The surface area of a rectangular solid is just the sum of the areas of all 6 faces. The formula is
where , and are the length, width and height of the rectangular solid, respectively.
In particular, the surface area of a cube is where is the length of a side of the cube.
Try to answer the following question about the surface area of a cube.
Do not check the solution until you have attempted this question yourself.
L EVEL 4: G EOMETRY
4. In the figure above, segment ̅̅̅̅ joins two vertices of the cube.
If the length of ̅̅̅̅ is √ , what is the surface area of the cube?
Solution
* The area of one of the faces is s2 = 9 (see below for several methods of computing this). Thus, the surface area is A = 6s2 = 6·9 = 54.
157
Methods for computing the area of a face:
(1) Since all sides of a square have equal length, an isosceles right triangle is formed. An isosceles right triangle is the same as a 45, 45, 90 triangle. So we can get the length of a side of the triangle just by looking at the formula for a 45, 45, 90 right triangle. Here s is 3. The area of the square is then (3)(3) = 9.
(2) If we let s be the length of a side of the square, then by the Pythagorean Theorem
s2 + s2 = (3√ )2 2s2 = 18
s2 = 9
(3) The area of a square is A = where d is the length of the diagonal of the square. Therefore in this problem
A = = ( √ ) = = 9.
You’re doing great! Let’s just practice a bit more. Try to solve each of the following problems. The answers to these problems, followed by full solutions are at the end of this lesson. Do not look at the answers until you have attempted these problems yourself. Please remember to mark off any problems you get wrong.
L EVEL 3: G EOMETRY
5. How many spherical snowballs with a radius of 4 centimeters can be made with the amount of snow in a spherical snowball of radius 8 centimeters? (the volume Vof a sphere with radius
ris given by 3 3 4r .)
158
6. How many figures of the size and shape above are needed to completely cover a rectangle measuring 80 inches by 30 inches?
L EVEL 4: G EOMETRY
7. Cube has surface area . The edges of cube are 4 times as long as the edges of cube . What is the surface area of cube in terms of ?
(A) (B) (C) (D) (E)
8. If a 2-centimeter cube were cut in half in all three directions, then in square centimeters, the total surface area of the separated smaller cubes would be how much greater than the surface area of the original 2-centimeter cube?
L EVEL 5: G EOMETRY
9. For any cube, if the volume is cubic centimeters and the surface area is square centimeters, then is directly proportional to for
159
10. ̅̅̅̅, ̅̅̅̅, and ̅̅̅̅ are diameters of the three circles shown above. If and , what is the area of the shaded region?
(A) (B) (C) (D) (E)
11. How many solid wood cubes, each with a total surface area of 294 square centimeters, can be cut from a solid wood cube with a total surface area of 2,646 square centimeters if no wood is lost in the cutting?
(A) 3 (B) 9 (C) 27 (D) 81 (E) 243
12. In the figure above, , , and . What is the length of line segment ̅̅̅̅?
160
Answers
1. 45 5. 8 9. 2/3, .666 or .667 2. A 6. 800 10. B
3. 75 7. D 11. C 4. 54. 8. 24 12. 6
Full Solutions
5.
* Solution using strategy 24: We divide the volumes.
= = = 8.
6.
* Solution using strategy 24: The area of the given figure is 3 inches2 and the area of the rectangle is 80·30 =2400 inches2. We can see how many of the given figures cover the rectangle by dividing the two areas.
= 800.
Note: We can get the area of the given figure by splitting it into 3 squares each with area 1 inch2 as shown below. Then 1 + 1 + 1 = 3
The area of the big square is 2·2 = 4 inches2, and the area of the little square is 1 1 = 1 inch2. So the area of the given figure is 4 – 1 = 3 inches2.
7.
Solution using strategy 4: Let’s choose a value for the length of an edge of cube X, say s = 1. Then the surface area of X is A = 6s2 = 6(1)2 = 6. The length of an edge of cube Y is 4(1) = 4, and so the surface area of Y is 6(4)2 = 6·16 = 96. Now we plug in A = 6 into each answer choice and eliminate any choice that does not come out to 96.
161
Since (A), (B), (C), and (E) all came out incorrect we can eliminate them.
Therefore the answer is choice (D).
* Algebraic solution: Let s be the length of an edge of cube X. Then we have A = 6s2. Since an edge of cube Y is 4 times the length of an edge of
Solution using strategy 4: Let’s choose a value for the length of a cube, say s = 1, Then S = 6(1)2 = 6 and V = 13 = 1. Now let’s try s = 2. Then we
* Algebraic solution: If s is the length of a side of the cube, then we have S = 6s2 and V = s3. Solving the second equation for s, we have s = V1/3. Therefore S = 6(V1/3)2 = 6V2/3. So S is directly proportional to V2/3 (with constant of proportionality 6). So n = 2/3.
10.
* Solution using strategy 23: We first find the radius of each of the three circles. The diameter of the small circle is 4, and so its radius is 2. The diameter of the medium-sized circle is 5 4 = 20, and so its radius is 10.
The diameter of the largest circle is 20 + 4 = 24, and so its radius is 12.
We can now find the area of the shaded region as follows.
162
A = (Area of big circle) – (Area of medium circle) + (Area of small circle)
= (π·122) – (π·102) + (π·22)
= (π·144) – (π·100) + (π·4)
= ·48π
= 24π Thus, the answer is choice (B).
11.
* Solution using strategy 24: We first find the length of a side of each cube.
6s2 = 294 and 6s2 = 2646 s2 = 49 s2 = 441 s = 7 s = 21
Thus the volume of each cube is 73 = 343 and 213 = 9261, respectively.
We can see how many smaller cubes can be cut from the larger cube by dividing the two volumes: = 27, choice (C).
12.
* Solution using strategy 22: The problem becomes much simpler if we
“move” ̅̅̅̅ to the left and ̅̅̅̅ to the bottom as shown below.
We now have a single right triangle and we can either use the Pythagorean Theorem, or better yet notice that 26 = (13)(2) and 24 = (12)(2). Thus the other leg of the triangle is (5)(2) = 10. So we see that ̅̅̅̅ must have length 10 – 4 = 6.
Remark: If we didn’t notice that this was a multiple of a 5-12-13 triangle, then we would use the Pythagorean Theorem as follows.
(x + 4)2 + 242 = 262 (x + 4)2 + 576 = 676
(x + 4)2 = 100 x + 4 = 10
x = 6