Surface area of three dimensional objects

Find the total surface area of a cone with diameter 8 m and slant height of 10 m. Use $\pi = 3.14$.

Solution:

$A = \pi r(r + s)$

$d = 8$, so $r = 4$

$A = 3.14 \times 4(4 + 10)$ $= 3.14 \times 4 \times 40$ $= 175.84 \, \text{m}^2$

Therefore the total surface area is $175.84$ m$^2$

If you want to know the amount of the covering the surface of a blue band margarine can, then you are finding the surface area of a right cylinder. Total surface area of the can is the sum of the areas of the top and bottom circular surfaces plus the area of the curved surface.

Now, consider a right cylinder of radius $r$ and height $h$.

If the cylinder is opened up, the curved surface flattens out to form a rectangle. The length of the rectangle is $2\pi r$ (the circumference of the circular base) and the width is $h$ (the height of the cylinder).

Total surface area of cylinder:

$= \text{Area of curved surface} + \text{Area of two bases}$

$\text{Area of curved surface} = 2\pi rh + 2\pi r^2$

$= 2\pi r(r + h)$

$\therefore \text{The total surface area of a right cylinder is given by} \quad \boxed{A = 2\pi r(r + h)}$

Find the total surface area of a cylinder with radius of 3 m and height of 10 m. Use $\pi = 3.14$.

Solution:

$A = 2\pi r(r + h)$ $r = 3, \quad h = 10$

Substituting: $A = 2 \times 3.14 \times 3(3 + 10)$ $= 6 \times 3.14 \times 13$ $= 244.92 \, \text{m}^2$

$\therefore$ Total surface area is $244.92$ m$^2$

A right pyramid is one in which the slant edges joining the vertex to the corner of the base are equal.

A right pyramid with a square base.

$\text{Total surface area} = \text{area of lateral surfaces} + \text{area of base}$

A right rectangular pyramid is such that the rectangle is 12 cm by 8 cm and each slant edge is 12 cm. Find the total surface area of the pyramid.

Solution

By Pythagoras, a slant edge from the midpoint of the base length to the common vertex of the pyramid is $\sqrt{12^2 - 6^2} = 6\sqrt{3}$ and that from the midpoint of the base width is $\sqrt{12^2 - 4^2} = 8\sqrt{2}$.

Area of lateral surfaces = $2\left(\frac{1}{2} \times 12 \times 6\sqrt{3}\right) + 2\left(\frac{1}{2} \times 8 \times 8\sqrt{2}\right)$

$= 72\sqrt{3} + 64\sqrt{2}$ $= 124.71 + 90.51$ $= 215.22 \, \text{cm}^2$

Area of rectangular base = $l \times w$

$= 12 \times 8 \, \text{cm}^2$ $= 96 \, \text{cm}^2$

Total surface area = area of lateral surfaces + area of the base

$= (215.22 + 96) \, \text{cm}^2$ $= 311.22 \, \text{cm}^2$

$\therefore$ Area = 311.22 cm$^2$

A full brick or concrete block is an example of a right rectangular prism.

A right prism is a prism in which each of the vertical edges is perpendicular to the plane of the base.

The figure above shows a rectangular right prism in which there are 6 faces, though only three of them can be seen easily.

Surface area = total or sum of the areas of each face.

Generally for any right prism,

The height of a right prism is 4 cm and the perimeter of its base is 30 cm. Find the area of its lateral surface.

Solution:

Area of lateral surface = perimeter of base $\times$ height

$\text{Area of lateral surface} = \text{perimeter of base} \times \text{height}$ $= 30 \times 4 \, \text{cm}^2$ $= 120 \, \text{cm}^2$

Find the total surface area of a rectangular prism 12 cm by 8 cm by 6 cm high.

Solution:

\text{Lateral surface area} & = 2 \times 2(12 + 8) \, \text{cm}^2 \\ & = 240 \, \text{cm}^2 \\ \text{Area of base} & = 2(12 \times 8) \\ & = 192 \, \text{cm}^2 \\ \text{Total surface area} & = (240 + 192) \, \text{cm}^2 \\ & = 432 \, \text{cm}^2 \\ \therefore \text{Total surface area} & = 432 \, \text{cm}^2 \end{aligned}$$

The figure above shows a sphere (ball) with radius $r$.

The surface area of a sphere is four times the area of a circle with the same radius. The area of a circle is $\pi r^2$. Hence, the surface area of a sphere is equal to $4\pi r^2$.

Find the surface area of a sphere of radius 5 cm. ($\pi = 3.14$)

Solution:

Surface area of sphere $= 4\pi r^2$ $= 4 \times 3.14 \times 5 \times 5$ cm$^2$ $\therefore$ The surface area is $314$ cm$^2$.

Find the surface area of a tennis ball, given that its radius is 3.3 cm. Use $\pi = 3.14$. Express your answer to the nearest tenth.

Solution:

$A = 4\pi r^2$ $\text{So } A = 4 \times 3.14 \times (3.3)^2$ $= (12.56)(10.89)$ $= 136.7784$ $\therefore$ The surface area is $136.8$ cm$^2$.