Determine the volume of a prism whose bases are right triangles with legs 5cm and 12cm, and whose height is 15
460515
i think the volume must be, ((12*5)/2)*15 \n\nin which (12*5)/2 is the area of the one of the bases and 15 is the height since the volume is the base area times the height
559607
A rectangular prism is simply made up of 6 rectangles.  One pair of rectangles would be 3cm x 3cm (the ends).  The other 4 rectangles would be 3cm x 20cm.  So the total surface area would be the sum.\n\n3cm x 3cm = 9 sq. cm\n3cm x 20cm = 60 sq. cm\n\nSo 9 + 9 + 60 + 60 + 60 + 60 = 258 sq. cm.\n\nIn this case two of the dimensions were the same, but if the sides were h,w,l the total surface area would be:\n= hw + hw + hl + hl + lw + lw\n= 2hw + 2hl + 2lw\n= 2 (hw + hl + lw)\n\nIn your example:\n= 2 (3x3 + 3x20 + 3x20)\n= 2 (129)\n= 258 sq. cm.   (Don't forget the units are sq. cm)
599733
I believe it would be one-half times the length of each diagonal.\n\nIn other words, if the diagonals are 4 & 6 the area would be:\n\nA = 1/2 * 4 * 6 = 12\n\nIf you don't know the diagonals, but you know the length of the sides, you could construct the diagonals and calculate their dimensions using the Pythagorean theorem:  c^2 = a^2 + b^2, and then solve from there.  Keep in mind that you'll have to do some extra geometric analysis to find the length of each leg of the right triangles that are created by the intersection of the diagonals before you can use the theorem.
1124064
Because you have a fixed height, h, of 30m, and the Volume  of a triangular prism is given by: V = h*A where A is the area of the triangular base, your problem reduces to maximizing A, given the perimeter of A = 210m\nAn equilateral will enclose the maximum area(see link below) So the length of each side will be 210/3 = 70 m. The area of an equilateral triangle =  S^2 x sqrt(3)/4 where S = side length. So, in your case the area = 70^2*sqrt(3)/4 = 2,121.18 m^2 and the maximum volume for your prism is 30m*2121.18 m^2 = 63,652.86 m^3\n\nFor a polygon, the general rule is: the 'closer' the shape of a polygon is to a circle, the greater the area of the interior of the polygon. This implies that a equilateral triangle, a square(see below), a regular pentagon; and in general a regular n-gon will enclose the most area. \n\nConsider a square with side length X, then the area of the square = X^2 and the perimeter is 4X. Let's keep the perimeter fixed at 4X and see what happens to the area if we let one pair of opposing sides of the rectangle be X-y and the other two be X+y (Note the perimeter will still be 4X). However, the area is now: (X+y)*(X-y) = X^2 - y^2. We see that if y>0 then area decreases, while the perimeter stays the same. We see that a square encloses the maximum area for a fixed perimeter. \n\nHowever, if you allow the height, h, to vary as well then you have 3D maximization problem; but the solution is not difficult in this case either. In general, right-angle prisms with regular polygon bases will have a maximum volume for a fixed perimeter or surface area\n\nhttp://mathforum.org/library/drmath/view/53789.html\n\nhttp://mathworld.wolfram.com/Prism.html\n\nhttp://mathworld.wolfram.com/EquilateralTriangle.html
881588
First, the isosceles triangle has sides of 10, 12, 12.  You can find the angle by law of cosines or, as I prefer, just divide the triangle into two right triangles.  The short side is 5 and the hypotnuese is 12 so the angle at G is arcsin(5/12) and the total angle FGA is twice that (since we split the original triangle in two) <FGA = 2*arcsin(5/12) = 49.249 deg.\n\nIf you make a pyramid, The radius of the hexagon (10 cm), the 12 cm side and the height form a right triangle so the height is:\n\nh = sqrt(12^2 - 10^2) = sqrt(44) = 6.633\n\nFinally, for the area, we need the height of the isosceles triangle which is the tjhird side of the right triangle in the first part. This height is:\n\nc = sqrt(12^2 - 5^2) = sqrt(119) = 10.909\n\nSo the area is: A = 3*10*sqrt(119) = 327.261
273778
The lengths of a right triangle are a^2 + b^2 = c^2 (Pythagorean theorem).  For 6, 7, 8, this would mean that 36 + 49 = 64, which is false.  For 10, 15, 20 you would need 100+225 = 400, also false.\n\nThe one common ration that works is 3, 4, 5  (9+16 = 25).  Multiples also work, such as 15, 20, 25 (225 + 400 = 625).
