Answer to Question #133038 in Geometry for Joseph Se

Question

D. Give the formula for the following and briefly explain the formula concept:
1. Area of a regular polygon
2. Volume of a Prism
3. Volume of a Pyramid

Answer 1

1. Area of a regular polygon

A regular polygon is a polygon that has all its sides of same length and all its angles are equal.

The general formula for finding area of any polygon is;

$Area, A=\frac{1}{2}ap$ where a is the apothem and p is the perimeter of the given polygon

Explanation

Given the polygon below, the number of sides is five.

For us to determine area of the polygon, we create a center and draw line to each of the vertices of the polygon, in this case 5 vertices, creating five congruent triangles.

to find area of the polygon, consider one of the triangles and find its area;

Area of a triangle $A_Δ=$ $\frac{1}{2}bh$

Now, replacing base, b with side, s, and height, h with apothem, a, which is the line from the center to one of the sides, we have;

$Area, A = \frac{1}{2}sa$ this is multiplied by the total number of sides, N, i.e.

$A = \frac{1}{2}saN$ which can be written as

$A = \frac{1}{2}a(sN)$

$A = \frac{1}{2}ap$ where p is the perimeter of the polygon

Therefore area of any given polygon is given by

$A = \frac{1}{2}ap$

2. Volume of a prism

A prism is a polyhedron with two congruent, parallel polygon faces known as bases. A prism can either be triangular or rectangular.

Consider the prism below;

The volume of a 3-D object is the amount of space it occupies. From the figure above; the volume of the prism is given by

V = Bh,

where V = Volume, B= Base area, h=height.

3. Volume of a Pyramid

A pyramid is a polyhedron with one base of any polygon, and the rest of the sides are triangles.

Just like the prism, volume of a 3-D object is the amount of space it occupies.

Therefore the volume of a pyramid is given by 1/3 the base area of the pyramid by height

Volume, V = 1/3 Bh

Why the 1/3 of the base?

As from the figure above, we have square based pyramid; asume the base sides has length x and the slant height x/2, then 6 of these pyramids can be put together to form a cube.

The volume of the cube will be x³ and therefore volume of the pyramid 1/6 x³

But, 1/6 x³ = 1/3 × x² × 1/2 x

= 1/3 × (base area) × height

Thus, the Volume V of pyramid = 1/3 Bh