Higher June 2023 Paper 2 Q24
24 The diagram shows 8 identical regular octagons joined to enclose a shaded shape.
Each octagon has sides of length \(a\).
Find, in terms of \(a\), an expression for the area of the shaded shape.
Give your answer in the form \(p\left(2 + \sqrt{2}\right)a^2\) where \(p\) is an integer.
You must show all your working. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(4\left(2 + \sqrt{2}\right)a^2\) | P1 | for process to find area required and identifying \(135^\circ\) or \(45^\circ\) or \(90^\circ\) angle(s), eg splitting shape into square and 4 triangles and an angle relevant to the triangle clearly identified. |
| P1 | for process to find the area of a relevant triangle using \(45^\circ\), eg \(\dfrac{1}{2} \times a \times \left(2 \times \dfrac{a}{\sqrt{2}}\right) \times \dfrac{1}{\sqrt{2}}\ \left(= \dfrac{a^2}{2}\right)\) or using \(90^\circ\), eg \(\dfrac{1}{2} \times a \times a\ \left(= \dfrac{a^2}{2}\right)\) or process to find the area of a square made from 2 small triangles, eg \(a \times a\ (= a^2)\) | |
| P1 | for process to find the length of the square, eg \(a + a + \sqrt{a^2 + a^2}\ \left(= 2a + a\sqrt{2}\right)\) | |
| P1 | for process to find the total area, eg \(\left(\text{``}2a + a\sqrt{2}\text{''}\right)^2 + 4 \times \text{``}\dfrac{a^2}{2}\text{''}\) | |
| A1 | (dep on P3) for \(4\left(2 + \sqrt{2}\right)a^2\) |
Additional guidance
\(90^\circ\) must be in a triangle to gain credit.
May be seen on diagram.
Accept \(0.49\ldots a^2\)
May be seen as the area of 2 squares (from 4 small triangles)
Accept \(3.41a\)
Accept \((11.655 + 4 \times 0.49)a^2\)
Answer only award no marks.
If working in decimals accept \(\dfrac{13.656}{2 + \sqrt{2}}\) leading to 4
Accept \(p = 4\) if supported by correct working