs are the slant height.
Calculate the total area of a frustum of a regular square pyramid whose bases are squares with sides $12\text{ cm}$ and $6\text{ cm}$ respectively, and whose lateral…
Calculate the total area of a frustum of a regular square pyramid whose bases are squares with sides $12\text{ cm}$ and $6\text{ cm}$ respectively, and whose lateral face height (slant height of the frustum) is $8\text{ cm}$. What is the minimum amount of wrapping paper needed to wrap an object of this shape?
A luxury perfume brand has designed a bottle in the shape of a frustum of a regular square pyramid. The edge of the larger base measures $8\text{ cm}$ and the edge of…
A luxury perfume brand has designed a bottle in the shape of a frustum of a regular square pyramid. The edge of the larger base measures $8\text{ cm}$ and the edge of the smaller base measures $4\text{ cm}$. If the height of the bottle is $10\text{ cm}$, calculate the total surface area of glass needed to manufacture the bottle (including both bases). To do this, you will first need to calculate the apothem of the frustum using the Pythagorean theorem in 3D space.
Regular Polyhedra
In a regular polyhedron all the faces are identical regular polygons. There are only five of them.
| Solid | Face | Faces | Vertices | Edges |
|---|---|---|---|---|
| Tetrahedron | triangle | 4 | 4 | 6 |
| Cube | square | 6 | 8 | 12 |
| Octahedron | triangle | 8 | 6 | 12 |
| Dodecahedron | pentagon | 12 | 20 | 30 |
| Icosahedron | triangle | 20 | 12 | 30 |
Since all faces are equal, $SA = (\text{number of faces}) \times (\text{area of one face})$. Three of them have equilateral triangle faces. For side $s$, the height of the triangle is $\frac{\sqrt{3}}{2}s$, so
$$A_{\text{triangle}} = \frac{\sqrt{3}}{4}s^2$$If a problem adds a cost, such as plating at a price per square inch, find the area first and then multiply by the price.
A glazier must manufacture a glass for a lamp in the shape of a regular tetrahedron (all its faces are equal equilateral triangles). If each face has an area of…
A glazier must manufacture a glass for a lamp in the shape of a regular tetrahedron (all its faces are equal equilateral triangles). If each face has an area of $15,6\text{ cm}^2$, how much glass surface area will the complete lamp have?
A trophy has the shape of a regular octahedron with an edge of $6\text{ in}$. The trophy is going to be silver-plated, and the cost of the plating is $0,05\text{ \$}…
A trophy has the shape of a regular octahedron with an edge of $6\text{ in}$. The trophy is going to be silver-plated, and the cost of the plating is $0,05\text{ \$}$ per square inch. What will be the total cost of plating the $8$ faces of the trophy?
(Given: The area of an equilateral triangle of side $a$ is $A = \frac{\sqrt{3}}{4} a^2$).
An ant walks over the surface of a regular octahedron with edge $12\text{ cm}$. If the octahedron can be considered as the union of two regular square-based pyramids…
An ant walks over the surface of a regular octahedron with edge $12\text{ cm}$. If the octahedron can be considered as the union of two regular square-based pyramids joined at their base, calculate the total area the ant can cover. To do this, remember that all the faces of a regular octahedron are equilateral triangles. Express the result in terms of $\sqrt{3}$.
Euler's Formula
Count the faces ($F$), vertices ($V$) and edges ($E$) of a cube: $F = 6$, $V = 8$, $E = 12$. Now look at the table in the last section. Notice that $F + V$ is always exactly 2 more than $E$.
Some problems call this Euler's theorem. It works for all convex polyhedra, not only the regular ones.
To count the parts of a prism with an $n$-gon base: $F = n + 2$, $V = 2n$, $E = 3n$. For a pyramid with an $n$-gon base: $F = n + 1$, $V = n + 1$, $E = 2n$. Notice that a pyramid always has as many faces as vertices.
Verify if Euler's theorem ($C + V = A + 2$) holds for a pentagonal prism. Clearly state the number of faces ($C$), vertices ($V$) and edges ($A$) you have counted.
Verify if Euler's theorem ($C + V = A + 2$) holds for a pentagonal prism. Clearly state the number of faces ($C$), vertices ($V$) and edges ($A$) you have counted.
A new polyhedron for a board game has been designed complying with Euler's theorem. We know that the die has $14$ vertices and that all its faces are equilateral…
A new polyhedron for a board game has been designed complying with Euler's theorem. We know that the die has $14$ vertices and that all its faces are equilateral triangles. Could you deduce how many faces this polyhedron has and what the total number of edges is? Prove your answer using Euler's mathematical relationship.
"Euler's theorem" states that in convex polyhedra $C + V = A + 2$. Imagine a polyhedron that has the same number of faces as vertices. If the number of edges is $12$,…
"Euler's theorem" states that in convex polyhedra $C + V = A + 2$. Imagine a polyhedron that has the same number of faces as vertices. If the number of edges is $12$, deduce how many faces it has and suggest the name of a regular polyhedron that meets this condition. Once identified, calculate its total area if its edge measures $\sqrt{10}\text{ cm}$.
Composite Solids
A composite solid is built by gluing solids together or by cutting pieces out of one. The golden rule: the surface area is the area of the faces you can touch from outside (or from inside a hollow).
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Gluing: the joined faces disappear. Do not count them.
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Carving a hollow: you lose the face you carved into, but you gain the new inner faces.
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Cutting off a corner: you lose the pieces of the old faces that were removed, but you gain the new cut face.
We are designing a decorative wooden piece consisting of a right square-based prism with base $8\text{ cm}$ and height $10\text{ cm}$ from which a pyramid with the same…
We are designing a decorative wooden piece consisting of a right square-based prism with base $8\text{ cm}$ and height $10\text{ cm}$ from which a pyramid with the same base and the same height has been removed. Calculate the total surface area of this piece (including the inner "hollow" left by the pyramid).
A metal part is to be built by joining a regular hexagonal prism and a regular hexagonal pyramid that fits perfectly on its top base. The side of the base hexagon…
A metal part is to be built by joining a regular hexagonal prism and a regular hexagonal pyramid that fits perfectly on its top base. The side of the base hexagon measures $6\text{ cm}$, the height of the prism is $15\text{ cm}$ and the height of the pyramid is $8\text{ cm}$. Calculate the outer surface area of the complete part (bear in mind that the base where both solids join does not count as an outer surface).
Consider a cube of edge $10\text{ cm}$. If we make a planar cut passing through the midpoints of three edges that meet at the same vertex, we obtain a small triangular…
Consider a cube of edge $10\text{ cm}$. If we make a planar cut passing through the midpoints of three edges that meet at the same vertex, we obtain a small triangular pyramid (called a truncated tetrahedron). Calculate the total area of the solid that remains after removing that small pyramid at that corner.
Back to the beginning: to wrap a closed gift box you need its whole surface area (lateral area plus both bases). For the glass pyramid, the base is open to the inside of the building, so you need only the lateral area.