Exercise 12 Page 859

Let's first calculate the lateral area and then the surface area.

Lateral Area

The given solid is a pyramid. We are also given that the measure of its base edge is $6\text{ mm}$ and the measure of its slant height is $9\text{ mm}.$

To calculate the lateral area of a pyramid we can use the known formula where $P$ is the perimeter of the base and $\ell$ is the slant height. We are given that the base edge of the hexagonal pyramid is $6\text{ mm}.$ Therefore, the base of the pyramid is a regular hexagon with a side length of $6\text{ mm}.$ Let's calculate its perimeter! Now we can substitute both the slant height $\ell =9$ and the perimeter $P=36$ into the formula for the lateral area and calculate it. The lateral area of the pyramid is $162.0{\text{ mm}}^2.$

Surface Area

To calculate the surface area of a pyramid we need to calculate the sum of the lateral area $L$ and the area of the base $B.$ Recall that the base of the pyramid is a hexagon with a side length of $6\text{ mm}.$ By drawing the six radii we can divide the hexagon into six isosceles triangles. Since the triangles are congruent and a full turn measures $360^{\circ },$ the central angles of the isosceles triangles formed by the radii measure $\frac{360}{6}=60^{\circ }.$

Now, let's consider just one of these isosceles triangles. We will also draw the apothem $a$ of the pentagon, which is perpendicular to the side. Note that the apothem bisects the central angle in the triangle and the side of the regular polygon.

With this information we can find $a$ using trigonometric ratios. We will write a tangent expression for the angle that measures $30^{\circ }$. Let's solve for $a$ to find the apothem. The apothem of the regular hexagon is about $4.129\text{ mm}.$ To find the area of the hexagon, we will substitute both the perimeter $P=36$ and the apothem $a=4.129$ in the formula $B=\frac{1}{2}aP.$ Let's do it! The area of the polygon is about $74.3{\text{ mm}}^2.$ Finally, let's substitute $L=162$ and $B=74.3$ into the formula for the surface area and calculate it. The surface area of the pyramid is about $236.3{\text{ mm}}^2.$