b The volume of the building is the area of the base multiplied by the height.
A
a 8 sides
B
b About 30 180feet^3
Practice makes perfect
a The sum of a polygon's interior angles can be calculated using the formula 180^(∘)(n-2), where n is the number of sides in the polygon. In a regular polygon the interior angles have equal measures. Therefore, we can find the measure of an individual angle by dividing the sum of the interior angles by n.
180^(∘)(n-2)/nSince each interior angle is 135^(∘), we can write the following equation.
180^(∘)(n-2)/n=135^(∘)
Let's solve this equation for n.
b The volume of the math building is the area of the building's base multiplied by its height. Let's illustrate the base of the building. To calculate its area we will draw diagonals between opposite vertices, creating 8 congruent triangles.
If we find the area of one triangle we can find the area of the base by multiplying this number by 8. Using the tangent ratio, we can calculate the height in one of the triangles.
The height of the triangle is about 30.18 feet. With this information, we can find the area of one of the triangles. If we multiply this by 8, we get the area of the base.
A=(1/2(25)(30.18))8 = 3018 feet^2
The base has an area of 3018 feet^2. To find the volume of the room, we multiply this number by the room's height.
V=3018(10)=30 180feet^3
