Exercises: Area of a polyhedron

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.

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).

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.

Calculate the lateral area of a regular square-based 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 $15\text{ cm}$.

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.

From the net of the figure, identify which polyhedron it is and calculate its total area knowing that each square has a side of $5\text{ cm}$.

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?

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 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 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).

An architecture company has designed a glass skylight in the shape of a regular square-based 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).

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.

Calculate the total area of a frustum of a regular square-based 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 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 $A = \frac{\sqrt{3}}{4} a^2$).

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).

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).

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.

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.

A luxury perfume brand has designed a bottle in the shape of a frustum of a regular square-based 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.

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).

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 $\sqrt{3}$.

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.

"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 $\sqrt{10}\text{ cm}$.

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 ($m^2$).

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.