Surface Area of Solids
How much wrapping paper do you need for a gift box? How much metal does it take to make a can of soup? To find out, you need the surface area of the solid: the total area of all its faces. The trick is always the same. Picture the solid unfolded, lying flat on a table.
What is surface area?
A net is the solid cut along some edges and unfolded flat. The surface area of the solid is just the area of its net. Here is the net of a cube.
You can also go the other way. Look at the faces in a net and you can tell which solid it makes. Count them, add their areas, and you have the surface area.
An artisan is making giant wooden dice for a toy library. If each die is a perfect cube with an edge of $a = 12 \text{ cm}$, calculate the total surface area of wood…
An artisan is making giant wooden dice for a toy library. If each die is a perfect cube with an edge of $a = 12 \text{ cm}$, calculate the total surface area of wood that will need to be painted on each of the dice.
Calculate the total area of a shoebox in the shape of a rectangular prism whose dimensions are $15 \text{ cm}$ wide, $30 \text{ cm}$ long and $10 \text{ cm}$ high. Draw…
Calculate the total area of a shoebox in the shape of a rectangular prism whose dimensions are $15 \text{ cm}$ wide, $30 \text{ cm}$ long and $10 \text{ cm}$ high. Draw a schematic of its net to help you with the calculation.
Look at the net of this geometric solid. If each of the equilateral triangles has an area of $15 \text{ cm}^2$ and the central square has a side of $6 \text{ cm}$, what…
Look at the net of this geometric solid. If each of the equilateral triangles has an area of $15 \text{ cm}^2$ and the central square has a side of $6 \text{ cm}$, what is the total area of the shape when it is assembled? What geometric solid is formed?
Prisms
Unfold the lateral faces of a right prism and they make one big rectangle. Its width is the perimeter of the base, and its height is the height of the prism.
$$LA = P_{base} \cdot h \qquad\qquad SA = LA + 2B$$Here $P_{base}$ is the perimeter of the base and $B$ is the area of one base.
Why does it work? The lateral faces are rectangles that all have height $h$. Their widths are the sides of the base. Add the widths and you get the perimeter.
A garden tent has the shape of a right regular triangular prism. The base is an equilateral triangle with a side of $l = 3 \text{ m}$ and a triangle height of…
A garden tent has the shape of a right regular triangular prism. The base is an equilateral triangle with a side of $l = 3 \text{ m}$ and a triangle height of $1,7 \text{ m}$. If the length of the tent (height of the prism) is $5 \text{ m}$, calculate the lateral area of the tent (excluding the two triangular bases).
Finding missing lengths with the Pythagorean Theorem
Often the problem does not give you the length you need. You find it with the Pythagorean Theorem, $a^2 + b^2 = c^2$, in a right triangle hiding inside the figure.
Regular polygon bases. The apothem $a$ of a regular polygon is the distance from its center to the midpoint of a side. The area of any regular polygon is
$$B = \tfrac{1}{2} \cdot a \cdot P.$$In a regular hexagon the distance from the center to a vertex equals the side $s$. That gives a right triangle with legs $a$ and $\tfrac{s}{2}$ and hypotenuse $s$.
Face diagonal of a square. The diagonal $d$ of a square with side $s$ is the hypotenuse of a right triangle with two legs $s$. So $s^2 + s^2 = d^2$, which gives $d^2 = 2s^2$ and $d = s\sqrt{2}$.
For a technology project, you have to make a wooden box in the shape of a regular hexagonal prism. The base edge measures $4\text{ cm}$ and the height of the prism is…
For a technology project, you have to make a wooden box in the shape of a regular hexagonal prism. The base edge measures $4\text{ cm}$ and the height of the prism is $15\text{ cm}$. Calculate the total surface area of wood you will need to build the box (including the two bases).
(Note: To find the area of the base, remember that you must first calculate the apothem of the hexagon using the Pythagorean theorem).
Calculate the total area of a cube knowing that the diagonal of one of its faces measures $10\text{ cm}$. (Hint: Use the Pythagorean theorem on one of the faces to find…
Calculate the total area of a cube knowing that the diagonal of one of its faces measures $10\text{ cm}$.
(Hint: Use the Pythagorean theorem on one of the faces to find the length of the cube's edge before calculating the area).
