Solid Mensuration

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The Regular Polygon
Rhombus is a quadrilateral with all sides equal (equilateral). Rectangle is a quadrilateral with all included angles are equal (equiangular). Square is both equilateral and equiangular, thus square is a regular polygon. Regular polygons are polygons with all sides equal and all included angles equal. Meaning, regular polygons are both equilateral and equiangular.  

Properties of regular polygons

1. The center of the circumscribing circle, the center of inscribed circle, and the center of polygon itself are coincidence. 2. All sides of regular polygon are equal in length; it is denoted by x in the figure. 3. All included angles are equal; it is denoted by β.

4. All external angles α, are equal.
5. Central angles of each segment are equal; it is denoted by θ. 6. The apothem is the radius of the inscribed circle, r.
7. The number of sides is equal to the number of vertices, both are denoted by n. 8. Diagonals of regular polygon will cross each other at the center. Length of diagonals is equal to the diameter of the circumscribing circle. 9. Segment of regular polygon is an isosceles triangle whose equal sides are radius of circumscribing circle; its area is denoted by A1.  

Formulas for a Regular Polygon
Area of one segment, A1

Total area, A

Perimeter, P


Central angle, θ

Exterior angle, α

Interior angle, β

A1 = area of one segment
A = total area
x = length of side
r = radius of the inscribed circle (apothem)
R = radius of the circumscribing circle
n = number of sides
θ = central angle
α = exterior angle
β = interior angle

Derivation of formula for volume of a frustum of pyramid/cone

Frustum of a pyramid and frustum of a cone

The formula for frustum of a pyramid or frustum of a cone is given by  

h = perpendicular distance between A1 and A2 (h is called the altitude of the frustum) A1 = area of the lower base
A2 = area of the...
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