Chapter 12.2, Problem 2PSA

a.

To determine

To calculate: The area of each lateral face of a triangular pyramid with dimensions as 16 and 17.

Answer to Problem 2PSA

The area of each lateral face of a triangular pyramid is $120$ .

Explanation of Solution

Given information:

A triangular pyramid with dimensions as 16 and 17.

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:

  

Since it is regular, all the lateral faces are equal.

Draw altitude perpendicular to base.

The altitude AE can be calculated by applying Pythagoras Theorem.

In right angle triangle AEC , we get

  $\begin{array}{l}EC=\frac{BC}{2}=\frac{16}{2}=8\\ {\left(AE\right)}^2+{\left(EC\right)}^2={\left(AC\right)}^2\\ {\left(AE\right)}^2+{\left(8\right)}^2={\left(17\right)}^2\\ {\left(AE\right)}^2+64=289\\ {\left(AE\right)}^2=289-64\\ {\left(AE\right)}^2=225\\ AE=\sqrt{225}=15\end{array}$

The altitude AE drawn perpendicular divides the triangle face into two right triangles of $8-15-17$ family.

Lateral face is triangle.

Area of lateral face $=\frac{1}{2}\times 16\times 15$

Area of lateral face = 120

b.

To determine

To find: The base area of a triangular pyramid with dimensions as 16 and 17.

Answer to Problem 2PSA

The base area of a triangular pyramid is $110.85$ .

Explanation of Solution

Given information:

A triangular pyramid with dimensions as 16 and 17.

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 equilateral triangle: $A=\frac{s^2}{4}\sqrt{3}$

s = side of equilateral triangle.

Calculation:

  

Because the figure is a regular polygon, the triangle base is also regular, which means it is equilateral.

Area of equilateral triangle:

  $\begin{array}{l}A=\frac{s^2}{4}\sqrt{3}\\ A=\frac{{\left(16\right)}^2}{4}\sqrt{3}\\ A=64\sqrt{3}\\ A=110.85\end{array}$

c.

To determine

To calculate: The total area of a triangular pyramid with dimensions as 16 and 17.

Answer to Problem 2PSA

The total area of a triangular pyramid is $470.85$ .

Explanation of Solution

Given information:

A triangular pyramid with dimensions as 16 and 17.

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 equilateral triangle: $A=\frac{s^2}{4}\sqrt{3}$

s = side of equilateral triangle.

Total area of pyramid = Area of lateral faces + Area of equilateral triangle

Calculation:

  

Since it is regular, all the lateral faces are equal.

Draw altitude perpendicular to base.

The altitude AE can be calculated by applying Pythagoras Theorem.

In right angle triangle AEC , we get

  $\begin{array}{l}EC=\frac{BC}{2}=\frac{16}{2}=8\\ {\left(AE\right)}^2+{\left(EC\right)}^2={\left(AC\right)}^2\\ {\left(AE\right)}^2+{\left(8\right)}^2={\left(17\right)}^2\\ {\left(AE\right)}^2+64=289\\ {\left(AE\right)}^2=289-64\\ {\left(AE\right)}^2=225\\ AE=\sqrt{225}=15\end{array}$

The altitude AE drawn perpendicular divides the triangle face into two right triangles of $8-15-17$ family.

Lateral face is triangle.

Area of lateral face $=\frac{1}{2}\times 16\times 15$

Area of lateral face = 120

Because the figure is a regular polygon, the triangle base is also regular, which means it is equilateral.

Area of equilateral triangle:

  $\begin{array}{l}A=\frac{s^2}{4}\sqrt{3}\\ A=\frac{{\left(16\right)}^2}{4}\sqrt{3}\\ A=64\sqrt{3}\\ A=110.85\end{array}$

Total area of pyramid = Area of lateral faces + Area of equilateral triangle

There are three lateral faces.

Total area of pyramid = (3 $\times$ Area of each lateral face) + Area of equilateral triangle

Total area of pyramid $=\left(3\times 120\right)+110.85$

Total area of pyramid $=470.85$