Geometry Notes: Circles, Polygons, and Volume Formulas

Geometry Notes: Circles, Polygons, and Volume Formulas

Circles in Geometry

1. Equation of a Circle:

• Standard Form:

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2(x−h)2+(y−k)2=r2

Where:

• (h,k)(h, k)(h,k) is the center of the circle

• rrr is the radius

• General Form:

x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0x2+y2+Dx+Ey+F=0

Where:

• D,E,FD, E, FD,E,F are constants.

2. Key Characteristics:

• Radius: Distance from the center to any point on the circle.

• Diameter: Double the radius (2r2r2r).

Polygons in Geometry

1. Types of Polygons:

• Regular Polygon: All sides and angles are equal.

• Irregular Polygon: Sides and angles are not equal.

2. Sum of Interior Angles of a Polygon:

Sum of Interior Angles=(n−2)×180∘\text{Sum of Interior Angles} = (n - 2) \times 180^\circSum of Interior Angles=(n−2)×180∘

Where nnn is the number of sides of the polygon.

3. Sum of Exterior Angles of a Polygon:

• Always equals 360∘360^\circ360∘ for any polygon.

4. Area of Polygons:

• Triangle:

A=12×Base×HeightA = \frac{1}{2} \times \text{Base} \times \text{Height}A=21​×Base×Height

• Rectangle/Square:

A=Length×WidthA = \text{Length} \times \text{Width}A=Length×Width

• Regular Polygon (Area for a polygon with nnn sides and side length sss):

A=n×s24×tan⁡(πn)A = \frac{n \times s^2}{4 \times \tan \left( \frac{\pi}{n} \right)}A=4×tan(nπ​)n×s2​

Surface Area and Volume Formulas

Surface Area & Volume of a Cube:

1. Surface Area:

SA=6a2SA = 6a^2SA=6a2

Where aaa is the length of a side.

2. Volume:

V=a3V = a^3V=a3

Surface Area & Volume of a Rectangular Prism:

1. Surface Area:

SA=2lw+2lh+2whSA = 2lw + 2lh + 2whSA=2lw+2lh+2wh

Where lll, www, and hhh are the length, width, and height of the prism.

2. Volume:

V=l×w×hV = l \times w \times hV=l×w×h

Surface Area & Volume of a Sphere:

1. Surface Area:

SA=4πr2SA = 4\pi r^2SA=4πr2

Where rrr is the radius of the sphere.

2. Volume:

V=43πr3V = \frac{4}{3} \pi r^3V=34​πr3

Surface Area & Volume of a Pyramid:

1. Surface Area:

SA=B+12PlSA = B + \frac{1}{2} P lSA=B+21​Pl

Where BBB is the base area, PPP is the perimeter of the base, and lll is the slant height.

2. Volume:

V=13BhV = \frac{1}{3} B hV=31​Bh

Where BBB is the base area and hhh is the height of the pyramid.

Surface Area & Volume of a Right Circular Cone:

1. Surface Area:

SA=πr2+πrlSA = \pi r^2 + \pi r lSA=πr2+πrl

Where rrr is the radius and lll is the slant height.

2. Volume:

V=13πr2hV = \frac{1}{3} \pi r^2 hV=31​πr2h

Where rrr is the radius and hhh is the height.

Surface Area of a Cylinder:

1. Surface Area:

SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi r hSA=2πr2+2πrh

Where rrr is the radius and hhh is the height.

Volume of a Prism:

1. Volume:

V=BhV = B hV=Bh

Where BBB is the area of the base and hhh is the height.

Quick Reference of Key Concepts:

• Radius (rrr): Distance from the center to the edge of a circle or sphere.

• Height (hhh): The perpendicular distance between the bases or the top and bottom of a solid figure.

• Slant Height (lll): The diagonal distance from the apex to the edge of a base in a cone or pyramid.

• Base Area (BBB): The area of the base of a solid.