Area, Surface Area, and Volume: math test solutions, Grade 6 – download the PDF

1 Areas of basic shapes / 4 pts

1. \( \dfrac{1}{2} \times 11 \times 6 = 33 \) cm². (1 pt)
2. \( 13 \times 5 = 65 \) cm². (1 pt)
3. \( \dfrac{8 + 12}{2} \times 7 = 10 \times 7 = 70 \) cm². (1 pt)
4. \( 45 = \dfrac{1}{2} \times 15 \times h \), so \( 7.5\,h = 45 \) and \( h = 6 \) cm. (1 pt)

2 A house wall / 3 pts

Rectangle: \( 12 \times 5 = 60 \) m². (1 pt)

Triangle: \( \dfrac{1}{2} \times 12 \times 4 = 24 \) m². (1 pt)

Total: \( 60 + 24 = 84 \) m². (1 pt)

3 Parallelogram on the grid / 3 pts

(a) AB is horizontal: \( 6 - 1 = 5 \) units. (1 pt)

(b) The two horizontal sides are on \( y = 2 \) and \( y = 6 \), so the height is \( 6 - 2 = 4 \) units. (1 pt)

(c) \( A = 5 \times 4 = 20 \) square units. (1 pt)

4 A pyramid net / 3 pts

(a) A square pyramid. (1 pt)

(b) Base: \( 8 \times 8 = 64 \). Each triangle: \( \dfrac{1}{2} \times 8 \times 5 = 20 \). (1 pt) \( SA = 64 + 4 \times 20 = 144 \) in². (1 pt)

5 Surface area of prisms / 4 pts

(a) Faces: \( 9 \times 4 = 36 \), \( 9 \times 3 = 27 \), \( 4 \times 3 = 12 \). (1 pt) \( SA = 2(36 + 27 + 12) = 2 \times 75 = 150 \) in². (1 pt)

(b) Two ends: \( 2 \times \dfrac{1}{2} \times 3 \times 4 = 12 \) cm². (1 pt) Rectangles: \( (3 + 4 + 5) \times 10 = 120 \) cm². \( SA = 12 + 120 = 132 \) cm². (1 pt)

6 Fractional volume / 3 pts

(a) \( V = \dfrac{9}{2} \times 2 \times \dfrac{5}{2} = 9 \times \dfrac{5}{2} = \dfrac{45}{2} = 22.5 \) ft³. (2 pts)

(b) Each cube has volume \( \dfrac{1}{8} \) ft³, so \( 22.5 \div \dfrac{1}{8} = 22.5 \times 8 = 180 \) cubes. (1 pt)