Re: cfm4life's questions at Yahoo! Answers regarding surface are of solids
Hello cfm4life,
1.) Find the surface area of a square prism with a side measure of 15.
The surface area of a cube is compose of 6 squares, and if the side measure is $s$, then the area of one square is $s^2$ and so the surface area $S$ of the cube is:
$$S=6s^2$$
Using the given value for $s$, we then find:
$$S=6(15)^2=1350$$
Thus, c) is the answer.
2.) Find the surface area of a rectangular prism with a height of 5, a length of 6, and a width of 3.
The surface area $S$ of a rectangular prism if width $W$, length $L$ and height $H$ is composed of 3 pairs of rectangles, taken from the three ways to choose two of the three dimensions at a time. And so we have:
$$S=2(WL+WH+LH)$$
Using the given data, we find:
$$S=2(3\cdot6+3\cdot5+6\cdot5)=2(18+15+30)=2(63)=126$$
Thus, d) is the answer.
3.) Find the surface area of a cylinder with a radius of 8 and a height of 8. Use 3.14 for π. Round your answer to the nearest tenth.
The surface area $S$ of a cylinder is composed of the circular top and bottom and the rectangular side. The area of a circle can be found by taking the concentric circles that make up the area, straighten them, lay them side by side in increasing size to get a right triangle whose base is $r$ and whose height is $2\pi r$, and so the area of a circle is $$\frac{1}{2}(r)(2\pi r)=\pi r^2$$. For the rectangular side, the area of a rectangle is $bh$. The base of the rectangle is the circumference of the circle, or $2\pi 2$. Thus, we have:
$$S=2\pi r^2+2\pi rh=2\pi r(r+h)$$
Using the given data, we find:
$$S=2\pi 8(8+8)=256\pi\approx256\cdot3.14\approx803.8$$
Thus, b) is the answer.
4.) Find the surface area of a cylinder with a radius of 13 and a height of 7. Use 3.14 for π. Round your answer to the nearest tenth.
Using the formula we developed in question 3.), and the given data, we find:
$$S=2\pi 13(13+7)=520\pi\approx520\cdot3.14=1632.8$$
Thus, a) is the answer.
5.) Find the surface area of a square pyramid with a slant height of 16 and a base side measure of 14.
The surface area of a square pyramid is composed of the square bottom, and it its side measure is $s$, then its area is $s^2$, and the four triangular sides, whose bases are $s$ and whose altitudes $h$ and the slant heights. And so we have:
$$S=s^2+4\left(\frac{1}{2}sh \right)=s^2+2sh=s(s+2h)$$
Using the given data, we find:
$$S=14(14+2\cdot16)=14\cdot46=644$$
Thus, c) is the answer.
6.) Find the surface area of a square pyramid with a slant height of 9 and a base side measure of 6.
Using the formula we developed in question 5.) and the given data, we find:
$$S=6(6+2\cdot9)=6\cdot24=144$$
Thus, c) is the answer.
7.) Find the surface area of a cone with a slant height of 20 and a radius of 6. Use 3.14 for π. Round your answer to the nearest tenth.
The surface area of a cone is composed of a circular base and the lateral surface. If we decompose this lateral surface into all of the radii of the circles that ring it, and lay them side by side in increasing order, we get a right triangle whose base is equal to the slant height $\ell$ and whose height is that of the circumference of the base of the cone, or $2\pi r$. Thus the lateral surface area is $\dfrac{1}{2}(\ell)(2\pi r)=\pi r\ell$. And so we find:
$$S=\pi r^2+\pi r\ell=\pi r(r+\ell)$$
Using the given data, we find:
$$S=6\pi(6+20)=156\pi\approx156\cdot3.14\approx489.8$$
Thus, b) is the answer.
8.) Find the surface area of a cone with a slant height of 19 and a radius of 7. Use 3.14 for π. Round your answer to the nearest tenth.
Using the formula we developed for question 7.) and the given data, we find:
$$S=7\pi(7+19)=182\pi\approx182\cdot3.14\approx571.5$$
Thus, c) is the answer.
9.) Find the surface area of a sphere with a radius of 8. Use 3.14 for π. Round your answer to the nearest tenth.
If we decompose the surface $H$ of a hemisphere into the circles that ring it, and pair them smallest with largest, next smallest with next largest, and so on and lay them side by side, we find that the area they fill is a rectangle of base $r$ and height $2\pi r$, hence:
$$H=2\pi r^2$$
And so the surface area $S$ of a sphere, being twice that of a hemisphere of equal radius, is:
$$S=2H=4\pi r^2$$
Using the given data, we find:
$$S=4\pi(8)^2=256\pi\approx256\cdot3.14\approx803.8$$
Thus, c) is the answer.
10.) Find the surface area of a sphere with a radius of 3. Use 3.14 for π. Round your answer to the nearest tenth.
Using the formula we developed in question 9.) and the given data, we find:
$$S=4\pi(3)^2=36\pi\approx36\cdot3.14\approx113.0$$
Thus, b) is the answer.
