Unit 12 Study Guide – Angles, Volume, Surface Area

  1. Parallel Lines Cut By a Transversal
  1. Vertical Angles – opposite angles formed by the intersection of two lines. Vertical angles are congruent.
  1. Corresponding Angles

  1. Alternate Interior and Alternate Exterior Angles

  1. Angles of Triangles
  1. Interior Angles of a Triangle
  1. Exterior Angle of a Triangle

  1. Angles of Polygons
  1. Interior Angles of a Polygon
  1. Sum of the interior angles

  1. Each interior angle – to find each interior angle of a REGULAR polygon, divide the sum by the number of sides
  1. Exterior Angles of a Polygon
  1. Sum of the exterior angles

  1. Each exterior angle - to find each exterior angle of a REGULAR polygon, divide 360 by the number of sides
  1. Volume of Prisms

In general, the formula for the Volume of a prism is:

V= Bh B = the area of the base h = height

a)Rectangular Prism:

V = lwh l = length, w = width, h = height

Example:

V = (4)(3)(2)

V = 24 m3

b)Triangular Prism:V = abh a = altitude (height), b = base, h = height

Example:

V = abh V = (6)(8)(18)

Therefore, the volume = 432 in3

  1. Volume of Composite Figures

3. Surface Area: the sum of the area of all faces of a solid shape.

a) Rectangular Prism:

Example: Find the surface area of the rectangular prisms.

All 6 faces of this rectangular prism are rectangles.

Face / Area / Multiply by 2
TOP and BOTTOM / /
FRONT and BACK / /
LEFT and RIGHT / /

Total Surface Area =

Total Surface Area = 52 m2

You can also use the formula: S.A. = 2 lw + 2 lh + 2wh

= 2(4)(3) + 2(4)(2) + 2(3)(2)

= 24 + 16 + 12

= 52m2

b)Triangular Prism

Find the area of each face and add.

  • The area of each triangle is • 4 • 3 or 6.

(REMEMBER: THERE ARE 2 TRIANGLES)

  • The area of two of the rectangles is 14 • 3.6 or 50.4.
  • The area of the third rectangle is 14 • 4 or 56.
  • The sum of the areas of the faces is

6 + 6 + 50.4 + 50.4 + 56 or 168.8 in2.

4) Composite Surface Area: To find the area of a composite figure, separate it into figures whose areas you know how to find, and then add the areas

For Example: Find the surface area of the composite figure.

The figure can be separated into a rectangular prism anda triangular prism. Find the area of the exposed surfaces.

The area of the exposed surfaces of the rectangular prism are:

Bottom: 2 ∙10= 20

Left and Right: 2(5 ∙ 2) = 20

Front and Back: 2(5 ∙ 10) = 100

The surface area of the rectangular prism (no top) is 20 + 20 + 100 = 140 m2.

The area of the exposed surfaces of the triangular prism:

Back: 2 ∙ 10 = 20

2 Triangles: 2(2 ∙ 2) = 4

Slant : 2.8 ∙ 10 = 28

The surface area of the exposed triangular prism is 20 + 4 + 28 = 52 m2.

So, the surface area of the above composite figure is 140 + 52 = 192 m2.

CYLINDER: a three-dimensional figure with two parallel congruent circular bases connected by a curved surface.

V = πr2h

CONE: a three-dimensional figure with one circular base connected by a curved surface to a single vertex

V = πr2h