3.7.2 Surface area
3.7.2 Surface area
Surface area is the total area that is exposed on the surface of an object. It includes the sum of all the areas of the individual faces or surfaces that make up the object. Surface area is typically measured in square units, such as square meters or square feet.
The calculation of surface area depends on the shape of the object.
For example, the surface area of a rectangular prism can be calculated by finding the area of each of its six faces and adding them together.
Similarly, the surface area of a sphere can be calculated using a different formula that takes into account its curved surface.
The surface area of other shapes, such as cones and cylinders, can also be calculated using specific formulas.
The method for calculating the surface area of an object depends on its shape. Here are some common shapes and their respective surface area formulas:
Rectangular Prism:
The surface area of a rectangular prism can be calculated by adding the area of each of its six faces.
The formula is
\( SA=2lw+2lh+2wh \)
where \( SA \) represents the surface area, \( l \) represents the length, \( w \) represents the width, and \( h \) represents the height.
Cube:
The surface area of a cube can be calculated by adding the area of each of its six faces.
The formula is
\( SA = 6s^2 \)
where \( SA \) represents the surface area, and s represents the length of one side.
Sphere:
The surface area of a sphere is calculated by using the formula
\( SA = 4\pi r^2 \)
where \( SA \) represents the surface area, and r represents the radius of the sphere.
Cylinder:
The surface area of a cylinder can be calculated by adding the area of its two circular bases to the area of its curved surface.
The formula is
\( SA=2 \pi r^2+2\pi rh \)
where \( SA \) represents the surface area, \( r \) represents the radius of the base, and \( h \) represents the height of the cylinder.
Cone:
The surface area of a cone can be calculated by adding the area of its circular base to the area of its curved surface.
The formula is
\( SA=\pi r^2+\pi r \sqrt{r^2+ h^2 } \)
where \( SA \) represents the surface area, \( r \) represents the radius of the base, and \( h \) represents the height of the cone.
Worked example:
The diagram shows the surface of a garden pond, made from a rectangle and two semicircles.
The rectangle measures \( 3 \) m by \( 1.2 \) m.
Calculate the area of this surface.
The shape is made up of a rectangle of length \( 3 \) m and width \( 1.2 \) m and a circle of diameter \( 1.2 \) m.
Calculate the area of rectangle by \( A=l\times w \)
\( A=3\times 1.2 \)
\( A=3.6 \)
Calculate the radius of the circle by \( r=\frac{d}{2} \)
\( r=\frac{d}{2} \) , \( r=\frac{1.2}{2}=0.6 \)
Calculate the area of the circle by \( A=\pi r^2 \)
\( A=\pi \times 0.6^2 \)
\( A=1.130973355… \)
Add the two areas together
\( 3.6+1.130973355…=4.730973355 \)
Round to \( 3 \) significant figures
\( 4.73 m^2 \)
Test yourself
Question 1:
The diagram shows a prism with a rectangular base, \( ABFE \).
The cross section, \( ABCD \), is a trapezium with \( AD=BC \).
\( AB=8 \) cm, \( GH=5 \) cm, \( BF=12 \) cm and angle \( ABC=70° \).
Calculate the total surface area of the prism.
[6]
Question 2:
A cone has radius \( x \) cm and slant height \( x\sqrt{5} \) cm. The volume of the cone is \( 1000 \) cm2.
Calculate the value of \( x \).
[The volume, \( V \), of a cone with radius \( r \) and height h is \( V=\frac{1}{2}\pi r^2h \).]
[4]
Question 3:
The diagram shows a hemispherical bowl of radius \( 5.6 \) cm and a cylindrical tin of height \( 10 \) cm.
Show that the volume of the bowl is \( 368 \) cm3, correct to the nearest cm3.
[The volume, \( V \), of a sphere with radius, \( r \), is given by \( V=\frac{4}{3}\pi r^3 \)]
[2]
The tin is completely full of soup.
When all the soup is poured into the empty bowl , \( 80% \) of the bowl is filled.
Calculate the radius of the tin.
[4]
Question 4:
A solid metal prism with volume \( 500 \) cm3 is melted and made into \( 6 \) identical spheres.
Calculate the radius of each sphere.
The volume, \( V \), of a sphere with radius, \( r \),is \( V=\frac{4}{3}r^3 \)
[3]
Question 5:
Show that the volume of a metal sphere of radius \( 15 \) cm is \( 14140 \) cm3, correct to \( 4 \)
significant figures.
The volume,\( V \), of a sphere with radius,\( r \),is \( V=\frac{4}{3}\pi r^3 \)
[2]
The sphere is placed inside an empty cylindrical tank of \( 25 \) cm and height \( 60 \) cm.
The tank is filled with water.
Calculate the volume of water needed to fill the tank.
[3]
The sphere is removed from the tank.
Calculate the depth, \( d \), of water in the tank.
[2]