Surface Area of a Polyhedron
Doña Altagracia wants to cover the roof of a pyramid-shaped shed with new zinc sheeting, and Wilkin needs to know how much cardboard he uses on a box for his school project. In both cases you have to calculate the surface area of the solid: the sum of the areas of all its faces. The trick is to unfold it and see it flat.
The Net
If you "open up" a polyhedron and lay it flat on the table, you get its net: all its faces joined along some of its edges. The surface area of the polyhedron is the area of that net. I recommend building one yourself with cardstock, a ruler, a set square, and scissors: it makes it much easier to understand.
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
The lateral faces of a right prism, when unfolded, form a rectangle whose base is the perimeter of the prism's base and whose height is the height of the prism.
When the base is not a rectangle, you first find its perimeter and area using what you already know (for example, the rhombus: $A = \frac{D \times d}{2}$, and its side using the Pythagorean Theorem with the half-diagonals). If you are given the volume, remember that $V = A_{base} \times h$ and solve for the height.
The Space Diagonal of a Rectangular Prism
The space diagonal joins two opposite corners of a rectangular prism with dimensions $a$, $b$, and $c$. You get it by using the Pythagorean Theorem twice: first the diagonal of the floor, $d_1^2 = a^2 + b^2$, and then a right triangle with legs $d_1$ and $c$ (the height):
$$D^2 = a^2 + b^2 + c^2$$If you are given $D$ and asked for a dimension, solve for it in $D^2 = a^2 + b^2 + c^2$. For example, if the length is twice the width ($a = 2b$), you get $D^2 = 4b^2 + b^2 + c^2 = 5b^2 + c^2$, an equation in which only $b$ is missing.
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 make the box (excluding flaps).
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).
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…
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 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…
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.
Surface Area of a Pyramid
The lateral faces of a regular pyramid are identical triangles with base equal to the side $l$ and height equal to the slant height of the pyramid $ap$:
$$A_L = \frac{P_{base} \times ap}{2}$$ $$A_T = A_L + A_{base}$$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
A frustum of a pyramid is what remains when you cut a pyramid with a plane parallel to the base and remove the top part. Its lateral faces are trapezoids and it has two different bases ($B$ the larger, $b$ the smaller).
$$A_L = \frac{(P_B + P_b) \times ap}{2}$$ $$A_T = A_L + A_B + A_b$$Here $ap$ is the slant height of the frustum (the height of each trapezoid). If you are given the height $h$ of the frustum, use the Pythagorean Theorem in a right triangle with legs $h$ and the difference of the apothems of the bases (for squares, $\frac{l_B - l_b}{2}$).
Regular polyhedra: all the faces are identical, so $A = \text{no. of faces} \times \text{area of one face}$. For an equilateral triangle with side $l$, the area of each face is $\frac{\sqrt{3}}{4}\,l^2$.
When a solid is composite (for example, a prism with a pyramid on top), add up the areas of the outer faces and subtract the faces that are glued together: they are not part of the outer surface.
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 $\text{RD\$}\ 3…
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 $\text{RD\$}\ 3$ 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 $A = \frac{\sqrt{3}}{4} a^2$).
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…
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.
Faces, Vertices, and Edges: Euler
In every convex polyhedron the Euler relation holds: counting faces ($C$), vertices ($V$), and edges ($A$),
$$C + V = A + 2$$It is useful for checking that you counted correctly, or for finding a missing piece of data. In a cube: $6 + 8 = 12 + 2$. It holds.
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…
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.
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.…
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.
"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 $6$,…
"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 $\sqrt{10}\text{ cm}$.
Composite Solids and Problems
The longer problems combine everything above. Follow this plan: understand which surface you are asked for (with a lid? without a bottom?), draw the net, calculate the missing data (with the Pythagorean Theorem if needed), carry out the sums, and check units and result.
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…
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…
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 $9.66\text{ cm}$.…
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.
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…
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…
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…
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.
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…
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…
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.
Back to the beginning: for Wilkin's (closed) box you calculate the total area, lateral plus two bases; for the zinc roof of doña Altagracia's pyramid-shaped shed you only need the lateral area, because the base is the floor.