Similarity/Congruency & Ratios of length, Area & Volume

Congruency

*3 tests required

• SSS (3 Sides) - All 3 sides are equal
• SAS (Side - Angle - Side) - 2 sides are equal, included angle equal
• AAS (Angle - Angle - Angle) - 2 angles equal, included side equal
• RHS (Right  Angle Hyp. Side) - Longest side of triangle equal, 1 angle equal

Similarity

*2 tests required

• RRR (3 Sides) -  3 sides equal proportions
• AAA (3 Angles) -  3 angles equal
• RAR (Ratio - Angle - Ratio) - 2 sides equal proportion, included angle equal

Area/Volume of similar solids

Area/Surface Area

Volume/Mass

It is important to note the units 

• of length: mm, cm, m
• of area: mm², cm², m²
• of volume: mm³, cm³, m³

• If the scale factor for lengths is x, then the scale factor for areas is x², and the scale factor for volumes is x³.
• All the length of the sides of the solid on the right are twice the lengths of corresponding sides on the left.
• If the area of the cuboid on the left is 20 units² and the volume is 50 units³
• the area of the cuboid on the right is 2² x 20 = 160 units²
• the volume of the cuboid on the right is 2³ x 50 units³ = 400 units³

• All the length of the sides on the right are 3 times the lengths of corresponding sides on the left
• If the area of the cuboid on the right is 180 units² and the volume is 540 units³
• the area of the cuboid on the left is 180/3² = 20 units²
• the volume of the cuboid on the left is 540/3³ = 20 units³