Area of a Polyhedron
How much cardboard does it take to make a box? How much glass for a skylight shaped like a pyramid? To find out, you need to calculate the area of the polyhedron: 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 spread it out on a table, you get its net: all of its faces joined along some of its edges. The 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 $5\text{ cm}$.
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.
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).
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).
Area of a Pyramid
The lateral faces of a regular pyramid are equal triangles with base equal to the side $l$ and height equal to the slant height of the pyramid $ap$:
$$A_L = \frac{P_{base} \times ap}{2} \qquad\qquad A_T = A_L + A_{base}$$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
-
Frustum of a pyramid: the lateral faces are trapezoids. $A_L = \dfrac{(P_B + P_b) \times ap}{2}$, and the total adds the two bases.
-
Regular polyhedra: all the faces are equal, so $A = n.º\ \text{of faces} \times \text{area of one face}$. 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 1 pesos per square…
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 $A = \frac{\sqrt{3}}{4} a^2$).
Nets of Cylinders and Cones
Round solids can also be cut out and assembled from paper, even though they are not polyhedra. To build them, you need to know what shape each piece has and how big it is.
-
Cylinder: two circles of radius $r$ and a rectangle with height $h$ and base $2\pi r$ (the length of the edge of the circle, so that it wraps around exactly).
-
Cone: a circle of radius $r$ and a circular sector whose radius is the slant height $g$ and whose arc measures $2\pi r$.
In Solids of Revolution you will use these nets to calculate the areas of the cylinder and the cone.
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…
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.
- a) If you peel off the label and lay it flat, what shape is it and what are its dimensions?
- 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…
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 party in Guadalajara, cone-shaped hats (open at the bottom) will be made with a radius of $5\text{ cm}$ and a height of $12\text{ cm}$. a) Find the slant height of…
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 $12\text{ cm}$.
- a) Find the slant height of the cone.
- b) What angle must the circular sector that is cut out have? Round to the nearest tenth of a degree.
- c) How much poster board is needed for each hat, not counting the tabs? Round to the nearest tenth.
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. a) What is the radius…
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.
- a) What is the radius of the base of the cone?
- b) Find the height of the cone.
- c) How many square centimeters of poster board does the cone have? Round to the nearest tenth.
- d) If it is used as a funnel with the tip plugged, how many liters does it hold? Round to the nearest hundredth.