Surface area of a polyhedron
How much cardboard do you need for a box? How much glass for a skylight shaped like a pyramid? To find out, you have to calculate the surface area of the polyhedron: the sum of the areas of all its faces. The trick is to unfold it and look at it flat.
The net
If you "open up" a polyhedron and lay it out flat on the table, you get its net: all its faces joined along some of the edges. The surface area of the polyhedron is the area of that net.
A young designer is creating a prototype of a smart speaker 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 prototype of a smart speaker 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 surface area knowing that each square has a side length of $5\text{ cm}$.
From the net of the figure, identify which polyhedron it is and calculate its total surface area knowing that each square has a side length of $5\text{ cm}$.
Surface area of a prism
When unfolded, the lateral faces of a right prism form a rectangle whose base is the perimeter of the base of the prism and whose height is the height of the prism.
A shoe box has the shape of a cuboid (rectangular prism) with the following dimensions: $30\text{ cm}$ long, $18\text{ cm}$ wide and $12\text{ cm}$ high. Calculate the…
A shoe box has the shape of a cuboid (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 make the box (excluding flaps).
A case for a musical instrument has the shape of a right triangular prism. The base is an equilateral triangle with side length $8\text{ cm}$ and the height of the prism…
A case for a musical instrument has the shape of a right triangular prism. The base is an equilateral triangle with side length $8\text{ cm}$ 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 surface area of the case.
Calculate the lateral surface 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:…
Calculate the lateral surface 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: You must first find the side length of the rhombus using Pythagoras' theorem in one of the interior right-angled triangles of the base).
Surface area of a pyramid
The lateral faces of a regular pyramid are identical triangles with base the side $l$ and height the slant height of the pyramid $ap$:
$$A_L = \frac{P_{base} \cdot ap}{2} \qquad\qquad A_T = A_L + A_{base}$$Calculate the lateral surface area of a regular square pyramid if we know that the side length of the base is $10\text{ cm}$ and the apothem of the pyramid (the height…
Calculate the lateral surface area of a regular square pyramid if we know that the side length of the base is $10\text{ cm}$ and the apothem of the pyramid (the height of its lateral faces) is $15\text{ cm}$.
An architecture firm 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…
An architecture firm 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).
Frustum of a pyramid and regular polyhedra
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Frustum of a pyramid: the lateral faces are trapezia. $A_L = \dfrac{(P_B + P_b) \cdot ap}{2}$, and the total adds the two bases.
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Regular polyhedra: all the faces are identical, so $A = \text{number of faces} \cdot \text{area of one face}$. For equilateral triangles of side $l$: area of each face $= \frac{\sqrt 3}{4}\,l^2$.
A glazier needs to make glass for a lamp in the shape of a regular tetrahedron (all its faces are identical equilateral triangles). If each face has an area of…
A glazier needs to make glass for a lamp in the shape of a regular tetrahedron (all its faces are identical equilateral triangles). If each face has an area of $15,6\text{ cm}^2$, how much total glass surface area will the complete lamp have?
Calculate the total surface 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…
Calculate the total surface 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 height of the lateral face (apothem of the frustum) is $8\text{ cm}$. What is the minimum amount of wrapping paper needed to wrap an object of this shape?
A trophy has the shape of a regular octahedron with an edge length of $6\text{ cm}$. The trophy is 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 length of $6\text{ cm}$. The trophy is to be silver-plated, and the cost of the plating is $0,05\text{ €}$ per square centimetre. What will be the total cost of silver-plating the $8$ faces of the trophy?
(Given: The area of an equilateral triangle with side length $a$ is $A = \frac{\sqrt{3}}{4} a^2$).
Back to the start: To work out the cardboard for a closed box you find the total surface area (lateral plus two bases), whereas for the glass of a pyramid-shaped skylight you only calculate the lateral surface area, since the base is open to the inside of the building.