Pyramids
The lateral faces of a regular pyramid are identical triangles. Each triangle has base $s$ (a side of the base polygon). Its height is the slant height $\ell$, measured along the face from the midpoint of a base edge up to the top vertex.
$$LA = \tfrac{1}{2} \cdot P_{base} \cdot \ell \qquad\qquad SA = LA + B$$Why? Each triangle has area $\tfrac{1}{2} s \ell$. Add up all the triangles and the sides $s$ add up to the perimeter $P_{base}$.
For a square base, $a$ is half the side of the base. For a hexagonal base, $a$ is the apothem of the hexagon.
Find the total area of a regular square-based pyramid if we know that the side of the square base measures $8 \text{ cm}$ and the slant height of its lateral faces…
Find the total area of a regular square-based pyramid if we know that the side of the square base measures $8 \text{ cm}$ and the slant height of its lateral faces measures $10 \text{ cm}$.
A tent for a youth event has the shape of a regular square-based pyramid. The side of the square base measures $10\text{ ft}$ and the height of the pyramid (from the…
A tent for a youth event has the shape of a regular square-based pyramid. The side of the square base measures $10\text{ ft}$ and the height of the pyramid (from the center of the base to the top vertex) is $12\text{ ft}$. Calculate how many square feet of canvas were needed to make the four lateral faces of the tent.
A design company wants to manufacture a ceiling lamp in the shape of a regular hexagonal pyramid. The side of the hexagonal base measures $10\text{ cm}$ and the height…
A design company wants to manufacture a ceiling lamp in the shape of a regular hexagonal pyramid. The side of the hexagonal base measures $10\text{ cm}$ and the height of the pyramid is $24\text{ cm}$. For the light to project correctly, they must cover the lateral faces with a special copper sheet. Calculate how many $\text{cm}^2$ of copper sheet are needed to cover the lateral surface of the lamp. Keep in mind that you will first need to find the apothem of the base and, with it, the apothem of the pyramid.
Cylinders
Cut a cylinder (a soup can) and unroll the curved side. You get a rectangle. Its height is $h$, and its width is the circumference of the base, $C = 2\pi r$. The bases are two circles.
$$LA = 2\pi r h = C \cdot h \qquad\qquad SA = 2\pi r h + 2\pi r^2$$For a technology project, you need to cover a cylindrical pencil holder with sticky paper. The holder has a base radius of $r = 4 \text{ in}$ and a height of…
For a technology project, you need to cover a cylindrical pencil holder with sticky paper. The holder has a base radius of $r = 4 \text{ in}$ and a height of $h = 12 \text{ in}$. Calculate how many square inches of paper you will need, knowing that the holder is open at the top (it only has one base). Take $\pi \approx 3,14$.
Calculate the lateral area of a cylinder whose base circumference measures $C = 25,12 \text{ cm}$ and its height is $h = 10 \text{ cm}$. (Hint: Remember that when you…
Calculate the lateral area of a cylinder whose base circumference measures $C = 25,12 \text{ cm}$ and its height is $h = 10 \text{ cm}$. (Hint: Remember that when you open the cylinder, the lateral area is a rectangle whose base is the length of the circumference).
An artisan candle maker wants to cover the lateral surface of a cylindrical candle with silver protective paper. The candle has a diameter of $8\text{ cm}$ and a height…
An artisan candle maker wants to cover the lateral surface of a cylindrical candle with silver protective paper. The candle has a diameter of $8\text{ cm}$ and a height of $20\text{ cm}$. If the paper must only cover the curved part (without the bases), what surface area of paper is needed for each candle? Use $\pi \approx 3,14$.
Cones
A right cone is like a pyramid with a circular base. The slanted side has length $\ell$, the slant height. The unrolled lateral surface is a slice of a circle with radius $\ell$, and its curved edge has the length of the base circle, $2\pi r$. That slice is the fraction $\frac{2\pi r}{2\pi \ell} = \frac{r}{\ell}$ of a whole circle of area $\pi \ell^2$. Multiply the two and you get
$$LA = \pi r \ell \qquad\qquad SA = \pi r \ell + \pi r^2.$$In a designer ice cream shop, they have created a wafer cone in the shape of a right cone whose height is $12\text{ cm}$ and its base radius is $5\text{ cm}$. Calculate…
In a designer ice cream shop, they have created a wafer cone in the shape of a right cone whose height is $12\text{ cm}$ and its base radius is $5\text{ cm}$. Calculate the total area of the wafer cone (lateral surface plus the base circle) to find out how much chocolate they need to coat it completely.
