# Similarity/Congruency & Ratios of length, area & volume

## Notes

### 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