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Math Help - Volume, Surface Area, and Lateral Surface Area

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    Wink Volume, Surface Area, and Lateral Surface Area

    Find the Volume, Surface Area, and Lateral Surface Area of the pyramids and cone. If someone could include their work and how they got their answer that would be great! Thanks.

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    Quote Originally Posted by peachgal View Post
    Find the Volume, Surface Area, and Lateral Surface Area of the pyramids and cone. If someone could include their work and how they got their answer that would be great! Thanks.
    Let a_B denote the base area, a_L the lateral surface area and H the height of the solid.
    Then the volume is calculated by:

    V = \frac13 \cdot a_B \cdot H

    to #a.: a_B is a rectangle (84) and the height is 19. Thus V = 532

    to #b.: Let s denote the side of the hexagon. Then a_B = 6 \cdot \frac12 \cdot s \cdot \frac12 \sqrt{3} \cdot s = \frac32 \cdot s^2 \sqrt{3}
    Now use the general formula to calculate the volume.

    to #c.: All areas are half rectangles. So I'm sure that you can do the calculations.

    to #d.: a_B is a circle with a_B = \pi r^2. Use the general formula to calculate the volume ( 256 \pi \approx 804.2477...)

    To calculate the surface area or the lateral surface area is a little bit more tricky. If the solid is a right solid and the base area is a regular polygon or a circle then the lateral surface consists of congruent triangles or a sector of a circle. In this case the lateral surface is calculated by:

    a_L = \frac12 \cdot (perimeter\ of \ a_B) \cdot (height\ of\ triangle). This formula can be used at #b. and #d.. In both cases you must calculate the height h of the triangle first.

    to #a.: The lateral surface area consists of 4 right triangles.

    a_L=\frac12 \left( 14 \cdot 19+6 \cdot 19 + 6\cdot\sqrt{14^2+19^2} + 14\cdot\sqrt{6^2+19^2} \right)

    The complete surface area consists of a_B + a_L

    to #d. Let s denote the line segment from the tip of the cone to the edge of the base area then

    a_L = \pi r s. The complete surface area is: a = \pi r^2 + \pi r s = \pi r(r+s)

    to #c.: All areas are right triangles (that means half rectangles) and I'm sure that you can do the calculations. Post your results so we can check.
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