Cylinder â volume
A cylinder is a A three-dimensional figure having two parallel bases that are circles equal in terms of radius, diameter, circumference and surface area congruent).. The cross-section of a cylinder is a circle.
For a cylinder of length h with radius r, the volume of a cylinder can be calculated using the formula: \(\text{volume of a cylinder} = \pi r^2 h\)
Example
Calculate the volume of the cylinder.
The measurement shown is the diameter of the circular face, so the radius is 3.5 cm.
Substituting the values h = 16cm and r = 3.5cm into the formula for the volume of the cylinder gives:
Volume = \(\pi \times 3.5^2 \times 16 = 615.8~\text{cm}^3\)
Cylinder â Surface area
A cylinder has 2 flat circular faces and a curved surface.
Imagine a tin with a label around the curved surface. Taking the label off the tin and flattening it out gives a rectangle.
The length of the rectangle will equal the circumference of the circular faces as the edge of the rectangle must fit exactly round the edge of the circles. The sides of the rectangle are therefore h and 2Ï°ù.
The area of each of the two circles is \(\pi r^2\)and the area of the rectangle is \(2 \pi r \times h\).
The surface area of a cylinder is the area of each of its faces added together.
\(\text{Total~surface~area~of~a~cylinder} = 2 \pi r^2 + \pi d h\)
This can also be written as \(\text{Total~surface~area~of~a~cylinder} = 2 \pi r^2 + 2 \pi r h\)
Question
Calculate the total surface area of the cylinder.
Answer:
Draw each face.
The area of a circle \(= \pi \times r^2\)
Circle area \(= \pi \times 3^2 = 28.3~\text{cm}^2\) (1d.p.).
The rectangle surrounds the circular face. The length of the rectangular face is the circumference of the circular face.
Remember that the circumference of a circle \(= \pi \times d\) and the diameter \(= 2 \times \text{radius}\).
Circumference \(= \pi \times 6 = 18.8~\text{cm}\) (1d.p.).
Rectangle \(= 18.8 \times 10 = 188~\text{cm}^2\)
Total surface area \(= 28.3 + 28.3 + 188 = 244.6~\text{cm}^2\) (1d.p.).
Cone â Volume
If a cone is filled with water it is possible to pour 3 full cones into a cylinder of the same diameter and height.
Remember the volume of a cylinder is \(\pi r^2 h\).
The volume of the cone is one third of the volume of the cylinder.
The formula for the volume of a cone is: \(\text{volume of a cone} = \frac{1}{3} \pi r^2 h\).
In the exam \(V = \frac{1}{3} \pi r^2 h\) is given on the formula sheet.
Cone â Surface area
The What a 3-dimensional object would look like if it was opened out and laid flat. of a cone is a circle and a A slice of the circle, cut off by two radii..
The sector creates the curved surface of the cone.
The curved surface area of a cone can be calculated using the formula:
\(\text{curved surface area} = \pi r l\)
\(l\) is the slanted height.
The total surface area is the area of the circular base and the curved surface area.
\(\text{total surface area of a cone} = \pi r^2 + \pi r l\)
In the exam Curved Surface Area = Ï°ù±ô is given on the formula sheet
Example
Calculate the volume and total surface area of the cone.
Answer:
\(\begin{array}{rcl} \text{Volume} & = & \frac{1}{3} \pi r^2 h \ & = & \frac{1}{3} \times \pi \times 3^2 \times 4 \ & = & 37.7~\text{cm}^3 \end{array}\)
\(\begin{array}{rcl} \text{Total surface area} & = & \pi r^2 + \pi r l \ & = & (\pi \times 3^2) + (\pi \times 3 \times 5) \ & = & 75.4~\text{cm}^2 \end{array}\)
Sphere â Volume and surface area
The volume of a sphere can be calculated using the formula:
\(\text{volume of a sphere} = \frac{4}{3} \pi r^3\)
The surface area of a sphere can be calculated using the formula:
\(\text{surface area of a sphere} = 4 \pi r^2\)
Example
Calculate the volume and surface area of a football with a radius of 12 cm.
\(\begin{array}{rcl} \text{Volume} & = & \frac{4}{3} \pi r^3 \ & = & \frac{4}{3} \times \pi \times 12^3 \ & = & 7,238.2~\text{cm}^3 \end{array}\)
In the exam \(\text{Volume} = \frac{4}{3} \pi r^3\) is given on the formula sheet
\(\begin{array}{rcl} \text{Surface area} & = & 4 \pi r^2 \ & = & 4 \times \pi \times 12^2 \ & = & 1,809.6~\text{cm}^2 \end{array}\)
In the exam \(\text{Surface area} = 4 \pi r^2\) is given on the formula sheet
Hemisphere â Volume and surface area
A hemisphere is half of a sphere.
The volume can be found by calculating the volume of the whole sphere and dividing by 2 or by halving the formula:
Volume of hemisphere \(= \frac{2}{3}Ïr^3\)
The surface area of a hemisphere is a curved surface which is half of the surface area of the sphere, \(2Ï°ù^2\) and a flat circular surface, \(Ïr^2\)
Surface area of hemisphere \(= 2Ï°ù^2 + Ïr^2 = 3Ïr^2\)
Example
Calculate the volume and surface area of this hemisphere with a diameter of 18 cm. Give answers to the nearest integer.
d = 18 cm ââ r = 9 cm
Volume of hemisphere = \(\frac{2}{3}Ïr^3 \)
\(V = \frac{2}{3}Ï \times 9^3 = 1527 cm^3 \) (to the nearest integer).
Surface area of hemisphere \(= 3Ïr^2\)
\(= 3Ï \times 9^2= 763~cm^2\) (to the nearest integer).
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