Question
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  1. The figure shows a cube of side and
    .
    (a) Find the volume of the pyramid .
    (b) Find the total surface area of the pyramid .

Ask by Watson Pearson. in Hong Kong
Mar 16,2025

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Answer

(a) 金字塔的體積為約170.67立方厘米。
(b) 金字塔的總表面積為 平方厘米。

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The Deep Dive

To solve the problem, let’s start with the calculations step-by-step.
(a) To find the volume of the pyramid , we can use the formula for the volume of a pyramid, which is:
The base of the pyramid is the square . Since it’s a cube with side length cm, the area of the base is:
The height of the pyramid from the base is equal to the side of the cube, which is cm.
Now, substitute these values into the volume formula:
(b) To find the total surface area of the pyramid , we need to calculate the area of the base and the areas of the triangular faces.
The area of the base is .
Now, we will find the area of the triangular faces . Each triangular face has a base of and a height (the slant height from to the base) that we will calculate. The height from to the midpoint of any side of the square base is given by the Pythagorean theorem.
The distance from to any vertex of the base (like ) is:
Each triangle has a height from to the base of:
The area of each triangle is , so for one triangle:
To find :
The area of one triangular face is:
Now, since there are four triangular faces:
Finally, the total surface area of the pyramid is:
So, the final answers are approximately:
(a) or ,
(b) or .

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