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Either paper 4 marks
Here is a rectangle.
(a)Find the perimeter of the rectangle.
(b)Find the area of the rectangle.
Show answer
(a) \(46\ \text{cm}\)
(b) \(112\ \text{cm}^2\)The two unlabelled sides are the same as the two labelled ones.
\(16 + 7 + 16 + 7 = 46\)
The perimeter is \(46\) cm.
\(\text{Area} = \text{length} \times \text{width}\)
\(16 \times 7 = 112\)
The area is \(112\) cm².
- (a) M1adding all four sides, \(16 + 7 + 16 + 7\), or \(2 \times (16 + 7)\)
- (a) A1perimeter \(46\) cm
- (b) M1\(16 \times 7\)
- (b) A1area \(112\) cm²
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Either paper 2 marks
Here is a triangle. The dashed line is the perpendicular height.
Work out the area of the triangle.
Show answer
\(100\ \text{cm}^2\)\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}\)
\(= \frac{1}{2} \times 20 \times 10 = \frac{ 200 }{2} = 100\)
The area is \(100\) cm².
- M1uses \(\frac{1}{2} \times 20 \times 10\) (or \(200\) halved)
- A1area \(100\) cm²
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Either paper 4 marks
Here is a rectangle with an area of \(12\) cm².
(a)Work out the length of the side marked \(x\).
(b)Work out the perimeter of the rectangle.
Show answer
(a) \(3\ \text{cm}\)
(b) \(14\ \text{cm}\)The area of a rectangle is length times width, so divide the area by the side you know.
\(x = 12 \div 4 = 3\)
\(x = 3\) cm
The rectangle is \(3\) cm by \(4\) cm.
\(2 \times (3 + 4) = 14\)
The perimeter is \(14\) cm.
- (a) M1\(12 \div 4\)
- (a) A1\(x = 3\) cm
- (b) M1\(2 \times (3 + 4)\), or all four sides added
- (b) A1perimeter \(14\) cm
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Either paper 4 marks
The diagram shows a right-angled triangle.
(a)Find the perimeter of the triangle.
(b)Find the area of the triangle.
Show answer
(a) \(48\ \text{cm}\)
(b) \(96\ \text{cm}^2\)The perimeter is the total distance round the outside.
\(12 + 16 + 20 = 48\)
The perimeter is \(48\) cm.
The two sides that meet at the right angle are the base and the height.
\(\frac{1}{2} \times 12 \times 16 = \frac{ 192 }{2} = 96\)
The area is \(96\) cm².
- (a) M1adding the three sides \(12 + 16 + 20\)
- (a) A1perimeter \(48\) cm
- (b) M1\(\frac{1}{2} \times 12 \times 16\)
- (b) A1area \(96\) cm²
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Either paper 2 marks
The diagram shows an isosceles triangle.
Find the perimeter of the triangle.
Show answer
\(46\ \text{cm}\)The dashes show that the two sloping sides are equal, so the unlabelled side is also \(16\) cm.
\(16 + 16 = 32\)
\(32 + 14 = 46\)
The perimeter is \(46\) cm.
- M1third side equal to the marked side, \(16\) cm, and adding all three sides
- A1perimeter \(46\) cm
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Either paper 4 marks
The diagram shows a parallelogram.
(a)Work out the area of the parallelogram.
(b)Work out the perimeter of the parallelogram.
Show answer
(a) \(204\ \text{cm}^2\)
(b) \(64\ \text{cm}\)\(\text{Area} = \text{base} \times \text{perpendicular height}\)
The perpendicular height is the dashed line, \(12\) cm. The \(15\) cm side is sloping, so it is not the height.
\(17 \times 12 = 204\)
The area is \(204\) cm².
Opposite sides of a parallelogram are equal, so the two unlabelled sides are \(17\) cm and \(15\) cm.
\(2 \times (17 + 15) = 2 \times 32 = 64\)
The perimeter is \(64\) cm.
