Question 1: Calculate the volume of a square-based pyramid with base length 6 cm and perpendicular height 9 cm.

Answer: 324

Question 2: Determine the total surface area of a pyramid with a square base of side 8 cm and a slant height of 10 cm.

Answer: 192 + 256 = 448

Question 3: Plot the graph of y=2x for x values from 0 to 10. Use the grid to accurately draw the line.

Answer: Line passing through points (0,0) to (10,20)

Question 4: Construct a pyramid with a rectangular base measuring 4 cm by 6 cm and a height of 7 cm. Draw the net on the grid.

Answer: Net with rectangle and four triangles

Question 5: A pyramid has a base perimeter of 20 cm and a slant height of 12 cm. Its base is a regular pentagon. Calculate its lateral surface area.

Answer: Lateral area = 5 × (1/2 × side length × slant height); side length = 20/5=4cm; lateral area = 5 × (0.5×4×12)= 5×24=120cm²

Question 6: Explain why a pyramid with a square base and a height equal to its side length has a volume of 1/3 × base area × height. Use a diagram or reasoning.

Answer: Because the pyramid can be considered a scaled version of the cube divided into three equal parts in volume.

Question 7: A student calculates the volume of a pyramid as 216 cm³, base length 6 cm, and height 12 cm. Identify and correct their mistake.

Answer: Incorrect volume formula used; correct volume is (1/3)×base area×height = (1/3)×36×12=144cm³

Question 8: A pyramid with a triangular base has base sides of 5 cm, 7 cm, and 8 cm, and a perpendicular height of 6 cm. Calculate its volume.

Answer: Area of triangle using Heron's formula: s=, then area; volume= (1/3)×area×height

Question 9: A pyramid's slant height is 10 cm, and the base is a regular hexagon with a side length of 6 cm. Calculate the lateral surface area.

Answer: Lateral area = 6 × (1/2 × side length × slant height) = 6 × (0.5×6×10) = 6×30=180cm²

Question 10: Challenge: Derive the formula for the surface area of a pyramid with a regular polygon base in terms of side length, number of sides, and slant height.

Answer: Surface area = base area + lateral area; base area depends on polygon; lateral area = (perimeter × slant height)/2