Geometry/Chapter 8

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Perimeter[edit]

The perimeter of a particular shape is the total length of its sides.

  • For a triangle:

P=l_a+l_b+l_c The perimeter is equal to the length of side a, l_a, plus the length of side b, l_b, plus the length of side c, l_c.

  • For a square:

P=4 l The perimeter is equal to 4 times the length (l) of a side.

  • For a rectangle:

P=2 (b+h) The perimeter is equal to 2 times the sum of the base plus the height.

  • For regular polygons

P=n l The perimeter is equal to the number of sides (n) times the length (l) of a side.

Circles do not have sides made of line segments like polygons do but they do have a perimeter known as a circumference. C=2*\pi*r The circumference is equal to 2 times pi times the radius (r).

Area[edit]

Area of a shape is how much space is inside the perimeter.

  • For a triangle:

A={b h \over 2} The area is equal to the product of the base (b) times the height (h) divided by 2.

  • For a square:

A=l^2 The area is equal to the length (l) of a side squared.

  • For a rectangle:

A=b h The area is equal to the length of the base (b) times the base of the height (h).

  • For a circle:

A=\pi r^2 The area is equal to pi times the radius (r) squared.

  • For polygons with irregular shapes a sum of smaller areas can be used. The smaller area must completely compose the polygon. Useful smaller areas can be squares, triangles, or rectangles.
  • There is another method to calculate the area of a polygon located in an 2D coordinate system:

A={1 \over 2} |x_1 y_2 - x_2 y_1 + x_2 y_3 - x_3 y_2 + x_3 y_4 - x_4 y_3 + ... + x_n y_1 - x_1 y_n| where (x_i, y_i) is the ith vertex of the polygon, they have to be given in correct order, clockwise and counter clockwise is both ok. The polygon NEED NOT to be convex.

Volume[edit]

Volume is the amount of space an object occupies. Only shapes with 3 dimensions have a volume. This is because a 2 dimensional object has no thickness, and, therefore, takes-up no space.

  • For a cube:

V=l^3 The volume is equal to the length of a side (l) cubed.

  • For a rectangular prism

V=b w h The volume is equal to the base (b) times the width (w) times the height (h).

  • For a sphere

V={4 \over 3}\pi r^3 The volume is equal to four-thirds pi times the radius cubed.

  • For a cone or pyramid

V={1 \over 3}Bh The volume is one-third the area of the base times the height.

  • For a cylinder with a base of any shape (as long as the cross sectional area is constant),

 V = A_{base} * h where h is the height (not slant height) of the cylinder and  A_{base} is the area of the base. For example, the volume of a circular cylinder is  \pi r^2 h

Surface Area[edit]

For most shapes you can find the surface area by adding up the area of all its sides. For example,

  • (closed) Box with dimensions w, l, and h:  SA = 2lw + 2lh + 2wh
  • Closed cube:  SA = 6s^2
  • Closed Cylinder with base area A and base perimeter P:  SA = P*h + 2A
    For a circular cylinder,  SA = 2\pi r h + 2 \pi r^2

Spheres are special because they have no sides but using calculus it's possible to show that:

  • Sphere:  SA = 4\pi r^2

Exercises[edit]

Links[edit]

Geometry

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Chapter 7 · Chapter 9

Chapter 7 · Geometry · Chapter 9