Surface Area of Solids
Don Wilkin runs a corner store and wants to wrap a cardboard box in paper to give away a cake, and also cover an oil can with sheet metal. How much material does he need in each case? To find out, we need the surface area of the solid: the sum of the areas of all its faces. The trick is always the same: imagine the solid unfolded, lying flat on the table.
Lateral Area and Total Surface Area
An artisan is making giant wooden dice for a games center. If each die is a perfect cube with an edge length of $a = 12 \text{ cm}$, calculate the total surface area of…
An artisan is making giant wooden dice for a games center. If each die is a perfect cube with an edge length of $a = 12 \text{ cm}$, calculate the total surface area of wood they will have to paint on each of the dice.
Calculate the total surface area of a shoebox in the shape of a rectangular prism whose dimensions are $15 \text{ cm}$ wide, $30 \text{ cm}$ long and $10 \text{ cm}$…
Calculate the total surface 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 sketch of its net to help you with the calculation.
Surface Area of a Prism
When unfolded, the lateral faces of a right prism form a rectangle whose base is the perimeter of the prism's base and whose height is the height of the prism.
$$A_L = P_{base} \times h \qquad\qquad A_T = A_L + 2 \times A_{base}$$If the base is a regular hexagon with side $l$, its apothem is found with the Pythagorean theorem: $ap^2 = l^2 - (l/2)^2$. Then, $A_{base} = \dfrac{P \times ap}{2}$.
A garden tent is shaped like a right regular triangular prism. The base is an equilateral triangle with side length $l = 3 \text{ m}$. If the length of the tent (height…
A garden tent is shaped like a right regular triangular prism. The base is an equilateral triangle with side length $l = 3 \text{ m}$. 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).
For a design and 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…
For a design and 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 Pythagoras' theorem).
Calculate the total surface area of a cube knowing that the diagonal of one of its faces is $10\text{ cm}$. (Hint: Use Pythagoras' theorem on one of the faces to find…
Calculate the total surface area of a cube knowing that the diagonal of one of its faces is $10\text{ cm}$.
(Hint: Use Pythagoras' theorem on one of the faces to find the edge length of the cube before calculating the area).
Surface Area of a Pyramid
The lateral faces of a regular pyramid are equal triangles, with base equal to the side $l$ of the base and height equal to the slant height of the pyramid $ap$ (which is not the same as the height $h$).
$$A_L = \frac{P_{base} \times ap}{2} \qquad\qquad A_T = A_L + A_{base}$$Frustum of a Pyramid
If you cut off the tip of a pyramid with a plane parallel to the base, you get a frustum. Its lateral faces are equal trapezoids, whose height is the slant height $ap$ of the frustum.
$$A_L = \frac{(P_{larger} + P_{smaller}) \times ap}{2} \qquad A_T = A_L + A_{larger} + A_{smaller}$$If you are given the height $h$ of the frustum, find $ap$ with the Pythagorean theorem:
$$ap^2 = h^2 + x^2$$Here $x$ is half the difference between the sides of the bases:
$$x = \frac{l_{larger} - l_{smaller}}{2}$$Find the total surface area of a regular square pyramid if we know that the side length of the square base measures $8 \text{ cm}$ and the slant height of its side faces…
Find the total surface area of a regular square pyramid if we know that the side length of the square base measures $8 \text{ cm}$ and the slant height of its side faces is $10 \text{ cm}$.
A tent for a youth event is in the shape of a regular square pyramid. The side of the square base measures $10\text{ m}$ and the height of the pyramid (from the center…
A tent for a youth event is in the shape of a regular square pyramid. The side of the square base measures $10\text{ m}$ and the height of the pyramid (from the center of the base to the top vertex) is $12\text{ m}$. Calculate how many square meters of canvas were needed to make the four side faces of the tent.
A modern planter is shaped like a frustum of a regular square pyramid. The bases are two squares with sides $20\text{ cm}$ and $12\text{ cm}$ respectively. If the height…
A modern planter is shaped like a frustum of a regular square pyramid. The bases are two squares with sides $20\text{ cm}$ and $12\text{ cm}$ respectively. If the height of the side faces (the slant height of the frustum) is $10\text{ cm}$, calculate the total outer surface area of the planter (including upper and lower bases).
A design company wants to manufacture a ceiling lamp in the shape of a regular hexagonal pyramid. The side length of the hexagonal base is $10\text{ cm}$ and the height…
A design company wants to manufacture a ceiling lamp in the shape of a regular hexagonal pyramid. The side length of the hexagonal base is $10\text{ cm}$ and the height of the pyramid is $24\text{ cm}$. For the light to project correctly, they must line the side faces with a special copper sheet. Calculate how many $\text{cm}^2$ of copper sheet are needed to cover the lateral surface area of the lamp. Bear in mind that you will first need to find the apothem of the base and, using it, the slant height of the pyramid.
An architect has designed a glass skylight for a museum consisting of a frustum of a regular square pyramid. The lower base has a side length of $4\text{ m}$, the upper…
An architect has designed a glass skylight for a museum consisting of a frustum of a regular square pyramid. The lower base has a side length of $4\text{ m}$, the upper base a side length of $2\text{ m}$ and the height of the frustum is $1.5\text{ m}$. The glass used for the side faces costs $85\text{ \text{RD\$}\ /m}^2$. Calculate the budget required to buy the glass for the lateral surface area, rounding the result to two decimal places.