(Hint: You will first need to find the slant height of the cone).
Spheres and hemispheres
A sphere cannot be unfolded flat, because it curves in every direction at once. Its surface area still has a very simple formula: four times the area of its biggest circle (the one through the center).
$$SA = 4\pi r^2$$A hemisphere is half a sphere. Its curved surface is half of $4\pi r^2$, which is $2\pi r^2$. If the flat circle is included (a bowl with a lid), add $\pi r^2$ to get $3\pi r^2$.
A basketball has a radius of $r = 12 \text{ cm}$. Calculate the surface area of leather needed to make it, rounding the result to two decimal places. (Remember that the…
A basketball has a radius of $r = 12 \text{ cm}$. Calculate the surface area of leather needed to make it, rounding the result to two decimal places. (Remember that the area of a sphere is $A = 4 \cdot \pi \cdot r^2$).
The dome of an astronomical observatory, which has the shape of a hemisphere with a radius of $6\text{ meters}$, is to be restored. If the budget for the paint is…
The dome of an astronomical observatory, which has the shape of a hemisphere with a radius of $6\text{ meters}$, is to be restored. If the budget for the paint is $15\text{ \$}$ per square foot, how much will it cost to paint the outer surface of the dome? (Use $\pi \approx 3,14$).
A manufacturer of gym balls produces rubber spheres in two sizes. The small ball has a surface area of $5024\text{ cm}^2$. The large ball has a radius that is exactly…
A manufacturer of gym balls produces rubber spheres in two sizes. The small ball has a surface area of $5024\text{ cm}^2$. The large ball has a radius that is exactly double that of the small ball. Calculate the surface area of the large ball. What numerical relationship (ratio) is there between the area of the large ball and that of the small one? Use $\pi = 3,14$.
Frustums
A frustum is what is left of a pyramid after you slice off the top with a cut parallel to the base. Its lateral faces are trapezoids, and it has two bases of different sizes.
For a regular frustum with $4$ lateral faces, each trapezoid has area $\tfrac{1}{2}(b_1 + b_2)\,\ell$, where $b_1$ and $b_2$ are the sides of the two bases and $\ell$ is the slant height. Adding all of them gives
$$LA = \tfrac{1}{2}(P_1 + P_2)\,\ell \qquad\qquad SA = LA + B_1 + B_2.$$To get $\ell$ from the height $h$, use a right triangle. Its legs are $h$ and the horizontal gap between the edges of the two bases, which is $\frac{b_1 - b_2}{2}$.
A modern planter has the shape of a frustum of a regular square-based pyramid. The bases are two squares with sides $20\text{ cm}$ and $12\text{ cm}$ respectively. If…
A modern planter has the shape of a frustum of a regular square-based pyramid. The bases are two squares with sides $20\text{ cm}$ and $12\text{ cm}$ respectively. If the height of the lateral faces (the slant height of the frustum) is $10\text{ cm}$, calculate the total outer surface area of the planter (including the top and bottom bases).
An architect has designed a skylight for a museum consisting of a frustum of a regular square pyramid made of glass. The bottom base has a side of $4\text{ m}$, the top…
An architect has designed a skylight for a museum consisting of a frustum of a regular square pyramid made of glass. The bottom base has a side of $4\text{ m}$, the top base a side of $2\text{ m}$ and the height of the frustum is $1,5\text{ m}$. The glass used for the lateral faces costs $85\text{ \$/m}^2$. Calculate the budget needed to buy the glass for the lateral surface, rounding the result to two decimal places.
Composite solids
Many real objects are two solids put together: a silo, a lighthouse, a tent. The rule is simple: add only the surfaces you can touch from the outside. Where two solids are glued together, those faces disappear.
Half of a solid. A half-cylinder lying on its flat side, like a greenhouse tunnel, has a curved surface that is half of the lateral area, $\tfrac{1}{2}(2\pi r L) = \pi r L$, where $L$ is its length. Its two ends are semicircles, and together they make one full circle, $\pi r^2$. The floor is not part of the outside.
