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 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).

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).

Aoife wants to varnish a solid wooden decorative piece shaped like a regular square pyramid. The base edge measures $6\text{ cm}$ and the height of the pyramid is $4\text{ cm}$. Aoife made the following calculations in her notebook:

"The perimeter of the base is $4 \times 6 = 24\text{ cm}$. The lateral area is $\frac{24 \times 4}{2} = 48\text{ cm}^2$. The base area is $6^2 = 36\text{ cm}^2$. Therefore, the total area to varnish is $48 + 36 = 84\text{ cm}^2$".

1. a) Identify and explain in detail the geometric error Aoife made in her approach. Which two distinct segments did she confuse?
2. b) Correct the procedure by calculating the true slant height of the pyramid using the Pythagorean theorem, and determine the actual total area she needs to varnish.

A courier company is designing a new model of recycled cardboard box shaped like a right prism with a square base. For storage and stability reasons, the edge of the square base must strictly measure $8\text{ cm}$. To make the complete net of the box (counting its two bases and the four lateral faces, without considering glue flaps), exactly $448\text{ cm}^2$ of cardboard is used.

1. a) Write an algebraic equation involving the unknown height $h$ of the prism and the given total area.
2. b) Solve this equation to find how many centimetres high the box will be.

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 centimetres 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 Pythagoras' 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 €0.05 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 $A = \frac{\sqrt{3}}{4} a^2$).

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.

An individual tent is going to be made in the shape of a regular square pyramid. The base is a square with side length $2\text{ m}$ and the central vertical pole has a height of $1.5\text{ m}$. Only a special waterproof fabric will be used on the four lateral faces (the floor will be made from a different insulating material).

1. a) Without using a calculator or finding the slant height yet, make a reasoned estimate of the lateral area: will it be greater than, less than, or equal to $6\text{ m}^2$? (Hint: think about what area the four faces would have if they were vertical triangles with base $2\text{ m}$ and height $1.5\text{ m}$).
2. b) Use the Pythagorean theorem to find the lateral slant height of the pyramid and calculate the exact amount of fabric needed (round the result to two decimal places).
3. c) Compare your result with the estimate you made and justify geometrically why the slant height of a pyramid is always strictly greater than its height.

A municipal works brigade is going to restore and waterproof 10 concrete planters located on a pedestrian boulevard. Each planter is in the shape of a regular square frustum open at the top and resting on its smaller base. The interior dimensions are: edge of the upper square (open) $1.2\text{ m}$, edge of the lower square (bottom) $0.8\text{ m}$, and slant height of the lateral faces (height of the trapezoids) $0.5\text{ m}$.

1. a) Model the planter by identifying the polygonal shapes that make up the interior surface and calculate the total area that must be covered in a single planter.
2. b) The waterproofing resin is sold in $750\text{ ml}$ cans at a price of $14.50\text{ \euro}$ each, and the manufacturer certifies that each can covers exactly $2\text{ m}^2$. Determine how many full cans the town council needs to purchase to treat the 10 planters and calculate the total cost of the project.

An architecture firm is designing a loft in the shape of a cuboid. 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 $2\text{ metres}$, calculate the total exterior surface area of the container in square metres ($m^2$).

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.

For a cultural centre, a hollow decorative metal sheet column is planned in the shape of a right prism $3\text{ m}$ high. The base of the prism is an 'L'-shaped polygon with all right angles, whose six consecutive edges measure: $3\text{ m}$, $2\text{ m}$, $1\text{ m}$, $1\text{ m}$, $2\text{ m}$, and $1\text{ m}$. It is necessary to calculate the total surface area of metal sheet (lateral faces plus the two 'L'-shaped bases). Two technicians on the team propose different strategies:

• Technician A (Decomposition): Divides the 'L'-shaped base into two independent rectangles, calculates the total area of two rectangular prisms separately, and then subtracts the area of the fictitious interior faces where they join.

• Technician B (General formula): Finds the complete perimeter of the 'L'-shaped polygon to calculate the lateral area directly using $A_L = P_{base} \times h$, and adds twice the area of the 'L'-shaped base.

1. a) Work through both methods step by step and check numerically whether both lead to the same result.
2. b) Compare the two approaches, justifying which one is more efficient and less prone to errors when working with solids with complex polygonal bases.

An industrial designer creates a solid polyhedral trophy. The solid consists of a cube of edge length $a$ with a regular pyramid resting on its top face; the base of the pyramid fits exactly onto the top face of the cube and its four lateral faces are equilateral triangles of side length $a$.

1. a) Describe the net of the solid, stating the number and type of polygons that form its total exterior surface, specifying which faces remain hidden on the inside and must not be counted.
2. b) Knowing that the area of an equilateral triangle of side length $a$ is $\frac{\sqrt{3}}{4}a^2$ and that the total exterior surface area of the trophy is exactly $(500 + 100\sqrt{3})\text{ cm}^2$, deduce in a reasoned way the exact measurement of edge length $a$.
3. c) If a client orders a scaled replica tripling the edge length ($a' = 3a$), will the amount of material for the surface also triple? Justify your answer algebraically without calculating the new numerical value.