Chapter 12.2, Problem 7PSB

a.

To determine

To calculate: The perimeter of the base ABCD .

Answer to Problem 7PSB

The perimeter of base ABCD is $72$ .

Explanation of Solution

Given information:

Side EF = 12,

Side EB = 15.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of triangle: $A=\frac{1}{2}\times b\times h$

b = base of triangle

h = height of triangle

Calculation:

  

Side FB can be calculated by applying Pythagoras Theorem.

In right angle triangle EFB , we get

  $\begin{array}{l}{\left(EF\right)}^2+{\left(FB\right)}^2={\left(EB\right)}^2\\ {\left(12\right)}^2+{\left(FB\right)}^2={\left(15\right)}^2\\ {\left(FB\right)}^2+144=225\\ {\left(FB\right)}^2=225-144\\ {\left(FB\right)}^2=81\\ FB=\sqrt{81}=9\end{array}$

Base $AB=2\times FB=2\times 9=18$

In square all sides are equal.

Perimeter $=4\times AB$

Perimeter $=4\times 18$

Perimeter $=72$

b.

To determine

To find: The lateral area of pyramid EABCD .

Answer to Problem 7PSB

The lateral area of pyramid EABCD is $432$ .

Explanation of Solution

Given information:

Side EF = 12,

Side EB = 15.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of triangle: $A=\frac{1}{2}\times b\times h$

b = base of triangle

h = height of triangle

Calculation:

  

Side FB can be calculated by applying Pythagoras Theorem.

In right angle triangle EFB , we get

  $\begin{array}{l}{\left(EF\right)}^2+{\left(FB\right)}^2={\left(EB\right)}^2\\ {\left(12\right)}^2+{\left(FB\right)}^2={\left(15\right)}^2\\ {\left(FB\right)}^2+144=225\\ {\left(FB\right)}^2=225-144\\ {\left(FB\right)}^2=81\\ FB=\sqrt{81}=9\end{array}$

Base $AB=2\times FB=2\times 9=18$

Area of triangle:

  $\begin{array}{l}A=\frac{1}{2}\times AB\times EF\\ A=\frac{1}{2}\times 18\times 12\\ A=108\end{array}$

Lateral Area = 4 $\times$ Area of triangle

Lateral Area $=4\times 108=432$

c.

To determine

To calculate: The area of base ABCD .

Answer to Problem 7PSB

The area of base ABCD is $324$ .

Explanation of Solution

Given information:

Side EF = 12,

Side EB = 15.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of square: $A=s^2$

s = side of square

Calculation:

  

Side FB can be calculated by applying Pythagoras Theorem.

In right angle triangle EFB , we get

  $\begin{array}{l}{\left(EF\right)}^2+{\left(FB\right)}^2={\left(EB\right)}^2\\ {\left(12\right)}^2+{\left(FB\right)}^2={\left(15\right)}^2\\ {\left(FB\right)}^2+144=225\\ {\left(FB\right)}^2=225-144\\ {\left(FB\right)}^2=81\\ FB=\sqrt{81}=9\end{array}$

Base $AB=2\times FB=2\times 9=18$

Area of base ABCD $={\left(18\right)}^2=324$

d.

To determine

To calculate: The total area of pyramid EABCD .

Answer to Problem 7PSB

The total area of EABCD is $756$ .

Explanation of Solution

Given information:

Side EF = 12,

Side EB = 15.

Formula used:

The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of triangle: $A=\frac{1}{2}\times b\times h$

b = base of triangle

h = height of triangle Area of square: $A=s^2$

s = side of square

Calculation:

  

Side FB can be calculated by applying Pythagoras Theorem.

In right angle triangle EFB , we get

  $\begin{array}{l}{\left(EF\right)}^2+{\left(FB\right)}^2={\left(EB\right)}^2\\ {\left(12\right)}^2+{\left(FB\right)}^2={\left(15\right)}^2\\ {\left(FB\right)}^2+144=225\\ {\left(FB\right)}^2=225-144\\ {\left(FB\right)}^2=81\\ FB=\sqrt{81}=9\end{array}$

Base $AB=2\times FB=2\times 9=18$

Area of triangle:

  $\begin{array}{l}A=\frac{1}{2}\times AB\times EF\\ A=\frac{1}{2}\times 18\times 12\\ A=108\end{array}$

Lateral Area = 4 $\times$ Area of triangle

Lateral Area $=4\times 108=432$

Area of base ABCD $={\left(18\right)}^2=324$

Total Area = Lateral Area + Area of base ABCD

Total Area $=432+324$

Total Area $=756$