- (a) M1\(17 \times 12\) using the perpendicular height
- (a) A1area \(204\) cm²
- (b) M1\(2 \times (17 + 15)\) or \(2 \times 32\)
- (b) A1perimeter \(64\) cm
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Either paper 3 marks
The square below has an area of \(121\) cm².
(a)Find the length of the side marked \(x\).
(b)Find the perimeter of the square.
Show answer
(a) \(11\ \text{cm}\)
(b) \(44\ \text{cm}\)The area of a square is the side length multiplied by itself, so the side is the square root of the area.
\(x = \sqrt{ 121 } = 11\)
\(x = 11\) cm
All four sides of a square are equal.
\(4 \times 11 = 44\)
The perimeter is \(44\) cm.
- (a) M1square root of the area \(\sqrt{ 121 }\)
- (a) A1side length \(11\) cm
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Either paper 4 marks
Here is a regular decagon.
(a)Calculate the perimeter of the decagon.
(b)A different regular decagon has a perimeter of \(70\) cm.
Calculate the length of one of its sides.
Show answer
(a) \(50\ \text{cm}\)
(b) \(7\ \text{cm}\)Every side of a regular polygon is the same length, and a decagon has \(10\) of them.
\(10 \times 5 = 50\)
The perimeter is \(50\) cm.
The \(70\) cm is shared equally between \(10\) sides.
\(70 \div 10 = 7\)
Each side is \(7\) cm.
- (a) M1\(10 \times 5\)
- (a) A1perimeter \(50\) cm
- (b) M1\(70 \div 10\)
- (b) A1side \(7\) cm
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Either paper 2 marks
Here is a rectangular carpet that measures 11 m by 9 m.
Part of the carpet is covered by a square rug of side 3 m.
Show that \(\frac{ 1 }{ 11 }\) of the carpet is covered by the rug.
Show answer
\(\frac{ 3^2 }{ 11 \times 9 } = \frac{ 9 }{ 99 } = \frac{ 1 }{ 11 }\)Area of the rug: \(3 \times 3 = 9\) m²
Area of the carpet: \(11 \times 9 = 99\) m²
The fraction of the carpet covered is the rug area over the carpet area:
\(\frac{ 9 }{ 99 }\)
Divide the top and bottom by 9:
\(\frac{ 9 \div 9 }{ 99 \div 9 } = \frac{ 1 }{ 11 }\)
So \(\frac{ 1 }{ 11 }\) of the carpet is covered by the rug.
- M1rug area \(9\) m² and carpet area \(99\) m²
- A1\(\frac{ 9 }{ 99 }\) divided through by \(9\) to reach \(\frac{ 1 }{ 11 }\)
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Either paper 3 marks
Callum is trying to find the area of this rectangle.
Callum writes, "The area is 26 cm² because 5 + 8 + 5 + 8 = 26".
(a)Explain what is wrong with Callum's method.
(b)Sofia works out the area of a different shape. Her answer is 110 cm.
Explain why her answer cannot be fully correct.
(c)Calculate the correct area of Callum's rectangle.
Show answer
(a) Callum added all four sides, which gives the perimeter (the distance round the edge) and not the area. The area is length × width = 5 × 8 = 40 cm², not 26 cm².
(b) cm is a unit of length, but an area is measured in square units, so the answer should be 110 cm².
(c) \(40\ \text{cm}^2\)Callum's working is 5 + 8 + 5 + 8 = 26, which uses all four sides.
The area of a rectangle is length × width, so it uses two different sides once each: \(5 \times 8 = 40\), not \(26\).
Sofia wrote cm, which measures a length, not an area.
An area is a length multiplied by a length, so its unit is cm × cm = cm².
The answer should be written as \(110\) cm².
The rectangle is \(5\) cm by \(8\) cm.
\(5 \times 8 = 40\)
The area is \(40\) cm².