Surface Area of a Cylinder and a Cone
The cylinder and the cone unfold just like polyhedra, but their lateral surface is curved. The cylinder gives a rectangle with base $2\pi r$ and height $h$; the cone gives a circular sector of radius $g$.
$$A_{L,\text{cylinder}} = 2\pi r h \qquad A_{L,\text{cone}} = \pi r g$$For a design and technology project, you need to cover a cylindrical pencil holder with adhesive paper. The holder has a base radius of $r = 4 \text{ cm}$ and a height…
For a design and technology project, you need to cover a cylindrical pencil holder with adhesive paper. The holder has a base radius of $r = 4 \text{ cm}$ and a height of $h = 12 \text{ cm}$. Calculate how many square centimeters 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 curved surface area of a cylinder whose base circumference measures $C = 25.12 \text{ cm}$ and whose height is $h = 10 \text{ cm}$. (Hint: Remember that…
Calculate the curved surface area of a cylinder whose base circumference measures $C = 25.12 \text{ cm}$ and whose height is $h = 10 \text{ cm}$. (Hint: Remember that when you unfold the cylinder, the curved surface 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 only needs to cover the curved part (excluding the bases), what surface area of paper is needed for each candle? Use $\pi \approx 3.14$.
A decorative garden lighthouse is made up of two solids: a cylinder of radius $10\text{ cm}$ and height $30\text{ cm}$, topped by a cone with the same base and a slant…
A decorative garden lighthouse is made up of two solids: a cylinder of radius $10\text{ cm}$ and height $30\text{ cm}$, 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 on which it rests).
Surface Area of a Sphere
The sphere cannot be unfolded, but its surface area is found with a simple formula: four times the area of its great circle.
$$A = 4\pi r^2$$A hemisphere (dome) has half of that as its curved surface: $2\pi r^2$. If you also count its circular base, add $\pi r^2$.
A basketball has a radius of $r = 12 \text{ cm}$. Calculate the area of leather required 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 area of leather required to make it, rounding the result to two decimal places. (Remember that the surface area of a sphere is $A = 4 \times \pi \times r^2$).
The dome of an astronomical observatory in the shape of a hemisphere of radius $6\text{ meters}$ is to be restored. If the budget for painting is $\text{RD\$}\ 900…
The dome of an astronomical observatory in the shape of a hemisphere of radius $6\text{ meters}$ is to be restored. If the budget for painting is $\text{RD\$}\ 900$ per square meter, how much will it cost to paint the outer surface of the dome? (Use $\pi \approx 3.14$).
A gym ball manufacturer 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…
A gym ball manufacturer 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 is the numerical ratio between the area of the large ball and that of the small one? Use $\pi = 3.14$.
Composite Solids
When a solid joins together or hollows out other solids, only the surface that remains visible counts. Break the solid into parts, find the area of each part, and check which faces disappear or appear.
Observe the net of this geometric solid. If each of the isosceles triangles has an area of $15 \text{ cm}^2$ and the central square has a side of $6 \text{ cm}$, what is…
Observe the net of this geometric solid. If each of the isosceles 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 figure when assembled? What geometric solid is formed?
An acrylic trophy has been designed consisting of a right cylinder of height $10\text{ cm}$ and diameter $6\text{ cm}$, from which a perfect hemisphere has been removed…
An acrylic trophy has been designed consisting of a right cylinder of height $10\text{ cm}$ and diameter $6\text{ cm}$, from which a perfect hemisphere has been removed at the upper base, whose radius matches exactly the radius of the cylinder. Calculate the total surface area of the trophy (outer cylindrical surface, surface of the bottom base and the curved surface of the hemispherical hollow). Use $\pi \approx 3.14$.
An industrial storage tank is shaped like 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 storage tank is shaped like 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 surface (cone), but not the base on which it rests. If each tin of paint can cover $15\text{ m}^2$, what is the minimum number of tins of paint that must be bought?
A wooden cube of edge length $12\text{ cm}$ is cut by a plane passing through the midpoints of three edges meeting at the same vertex, removing a small triangular…
A wooden cube of edge length $12\text{ cm}$ is cut by a plane passing through the midpoints of three edges meeting at the same vertex, removing a small triangular pyramid (an irregular tetrahedron). Calculate the total surface area of the resulting solid left after removing the small pyramid. Hint: The cut creates a new face that is an equilateral triangle whose sides are the diagonals of squares with side length $6\text{ cm}$.
A greenhouse is to be built in the shape of a half-cylinder resting on the ground, as shown in the diagram. The length of the greenhouse is $15\text{ m}$ and the width…
A greenhouse is to be built in the shape of a half-cylinder resting on the ground, as shown in the diagram. The length of the greenhouse is $15\text{ m}$ and the width of the base (the diameter of the half-cylinder) is $6\text{ m}$. If the plastic to cover the curved part and the two front walls (semicircles) costs $12\text{ \text{RD\$}\ /m}^2$, what will be the total cost of the plastic required? (The floor is not included).
A mechanical steel part has the shape of a right square-based prism with a cylindrical hole running through it from base to base at the center. The side of the square…
A mechanical steel part has the shape of a right square-based prism with a cylindrical hole running through it from base to base at 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 beginning: the $30 \times 20 \times 12$ cm box needs $2{,}400\ \text{cm}^2$ of cardboard, and the can with a radius of $3$ cm and a height of $10$ cm needs about $245\ \text{cm}^2$ of sheet metal.