1104008
This forms a right triangle.  You can use the pythagorean theorem to find L.\nThe hypotenuse is 3L - 2, and the legs are L and 2L + 2\n(3L-2)^2 = L^2 + (2L+2)^2\n9L^2 - 12L + 4 = L^2 + 4L^2 + 8L + 4\n9L^2 - 12L + 4 = 5L^2 + 8L + 4\nSubtract 4 from each side.\n9L^2 - 12L = 5L^2 + 8L \nMove everything to one side.\n4L^2 - 20L = 0\nFactor\n4L(L-5) = 0\nL = 0 or L = 5\nThe height can't be zero; therefore, L = 5\ntan(x) = L/(2L+2)\ntan(x) = 5/(10+2)\nx = arctan(5/12)\nx = 22.619864948\nx = 23
175713
Your question is too complicated to appreciate the math in it, which is supposed to be the real beauty in the first place. Anyway, I have tried figuring it out and come with equations. I do not even want to try and begin substitution and end up with more complications. I just thought once you saw the relation, you would throw those 10 points my way.\n\nAssuming a to be sides of the squares of the prism and b to be the length of the rectangles, we know that the sum of all dimensions is 14a+5b. Now I do not even want to get started why you brought it the story of the equilateral triangle. You can just say the sum  of the dimensions is 61. Anyway, since any side of the trianle, as it is equiangular hence equilateral, is twice 14a+5b and 122, the sum of the dimensions will be 61. So your first equation is 14a+5b=61\n\nYou know the volume of the prism is the length X width X heiht = a x a x b = (a^2)b This is equal to the surface area. The surface area is the sum of all the areas on the prism and that is a^2+4ab. (How about you calculate how that came up?)\n\nSo there is your second equation, (a^2)b = a^2+4ab So then from 14a+5b=61, express a in terms of b or the other way around and plug it into the second equation and experience brutal complications and figure out what the variables are. I think you know it is too easy to calculate each dimension and area after that.
125346
Use Heron's formula to find the area of the base: 6 times the square root of 5.  Then multiply that by the height.\n\nThe volume is 30 times the square root of 5.
663486
Diagram here:\n\nhttp://mathworld.wolfram.com/TriangularPrism.html\n\nto solve for its volume, calculate for the area of the triangle bottom and multiply it by the height of the prism.\n\nTo compute for surface area, compute for the area of the 3 rectangular faces and add the area of the 2 triangular faces.
352673
Make right triangles with the height as one leg, a radius (segment from center of base to one of its vertices) as the other leg, and lateral edge as the hypotenuse.  The radius will equal the length of the base edge as it's a side of an equilateral triangle.  So 10^2 + 15^2 will equal lateral edge squared.\n\nSlant height would be the hypotenuse of a right triangle with legs of the pyramid's height, and the apothem (distance from center of base to midpoint of a side.)  You can get the apothem from Pythagorean theorem from the right triangle formed by half a base edge, a radius, and the apothem.  Then use the apothem in another Pythagorean theorem to get slant height.\n\nLateral area = 6 times the area of any one triangle.  Each triangle's area is 1/2 times 10 times slant height.\n\nVolume = 1/3 (area of base)(height).  The area of the base is 1/2 (apothem)(perimeter).\n\nThose are the steps.  Hope you can figure them out from this typing!
981841
area of the base times the height.
508552
For any prism, the formula for the volume is the formula for the corresponding polygon times h, the height.
374445
sphere has no vertices\n\narea of prism is:\n (area of base)*2 + (perimeter of base)*height.
12258
sorry i made a mistake....this is correct...\ndivide the base by 2.  Now you have a right triangle.  the height is the same, but the base is now 3.  pygathora (spelling, whatever) theorm is A squared + B squared = C squared.  c is the side opposite the right angle, a and b are the other sides.  so 3 squared + 4 squared is 9 + 16 = 25. The square root of 25 is 5, so that is the length of the side.  the volume = area of base times height.  Area of a triangle is 1/2 base times height.  so 3 (one half of 6) times 4 = 12.  times this by height of 8 = 96 cubic units. largest rectangular side is the 6 (the longest side) times 8 = 48 units squared. hope this helped.  the lateral area is the area of the sides added together. so C times 8 = 40 (area of one side)  add the other side of equal area (the C from the second triangle we got when we split the first one) 40 + 40 = 80.  Add the third side of length 6 times 8 = 48.  80 + 48 is 128 (in units squared).  or all sides added up 5 + 5 + 6 = 16 times 8 = 128.
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The surface area:\nS = 0.5*(B1+B2)*Hs  +  A1 + A2\nWhere B1 & B2 are the perimeters of the 2 bases\n      Hs = Slant height\n      A1 & A2 are the areas of the 2 bases\n\nThe volume:\nV = (1/3)*h*(A1+A2+ sqrt(A1*A2))\nWhere h = the height of the truncated tetrahedron
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Surface area:\nCylinder(lateral):2 times pie times radius times height\ncylinder(total): 2 times pie times radiuse times height + 2 times pie radius squared\nTriangular prism(total): find the area of every side and add all together to get surface area(hint:area of a triangle=1/2bh or bh/2)\nCone(lateral):pie times radius times length\nCone(total):pie times radius times length + pie squared\npyramid(total):find area of ever side and add all together \narea of a triangle=1/2bh bottom of the pyramid is length times width