Question

4. The area of the base of the regular quadratic pyramid SABCD is 25 cm2 and the area of the side wall SAB is 15 cm2. Calculate: 1) the length of the base edge; 2) the length of the diagonal of the base; 3) the length of the apothem; 4) the length of the side edge;

Answer 1

Answer:

1) 5 cm

2) 7.07 cm

3) 6 cm

4) 6.5 cm

Step-by-step explanation:

Part 1:

Since we're looking at a square pyramid, we'll have that the area of the base is the area of a square:

$A_b=L^2$

Since we know this area is 25 square centimiters, we can find L as following:

$\begin{gathered} 25=L^2 \\ \rightarrow L=\sqrt{25} \\ \\ \Rightarrow L=5 \end{gathered}$

Therefore, we can conlcude that the length of the base edge is 5 cm

Part 2:

The lenght of the diagonal of a square is given by the formula:

$D=\sqrt{2}\text{ }L$

Where L is the lenght of the sides of the square. Since we've already calculated this lenght, we can find the lenght of the diagonal as following:

$\begin{gathered} D=(\sqrt{2})(5) \\ \\ \Rightarrow D=7.07 \end{gathered}$

This way, we can conclude that the length of the diagonal of the base is 7.07 cm

Part 3:

Let's take a look at a drawing of side wall SAB:

Remember that the formula used to calculate the area of a triangle is:

$A_t=\frac{bh}{2}$

Where:

• b, is the base of the trianlge

,

• h, is the height of the triangle. In this case, the apothem ,(a)

Since we already know this area, we can find a as following:

$\begin{gathered} 15=\frac{5\times a}{2}\rightarrow30=5a\rightarrow\frac{30}{5}=a \\ \\ \Rightarrow a=6 \end{gathered}$

This way, we can conlcude that the length of the apothem is 6 cm

Part 4:

Now we know the apothem, let's take another look at side wall SAB:

We can extract from here the following right triangle:

Using the pythagorean theorem, we'll have that :

$l^2=2.5^2+6^2$

Solving for l,

$\begin{gathered} l=\sqrt{2.5^2+6^2} \\ \\ \Rightarrow l=6.5 \end{gathered}$

Therefore, we can conlcude that the length of the side edge is 6.5 cm