Geometry Trivia Quiz

Answer: obtuse

2^2 + 2^2 less than 3^2, so the triangle is obtuse.

Answer: 24 times the square root of 3

First, use the theorem to find the area of a regular polygon: A = 1/2 * A * P, where A = the apothem and P = the perimeter of the figure. Since it's a hexagon, it has six sides, which are all 4 (so the perimeter is 6 * 4 = 24). Put this into the equation and you now have A = 1/2 * a * 24. You can now reduce this to A = 12 * a, or A = 12a.

This is because 24 * 1/2 = 12. In hexagons, you can set up 6 equilateral triangles in its interior, with sides of 4. Divide any of these in half, and you are given a 30-60-90 triangle, whose side lengths are proportional 1:square root of 3:2, where 1 is the short leg, 2 is the hypotenuse, and the square root of 3 is the long leg. You know this is a 30-60-90 triangle because 2, the shorter leg, is half of 4, the hypotenuse.

Then you can multiply the shorter leg, 2, by the square root of 3 to get 2 times the square root of 3 for the apothem. This is because of the proportion mentioned above. Now, plug this into the equation A=12*2 times the square root of 3. Your answer, after simplifying, is 24 times the square root of 3.

Not too complicated.

Answer: 90

If the slopes of the 2 intersected lines are both 3/2, then they are parallel. The slope of the transversal is -2/3, or the opposite of the reciprocal of the other lines. This means that they are both perpendicular to the transversal, and all 8 angles are 90.

Answer: x=3, y=6

Pretty simple. Since ABCDEF is regular, the sides are all equal. Set 3x = 6 + x. Subtract x from both sides and 2x = 6. Divide both sides by 2 and x = 3. Next set x + y = 3x. Plug in x and you have 3 + y = 3 * 3, or 3 + y = 9. Subtract 3 from both sides and y = 6.

Answer: x=3

All right. Since GHI is equiangular, it is also equilateral. This means that GH = GI = HI. Set 30 + x - 3 = 12x - 6. Combine like terms and come out with 27 + x = 12x - 6. Add six to both sides and get 33 + x = 12x. Subtract x from both sides and get 33 = 11x. Divide both sides by 11 and get x = 3. Have fun?

Answer: 231

First, we'll set up a formula to find the surface area (which is what we're doing). This would be 2wh + 2lh + lw + (1/2 * P * L). Now, this may seem complicated, but let's break it down: w is the width (or depth) of the cube. h is the height of the cube, and l is the length of the cube.

The basic formula for finding the surface area of a cube is 2wh + 3lh + 2lw, but since the top of the cube will not be included, it is only lw. The P is the perimeter of the base, and the L is the slant height of the pyramid.

The standard formula for finding the surface area of a pyramid is B + 1/2 * P * L, where B stands for area of the base, but since the base is not included, we remove this from the problem. Now, we put in what we know. We know that all the sides of the cube are 6 (it was mentioned that it is square with a length of 6, with the height being 6).

This fills out about half the problem. Our formula is now 2 * 6 * 6 + (2 * 6 * 6) + (6 * 6) + (1/2 * P * L). It can be simplified to 72 + 72 + 36 + (1/2 * P * L), then to 180 + (1/2 * P * L). Now to find the perimeter of the base. Since it was said that the base is square, and a square is a quadrilateral, all 4 sides are congruent.

This is 6 * 4, or 24, since the side length is 6. Our formula is now 180 + (1/2 * 24 * L). This can be simplified to 180 + (12 * L). To find the slant height, both common sense, given information, and proportions for special triangles come into play. The given information: the height of the pyramid is 3. This will serve as a. Since the altitude drawn from the vertex should be right in the center of the square, you can divide the length of the side in half, giving you another 3. All right triangles with 2 of the same side lengths are special, and they're proportional. This proportion is 1:1:square root of 2. This is a 45-45-90 triangle. You can just use common sense and multiply 3 by the square root of 2 and get 3 times the square root of 2 as the slant height. Our formula is now nearly complete, as it stands 180 + (12 * 3 times square root of 2). This can be simplified to 180 + 36 time the square root of 2. This is 230.911..., but it is supposed to be rounded up, to 231, the correct answer.

Answer: 15.7079

The formula to find the surface area of a cone is ~ * r^2 + ~ * r * l, where ~ is pi, r is the radius, and l is the slant height. Also, the part to the left of the addition symbol is the area of the base, and the part to the right is the lateral area, which was what you were supposed to find. Since the radius is half the diameter, the radius is 1, as the diameter was given as 2.

This leaves you with ~ * 1 * l, or ~l. The slant height was given as 5, and the formula is now complete, except for the calculator math: 5~.

This comes out to 15.7079... .

Answer: 46656000

Pretty simple one. The volume of a cube is s^3, where s is the length of any side, since they are all congruent. Since you know the length of a diagonal, you can figure that the square was cut in half, leaving two right triangles. Pick either one of these (they're both congruent) and you can see that both legs are congruent, meaning that it's a 45-45-90 triangle. Since the proportion is 1:1:square root of 2, you can just divide the hypotenuse by the square root of 2 to find the length of a side.

This results in 30 as a side length. This cubed is 27000 cubic !FEET! The problem inquired that you find it in inches. That means multiply 30 by 12 and cube it, or 360^3, which comes out to 46656000, the right answer in inches.

Answer: 171.0423

The proportion to find the area of the sector of a circle:
Area of sector - - - Angle of sector
-------------- | = | ---------------
| |~r^2 | | | | | | | | 360
In this case, ~r^2 is 615.75216, and the angle of the sector is 100. Take the cross-products to get 61575.216 = 360x (x is the area of the sector). Then divide both sides by 360 to get the answer, 171.0423.

Answer: 30, 60, 90

40:40 times the square root of three:80 can be reduced to 1:square root of 3:2. This is the proportion for 30-60-90 triangles, meaning that it is a 30-60-90 triangle.