Exercises: Surface area of a polyhedron

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.

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 make the box (excluding flaps).

Verify whether Euler's Theorem ($C + V = A + 2$) holds for a pentagonal-base prism, like the one in the figure (the dashed edges are the ones that are hidden). Clearly state the number of faces ($C$), vertices ($V$) and edges ($A$) that you have counted.

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 $15\text{ cm}$.

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.

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 $5\text{ cm}$.

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 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).

A can of jalapeño peppers is a cylinder with a radius of $4\text{ cm}$ and a height of $11\text{ cm}$. Its paper label covers the whole lateral surface exactly, with no overlap.

1. a) If you peel off the label and lay it flat, what shape is it and what are its dimensions?
2. b) How many square centimeters of paper does the label have?

Use the $\pi$ key on your calculator and round to the nearest tenth.

You want to make a poster-board cone whose base has radius $r = 3\text{ cm}$ and whose slant height is $g = 8\text{ cm}$. What angle must the circular sector that forms its lateral surface have? What is the radius of that sector?

For a technology project, you have to build a toolbox in the shape of a regular hexagonal prism. The edge of the base measures $10\text{ cm}$ and the height of the prism is $25\text{ cm}$. If you want to cover all exterior faces (including the two bases) with a protective sheet, how many square centimeters 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 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).

A new polyhedron for a board game has been designed satisfying Euler's formula. 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 relation.

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 1 pesos per square centimeter. 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 $A = \frac{\sqrt{3}}{4} a^2$).

We design a decorative wooden piece consisting of a right square prism with base $8\text{ cm}$ and height $10\text{ cm}$ from which a pyramid with the same base and height has been removed. Calculate the total surface area of this piece (including the interior "hollow" left by the pyramid).

Calculate the lateral surface area of a regular pentagonal prism knowing that its volume is $450\text{ cm}^3$, the area of its base is $30\text{ cm}^2$ and the base side length is $4\text{ cm}$.

(Hint: First use the volume formula to find the height of the prism).

An open-topped metal bin in the shape of a regular octagonal prism is to be manufactured. 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.

Valeria cut out four sets of poster-board pieces for a project. Decide which sets can be assembled into a closed solid (a cylinder or a cone) with no material left over or missing, and justify each answer with a calculation:

1. a) A $18.85\text{ cm} \times 10\text{ cm}$ rectangle and two circles of radius $3\text{ cm}$.
2. b) A $15\text{ cm} \times 10\text{ cm}$ rectangle and two circles of radius $3\text{ cm}$.
3. c) A $120^\circ$ circular sector of radius $9\text{ cm}$ and a circle of radius $3\text{ cm}$.
4. d) A semicircle ($180^\circ$) of radius $9\text{ cm}$ and a circle of radius $3\text{ cm}$.

For a party in Guadalajara, cone-shaped hats (open at the bottom) will be made with a radius of $5\text{ cm}$ and a height of $12\text{ cm}$.

1. a) Find the slant height of the cone.
2. b) What angle must the circular sector that is cut out have? Round to the nearest tenth of a degree.
3. c) How much poster board is needed for each hat, not counting the tabs? Round to the nearest tenth.

An architecture firm is designing a loft in the shape of a rectangular prism. It is known that the length of the room is double 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 ceiling (excluding the floor) that must be covered with insulating paint.

A luxury perfume brand has designed a bottle shaped like 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 make the bottle (including both bases). To do so, you must first calculate the apothem of the frustum (the height of a lateral face) using the Pythagorean theorem. In the figure, $h$ is the height of the bottle and $a$ is the apothem.

A metal part is to be constructed by joining a regular hexagonal prism and a regular hexagonal pyramid that fits perfectly onto its top base. The side length of the base hexagon is $6\text{ cm}$, the height of the prism is $15\text{ cm}$ and the height of the pyramid is $8\text{ cm}$. Calculate the total surface area of the complete part (note that the base where both solids join does not count as exterior surface area).

An ant crawls over the surface of a regular octahedron with edge length $12\text{ cm}$. If the octahedron can be considered as the union of two regular square-based pyramids joined at their bases, calculate the total area of the surface the ant can crawl over. To do so, remember that all the faces of a regular octahedron are equilateral triangles. Give the result in terms of $\sqrt{3}$.

A right prism of height $20\text{ cm}$ has an isosceles trapezium as its base. The parallel sides (bases) of the trapezium measure $14\text{ cm}$ and $6\text{ cm}$, and its height is $3\text{ cm}$. Calculate the total surface area of the prism. Hint: You will need to calculate the length of the non-parallel sides of the base trapezium first using Pythagoras' 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 $6$, deduce how many faces it has and propose the name of a regular polyhedron that satisfies this condition. Once identified, calculate its total surface area if its edge measures $\sqrt{10}\text{ cm}$.

An industrial waste container in the shape of a right prism with a rhombus base has been manufactured. The diagonals of the base rhombus measure $160\text{ cm}$ and $120\text{ cm}$. If the height of the container is $2\text{ meters}$, calculate the total exterior surface area of the container in square meters ($m^2$).

Consider a cube with edge length $10\text{ cm}$. If we make a plane 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 surface area of the solid remaining after removing that small pyramid from that corner.

A $216^\circ$ sector is cut from a poster-board circle of radius $15\text{ cm}$, and by joining its straight edges a cone (with no base) is formed.

1. a) What is the radius of the base of the cone?
2. b) Find the height of the cone.
3. c) How many square centimeters of poster board does the cone have? Round to the nearest tenth.
4. d) If it is used as a funnel with the tip plugged, how many liters does it hold? Round to the nearest hundredth.

A $44\text{ cm} \times 30\text{ cm}$ rectangular sheet can form the lateral surface of a cylinder in two ways: rolling it so that the height is $30\text{ cm}$, or so that the height is $44\text{ cm}$.

1. a) Find the radius of the base in each case. Round to the nearest hundredth.
2. b) Which of the two cylinders has the greater volume? Find both volumes and round to the nearest whole number.
3. c) For the cylinder with the greater volume, how much poster board is needed for the two lids? Round to the nearest tenth.