What's total perimeter, surface area, volume formula for prismatoid (i.e. cuboid frustum [namely trapezoidal body, rectangle frustum] and wedge)?
Please give me the full formula and the train/stream of thoughts. Both of them I need. Thanks.
google can provide all of these
as, I suspect, can your course materials.
For Wedges:
Total perimeter: I don't know
Surface/Full Area: S = ab + 1 / 2 ( a + c ) √ ( 4h ^ 2 + b ^ 2 ) + b √ ( h ^ 2 + (a-c) ^ 2
Volume: V
Sorry, due to computer issues, I press 'enter' key by a mistake, and forgot:
Wedges: V = 1 / 6 bh ( 2a + c )
Cuboid Frustums: C = 2 ( a1 + a2 + b1 + b2 ) + 4c
S = a1b1 + a2b2 + ( a1 + a2 ) l1 + ( b1 + b2 ) l2
V = 1 / 6 h ( S1 + S2 + ( a1 + a2 ) ( b1 + b2 ) )
And: S1 = a1b1, S2 = a2b2, l1 = √ ( ( ( b1 - b2 ) / 2 ) ^ 2 + h ^ 2 ) , l1 = √ ( ( ( b1 - b2 ) / 2 ) ^ 2 + h ^ 2 ) , l2 = √ ( ( ( a1 - a2 ) / 2 ) ^ 2 + h ^ 2, ) c = √ ( ( ( a1 - a2 ) / 2 ) ^ 2 + l1 ^ 2 )
The total perimeter, surface area, and volume of a prismatoid can be calculated based on its specific shape. Let's break it down into the two shapes mentioned:
1. Cuboid Frustum (Trapezoidal Body, Rectangular Frustum):
- Perimeter: To calculate the total perimeter, we need to find the sum of all the sides. Since it has a trapezoidal base and a rectangle top, we will have four sides from the base and four sides from the top. To get the perimeter, add the lengths of all these sides.
- Surface Area: The surface area of a cuboid frustum can be found by summing the areas of all the individual faces. There are three pairs of identical faces: two rectangles (top and bottom) and four trapezoids (lateral faces). Find the area of each face and sum them up.
- Volume: The volume of a cuboid frustum can be calculated by multiplying its height by the sum of the areas of the top and bottom faces, and then dividing by two. The formula is: V = (h/2) * (A1 + A2), where h is the height of the frustum, A1 is the area of the top face, and A2 is the area of the bottom face.
2. Wedge:
- Perimeter: A wedge typically has a triangular base and three rectangular faces. To calculate the total perimeter, add the lengths of all the sides of the triangular base and the three rectangular faces.
- Surface Area: To determine the total surface area of a wedge, sum the areas of all the individual faces. There are two identical triangular faces (base and top) and three identical rectangular faces (lateral faces). Calculate the area of each face and add them together.
- Volume: The volume of a wedge can be calculated by multiplying the area of the triangular base by the height of the wedge. The formula is: V = (1/2) * b * h, where b is the base length of the triangle and h is the height of the wedge.
By following these formulas and concepts, you can calculate the total perimeter, surface area, and volume of both a cuboid frustum and a wedge. Make sure to plug in the appropriate values for each variable in the formulas to obtain the accurate results.