Question

practice a. find the surface area of this right rectangular prism. dimensions are in
b. find the surface area of this right triangular prism. dimensions are in feet
c. a right circular cylinder has a radius of 20 centimeters and a length of 200 cm as shown. find the surface area of the right circular cylinder.
d. a base of the right solid 10 meters high is shown. find the surface area of the ng dimensions are in meters. (remember that the lateral surface area of a right solid is to the perimeter of a base times the height.)

1. (a) what do we call the total area of all exposed surfaces of a geometric solid? (b) how do you find the lateral surface area of any right solid?

Explanation:

Step1: Recall surface - area formula for rectangular prism

The surface - area formula of a rectangular prism with length $l$, width $w$, and height $h$ is $SA = 2(lw+lh + wh)$. For the rectangular prism with $l = 4$, $w = 3$, and $h = 2$, we substitute these values: $SA=2(4\times3 + 4\times2+3\times2)$.

Step2: Calculate the products inside the parentheses

$4\times3 = 12$, $4\times2 = 8$, and $3\times2 = 6$. So, $SA = 2(12 + 8+6)$.

Step3: Add the numbers inside the parentheses

$12 + 8+6=26$. Then $SA = 2\times26=52$ square units.

Step4: Recall surface - area formula for right - triangular prism

The surface area of a right - triangular prism is $SA=2\times(\text{area of base})+\text{lateral surface area}$. The base is a right - triangle with legs $a = 8$ and $b = 6$, so the area of the base $A_{base}=\frac{1}{2}\times8\times6 = 24$ square feet. The lateral surface area: the three rectangles have areas $6\times12$, $8\times12$, and $10\times12$. The lateral surface area $A_{lateral}=(6 + 8+10)\times12=24\times12 = 288$ square feet. Then $SA = 2\times24+288=48 + 288=336$ square feet.

Step5: Recall surface - area formula for right - circular cylinder

The surface area of a right - circular cylinder is $SA = 2\pi r^{2}+2\pi rl$, where $r = 20$ cm and $l = 200$ cm. $2\pi r^{2}=2\pi\times20^{2}=2\pi\times400 = 800\pi$ square cm, and $2\pi rl=2\pi\times20\times200 = 8000\pi$ square cm. So, $SA=800\pi + 8000\pi=8800\pi\approx8800\times3.14 = 27632$ square cm.

Step6: Analyze the composite solid

The base of the solid has a perimeter $P$. The base is composed of a semi - circle and a right - triangle. The perimeter of the semi - circle is $\pi r=\pi\times3$ and the sides of the right - triangle are $8$, $10$. The perimeter of the base $P=8 + 10+3\pi$. The lateral surface area $A_{lateral}=(8 + 10+3\pi)\times10=180 + 30\pi$ square meters. The area of the semi - circle $A_{semicircle}=\frac{1}{2}\pi r^{2}=\frac{1}{2}\pi\times3^{2}=\frac{9\pi}{2}$ square meters, and the area of the right - triangle $A_{triangle}=\frac{1}{2}\times8\times3 = 12$ square meters. The total surface area $SA=2\times(\frac{9\pi}{2}+12)+180 + 30\pi=9\pi+24 + 180+30\pi=49\pi + 204\approx49\times3.14+204=153.86+204 = 357.86$ square meters.

Step7: Answer part 1(a)

The total area of all exposed surfaces of a geometric solid is called the surface area.

Step8: Answer part 1(b)

The lateral surface area of a right solid is the perimeter of a base times the height.

Answer:

a. 52 square units
b. 336 square feet
c. $8800\pi\approx27632$ square cm
d. $49\pi + 204\approx357.86$ square meters

1. (a) Surface area

(b) The perimeter of a base times the height.