Geometry Formulas

These formulas have been taken directly from a ninth grade geometry class...

Geometry Formulas...

Key:

B = Surface area of the base
A = Area
C = Circumference
P = Perimeter
L = Lateral Faces
S = Surface area
V = Volume
d = diameter
l = Length
l = Slant Height
h = Height
w = Width
r = Radius
π ≈ 3.14
a = Apothem


Changing Attributes:
When one attribute is multiplied by 2, the volume is also multiplied by 2.

When two attributes are multiplied by 2, the volume is multiplied by 2².

When all three attributes are multiplied by 2, the volume is multiplied by 2³ and the surface area is multiplied by 2².

Cavalieri's Principle: tells us that if two solids have equal heights and the cross sections formed by every plane parallel to the bases of both solids have equal areas, then the two solids have equal volume.

Pythagorean Theorem = a² + b² = c²
Perimeter of a Trapezoid = a + b₁ + b₂ + c

Area of a Trapezoid = ½h( b₁ + b₂ ) Where b₁ and b₂ are the lengths of the bases

Circumference of a Circle: C= πd or C= π2r

Area of a Circle: A= πr²

Note: These formulas may look confusing, but remember this little trick: Find ALL the measurements you will need FIRST. Then, use those numbers to plug into the formula and solve. For example, let's say you need to find the surface area of a pyramid (½lP + B). By looking at the formula, you may confused and not know where to start, but take another look at the formula and see what you NEED to solve it: you need a slant height (if this measurement is not given, you may need to use the Pythagorean Theorem to find it), the perimeter, and the surface area of the base. Once you find each of those numbers, then just plug them into the formula.If you do this EVERY time, you will find these formulas much easier to handle.




Prisms  
Surface area of a Right Prism: S= 2lw + 2lh + 2wh
*Note: Remember, prisms are named for their bases



Modified Formula for Finding the Surface Area of a Right Prism

Surface Area of a Right Prism: S= hP + 2B  where B=lw

Volume of a Right Prism: V= Bh

Surface Area of a BASE (B) of a Triangular Prism: B= ½bh,
then plug the B into the S=  hp + 2B formula

Volume of a Triangular Prism: Use the formula above to find the B, then plug into the V= Bh formula

Surface Area of the Hexagonal BASE (B): B=½aP




Cubes

Surface area of a Cube: 6x²

Volume of a Cube: V= x³



Pyramids
Surface area of a regular pyramid:  ½lP + B

Surface area of the BASE (B) of Square Pyramid: base measure²

Surface area of the BASE (B) of Triangular Pyramid: B = ½bh

Surface area of the BASE (B) of a Hexagonal Pyramid: B=½aP
Once you find the B, plug it into the regular ½lP + B formula

Finding slant height when height is given: use Pythagorean theorem...
A² + B² = C²

Finding Height when slant height is given: use Pythagorean theorem...
A² + B² = C²

The volume of a pyramid: V= ^1/3Bh.




Cylinders

Surface Area of a Cylinder: S = L + 2B

L= 2πrh or L= πdh

B= πr²

so, the S= L + 2B  becomes...

S= 2πrh + 2πr²

Volume of a Cylinder


V= Bh
By replacing the B with πr², you get

V= πr²h




Cones

Surface Area of a Cone S= L + B

L= πrl

B= πr²

So, the S= L + B becomes...

S= πrl+ πr²


Volume of a Cone V= ^1/3Bh

B=πr²

So, the V= ^1/3Bh becomes...

V= ^1/3πr²h

images courtesy of...
onlinemathlearning.com
hotmath.com
mathwarehouse.com
math.about.com