7.7 Shape and Measure

**Fencing a field.** A farmer wants to fence a rectangular field with area 72 m². She has exactly 34 m of fencing to use as the perimeter. Find the length and width of the field. Show all your working.

**Optimal pen design.** A farmer has exactly 40 m of fencing. She wants to make a rectangular enclosure divided into **three equal pens** by two internal fences parallel to one pair of sides (as shown below): $$\underbrace{\Big[\;\big|\;\big|\;\Big]}_{\text{3 pens}}$$ Let $W$ be the width of the whole enclosure (perpendicular to the dividers) and $L$ be the length. (a) Explain why the total fencing used is $2L + 4W = 40$. (b) Express $L$ in terms of $W$. (c) Write the total area $A$ as a function of $W$ alone, and complete the table: | $W$ (m) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |----------|---|---|---|---|---|---|---|---|---| | $A$ (m²) | | | | | | | | | | (d) What value of $W$ gives the maximum area? What is that area? (e) At the maximum, what is the ratio $L : W$? Does this surprise you?

**Trapezoidal swimming pool.** A swimming pool has a trapezoidal cross-section: it is 1 m deep at the shallow end and 3 m deep at the deep end. The pool is 25 m long and 12 m wide. (a) Sketch the cross-section and label all dimensions. (b) Find the area of the trapezoidal cross-section. (c) Find the volume of the pool in m³. (d) Convert the volume to litres and find how long (in hours) it takes to fill at 1 000 litres per minute. (e) A second pool is rectangular with the same length, width, and the same volume of water. How deep is the rectangular pool?

**Tiling a floor.** A rectangular kitchen floor is 3.6 m long and 2.4 m wide. Square tiles with side length 30 cm are to be laid with no gaps. How many tiles are needed? Show how you convert units consistently.

**Shape investigation.** A shape is made from a rectangle and a right-angled triangle. The rectangle is 12 cm long and 5 cm wide. The triangle is attached to one of the shorter ends (5 cm wide), and has a perpendicular height of 8 cm. (a) Find the total area of the compound shape. (b) Find the perimeter of the compound shape. The slant side of the triangle has length 8.5 cm (given).

**Nets.** A cube has side length 4 cm. (a) Draw a sketch of a valid net for this cube (describe it in words if you cannot draw). (b) Find the total area of the net. (c) Explain why a cross-shaped net with five squares in a column and one square to the right of the second square from the top is NOT a valid net for a cube.

**Algebra + area.** A rectangle has length $(2x + 4)$ cm and width $(x + 1)$ cm. Its area is 40 cm². (a) Show that $x^2 + 3x - 18 = 0$ and solve it to find $x$. (b) Write down the dimensions of the rectangle and find its perimeter.

**Perimeter puzzle.** The perimeter of an equilateral triangle equals the perimeter of a square. The square has side length 9 cm. (a) Find the side length of the triangle. (b) Find the area of the triangle. (Use the formula: area = $\frac{\sqrt{3}}{4} \times \text{side}^2$, or split into two right-angled triangles.) (c) Which shape has the larger area?

**Thinking about 3D shapes.** A toy factory uses cuboid boxes with dimensions 6 cm × 4 cm × 3 cm. (a) Find the volume and surface area of one box. (b) The boxes are packed into a larger cuboid crate. The crate is 24 cm × 20 cm × 12 cm. How many boxes fit in the crate? (c) What fraction of the crate's volume is taken up by the boxes? Simplify your answer.

**Area reasoning.** Two shapes have the same area. - Shape A is a triangle with base 16 cm and height $h$ cm. - Shape B is a trapezium with parallel sides 5 cm and 11 cm, and perpendicular height 8 cm. Find $h$.

**Staircase border investigation.** A "staircase" pattern is built from unit squares (each 1 cm × 1 cm). The $n$-step staircase has $n$ columns: column 1 has 1 square, column 2 has 2 squares, …, column $n$ has $n$ squares (like a rising staircase from left to right). A border of width 1 cm is painted around the outside of each staircase. (a) Draw (or describe) the 1-step, 2-step, and 3-step staircases. (b) For each of $n = 1, 2, 3$: - Count the number of unit squares in the staircase. - Count the perimeter of the staircase (in cm). - Calculate the area of the 1 cm border painted around it. (c) Complete the table: | $n$ | Squares in staircase | Perimeter (cm) | Border area (cm²) | |-----|----------------------|----------------|-------------------| | 1 | | | | | 2 | | | | | 3 | | | | | 4 | | | | (d) Find a formula for the number of unit squares in the $n$-step staircase. (e) Find a formula for the border area around the $n$-step staircase. (f) The border area of one staircase equals the number of squares in another staircase. Which two values of $n$ satisfy this? (There may be more than one answer.)

**Open-ended investigation.** A rectangle has a fixed perimeter of 24 cm. Complete the table of possible integer dimensions, calculate each area, and identify which dimensions give the maximum area. | Length (cm) | Width (cm) | Area (cm²) | |-------------|-----------|------------| | 11 | 1 | | | 10 | 2 | | | 9 | 3 | | | 8 | 4 | | | 7 | 5 | | | 6 | 6 | | What do you notice? What happens if the rectangle becomes a square?