How to find the surface area and volume of a trapezoidal prism?
The Correct Answer and Explanation is:
To find the surface area and volume of a trapezoidal prism, you need to break the shape into its components and apply appropriate formulas.
1. Volume of a Trapezoidal Prism
The volume VVV of a trapezoidal prism is calculated by multiplying the area of the trapezoidal base by the height (or length) of the prism.
The formula is: V=Ab×hV = A_b \times hV=Ab×h
Where:
- AbA_bAb is the area of the trapezoidal base.
- hhh is the height (length) of the prism.
To find the area of the trapezoidal base (AbA_bAb):
The formula for the area of a trapezoid is: Ab=12×(b1+b2)×hbA_b = \frac{1}{2} \times (b_1 + b_2) \times h_bAb=21×(b1+b2)×hb
Where:
- b1b_1b1 is the length of the top base.
- b2b_2b2 is the length of the bottom base.
- hbh_bhb is the height of the trapezoid (the perpendicular distance between the two bases).
Now, to find the volume, multiply this area by the length (or height) of the prism. V=(12×(b1+b2)×hb)×hV = \left( \frac{1}{2} \times (b_1 + b_2) \times h_b \right) \times hV=(21×(b1+b2)×hb)×h
2. Surface Area of a Trapezoidal Prism
The surface area of a trapezoidal prism is the sum of the areas of the two trapezoidal bases and the areas of the four lateral faces (rectangular sides).
To find the surface area:
- Area of the two trapezoidal bases:
Each trapezoidal base has an area AbA_bAb, which we found earlier. Since there are two bases, the total area for the two bases is: 2×Ab=2×(12×(b1+b2)×hb)2 \times A_b = 2 \times \left( \frac{1}{2} \times (b_1 + b_2) \times h_b \right)2×Ab=2×(21×(b1+b2)×hb)
- Area of the lateral faces:
The four lateral faces consist of rectangles. Two of them are the side faces, and two are the front and back faces. For each rectangular face:
- Two side faces: Each side face has an area of side height×h\text{side height} \times hside height×h, where “side height” is the slant height of the trapezoid (not the height between the two bases).
- Two front and back faces: These have an area of b1×hb_1 \times hb1×h and b2×hb_2 \times hb2×h, where b1b_1b1 and b2b_2b2 are the lengths of the top and bottom bases.
Total surface area:
The total surface area is the sum of the areas of the two trapezoidal bases and the lateral faces: A=2×Ab+Area of side faces+Area of front and back facesA = 2 \times A_b + \text{Area of side faces} + \text{Area of front and back faces}A=2×Ab+Area of side faces+Area of front and back faces
This will give you the complete surface area of the trapezoidal prism.
To summarize:
- Volume: Multiply the area of the trapezoidal base by the height (length) of the prism.
- Surface Area: Sum the areas of the two trapezoidal bases and the four lateral rectangular faces.