A decorative lighthouse for a garden is made up of two parts: a cylinder with a radius of $10\text{ cm}$ and a height of $30\text{ cm}$, topped by a cone with the same…
A decorative lighthouse for a garden is made up of two parts: a cylinder with a radius of $10\text{ cm}$ and a height of $30\text{ cm}$, topped by a cone with the same base and a slant height of $15\text{ cm}$. Calculate the total outer surface area of this lighthouse (excluding the base it rests on).
A greenhouse is to be built in the shape of a half-cylinder resting on the ground, as shown in the figure. The length of the greenhouse is $15\text{ m}$ and the width of…
A greenhouse is to be built in the shape of a half-cylinder resting on the ground, as shown in the figure. The length of the greenhouse is $15\text{ m}$ and the width of the base (the diameter of the semi-cylinder) is $6\text{ m}$. If the plastic to cover the curved part and the two front walls (semicircles) costs $12\text{ \$/m}^2$, what will be the total cost of the plastic needed? (The floor is not counted).
An industrial tank has the shape of a right cylinder topped by a cone with the same base. The height of the cylindrical part is $8\text{ m}$ and the height of the…
An industrial tank has the shape of a right cylinder topped by a cone with the same base. The height of the cylindrical part is $8\text{ m}$ and the height of the conical part is $3\text{ m}$. The diameter of the tank is $8\text{ m}$. It is desired to paint the entire outer lateral surface and the top base (cone), but not the base on which it rests. If each can of paint can cover $15\text{ m}^2$, what is the minimum number of cans of paint that must be bought?
Solids with a piece removed
When you drill, scoop or slice a solid, ask two questions:
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Which old surface disappears?
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Which new surface appears?
Then compute: $SA = SA_{\text{original}} - \text{(lost)} + \text{(new)}$.
A hole through a prism. You lose two circles (the openings on the bases). You gain the inside wall of the hole, a cylinder's lateral area.
A hemispherical dent. The flat circle $\pi r^2$ on top is replaced by the curved bowl $2\pi r^2$.
Cutting off a corner of a cube. Three faces each lose a small corner triangle, and a new triangular face appears. If that new face is an equilateral triangle with side $s$, its height comes from the Pythagorean Theorem: $\sqrt{s^2 - (s/2)^2} = \frac{\sqrt{3}}{2}s$. Then
$$A = \tfrac{1}{2} \cdot s \cdot \tfrac{\sqrt{3}}{2}s = \frac{\sqrt{3}}{4}s^2.$$An acrylic trophy has been designed consisting of a right cylinder with a height of $10\text{ cm}$ and a diameter of $6\text{ cm}$, from which a perfect hemisphere has…
An acrylic trophy has been designed consisting of a right cylinder with a height of $10\text{ cm}$ and a diameter of $6\text{ cm}$, from which a perfect hemisphere has been extracted at the top base, whose radius coincides exactly with the radius of the cylinder. Calculate the total surface area of the trophy (outer cylindrical surface, bottom base surface and the curved surface of the hemispherical hollow). Use $\pi \approx 3,14$.
A wooden cube with an edge of $12\text{ cm}$ is cut by a plane passing through the midpoints of three concurrent edges at the same vertex, separating a small triangular…
A wooden cube with an edge of $12\text{ cm}$ is cut by a plane passing through the midpoints of three concurrent edges at the same vertex, separating a small triangular pyramid (an irregular tetrahedron). Calculate the total area of the resulting figure left after removing the small pyramid. Hint: The cut generates a new face which is an equilateral triangle whose sides are the diagonals of the squares with side $6\text{ cm}$.
A mechanical steel part has the shape of a right square-based prism with a cylindrical hole passing through it from base to base through the center. The side of the…
A mechanical steel part has the shape of a right square-based prism with a cylindrical hole passing through it from base to base through the center. The side of the square base of the prism measures $10\text{ cm}$ and the height of the part is $20\text{ cm}$. The diameter of the cylindrical hole is $8\text{ cm}$. Calculate the total surface area of the part (including the inner surface of the hole).
Back to the start: the $10 \times 6 \times 4$ in. gift box needs $248$ in.² of wrapping paper, and the soup can uses about $51.8$ in.² of metal.