smc-1121-20-Perimeter and Area (Circular Measure)

v1 Measurement, STD2 M1 2009 HSC 11 MC

What is the area of the shaded part of this quadrant, to the nearest square centimetre?

68 m²
73 m²
95 m²
193 m²

Show Answers Only

`C`

Show Worked Solution

`text(Area)`	`=\ text(Area of Sector – Area of triangle)`
	`= (theta/360 xx pi r^2)-(1/2 xx bh)`
	`= (90/360 xx pi xx 12^2)-(1/2 xx 6 xx 6)`
	`= 113.097…-18`
	`= 95.097…\ text(m²)`

`=> C`

v1 Measurement, STD2 M1 2015 HSC 28a

The diagram shows a circular garden bed with a circular path around it.

The radius of the entire structure (garden + path) is 6 m, and the radius of the inner garden is 4 m.

Calculate the area of the path. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`≈ 62.83…\ text(m²)`

Show Worked Solution

`text(Area of path)`

`= pi(R^2 − r^2)`

`= pi(6^2 − 4^2)`

`= pi(36 − 16)`

`= 20pi\ \text(m²)`

`≈ 62.83…\ \text(m²)`

Measurement, STD1 M1 2024 HSC 18

A garden is made up of a right-angled triangle and a semicircle as shown.

What is the area of the garden, correct to the nearest square metre? (3 marks)

--- 7 WORK AREA LINES (style=lined) ---

Show Answers Only

\(35\ \text{m}^2\)

Show Worked Solution

\(\text{Base of triangle }=2\times\ \text{radius} = 6\ \text{metres}\)

\(\text{Total Area }\)	\(=\text{ Area of triangle + area of semicircle}\)
	\(=\dfrac{1}{2}\times 6\times 7 + \dfrac{1}{2}\times \pi\times 3^2\)
	\(=21+14.137\dots\)
	\(=35.137\dots\)
	\(=35\ \text{m}^2\ (\text{nearest m}^{2})\)

♦♦ Mean mark 33%.

Measurement, STD1 M1 2022 HSC 4 MC

The area `(A)` of a circle is given by the formula `A=\pi r^2`, where `r` is the radius.

What is the value of `A`, correct to three significant figures, if `r=3.55` ?

39.5
39.6
39.591
39.592

Show Answers Only

`B`

Show Worked Solution

`A`	`=pi\r^2`
	`=pi xx3.55^2`
	`=39.591…`
	`=39.6\ \ text{(3 sig fig)}`

`=>B`

♦♦ Mean mark 30%.
COMMENT: Significant figures warrant attention.

Measurement, STD1 M1 2019 HSC 25

The diagram shows a sector with an angle of 120° cut from a circle with radius 10 m.

What is the perimeter of the sector? Write your answer correct to 1 decimal place. (3 marks)

Show Answers Only

`40.9\ \ (text(1 d. p.))`

Show Worked Solution

`text(Arc length)`	`= 120/360 xx 2 xx pi xx 10`
	`= 20.94`

♦♦♦ Mean mark 16%.

`:.\ text(Perimeter)`	`= 20.94 + 2 xx 10`
	`= 40.94`
	`= 40.9\ \ (text(1 d. p.))`

Measurement, STD1 M1 2019 HSC 15

The diagram shows a shape made up of a square of side length 8 cm and a semicircle.

Find the area of the shape to the nearest square centimetre. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`89\ text(cm² (nearest cm²))`

Show Worked Solution

♦♦ Mean mark 26%.

`text(Area)`	`=\ text(Area of square + Area of semicircle)`
	`= 8 xx 8 + 1/2 xx pi xx 4^2`
	`= 89.13…`
	`= 89\ text(cm² (nearest cm²))`

Measurement, STD2 M1 2018 HSC 22 MC

A shape consisting of a quadrant and a right-angled triangle is shown.

What is the perimeter of this shape, correct to one decimal place?

28.6 cm
36.6 cm
66.3 cm
74.3 cm

Show Answers Only

`text(B)`

Show Worked Solution

`text(Using Pythagoras to find radius)\ (r):`

`r`	`= sqrt(10^2 – 6^2)`
	`= sqrt64`
	`= 8\ text(cm)`

`text(Arc length)`	`= 1/4 xx 2 pi r`
	`= 1/4 xx 2 xx pi xx 8`
	`= 12.56…\ text(cm)`

`:.\ text(Perimeter)`	`= 8 + 6 + 10 + 12.56…`
	`= 36.57…`

`=>\ text(B)`

Measurement, STD2 M1 2017 HSC 25 MC

In the circle, centre `O`, the area of the quadrant is 100 cm².

What is the arc length `l`, correct to one decimal place?

8.9 cm
11.3 cm
17.7 cm
25.1 cm

Show Answers Only

`C`

Show Worked Solution

`text(Find)\ r:`

♦ Mean mark 44%.

`text(Area)`	`= 1/4 pir^2`
`100`	`= 1/4 pir^2`
`r^2`	`= 400/pi`
`:. r`	`= 11.283…\ text(cm)`

`text(Arc length)`	`= theta/360 xx 2pir`
	`= 90/360 xx 2pi xx 11.283…`
	`= 17.724…`
	`= 17.7\ text(cm)`

`=> C`

Measurement, STD2 M1 2015 HSC 28a

The diagram shows an annulus.

Calculate the area of the annulus. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`≈ 50.26…\ text(cm²)`

Show Worked Solution

`text(Area of annulus)`

`= pi(R^2 − r^2)`

`= pi(5^2 − 3^2)`

`= pi(25 − 9)`

`= 16pi\ text(cm²)`

`≈ 50.26…\ text(cm²)`

Measurement, STD2 M1 2008 HSC 11 MC

The diagram shows the floor of a shower. The drain in the floor is a circle with a diameter of 10 cm.

What is the area of the shower floor, excluding the drain?

9686 cm²
9921 cm²
9969 cm²
10 000 cm²

Show Answers Only

`B`

Show Worked Solution

COMMENT: Students should see that answers are all in cm², and therefore use cm as the base unit for their calculations.

`text(Area)`	`=\ text(Square – Circle)`
	`= (100 xx 100)-(pi xx 5^2)`
	`= 10\ 000-78.5398…`
	`= 9921.46…\ text(cm²)`

`=> B`

Measurement, STD2 M1 2014 HSC 12 MC

A path 1.5 metres wide surrounds a circular lawn of radius 3 metres.

What is the approximate area of the path?

7.1 m²
21.2 m²
35.3 m²
56.5 m²

Show Answers Only

`C`

Show Worked Solution

`text(Area of annulus)`

`= pi (R^2-r^2)`

`= pi (4.5^2-3^2)`

`= pi (11.25)`

`=35.3\ text{m² (1 d.p.)}`

`=> C`

Measurement, STD2 M1 2009 HSC 11 MC

What is the area of the shaded part of this quadrant, to the nearest square centimetre?

34 cm²
42 cm²
50 cm²
193 cm²

Show Answers Only

`B`

Show Worked Solution

`text(Area)`	`=\ text(Area of Sector – Area of triangle)`
	`= (theta/360 xx pi r^2)-(1/2 xx bh)`
	`= (90/360 xx pi xx 8^2)-(1/2 xx 4 xx 4)`
	`= 50.2654…-8`
	`= 42.265…\ text(cm²)`

`=> B`

Measurement, STD2 M1 2012 HSC 27b

The sector shown has a radius of 13 cm and an angle of 230°.

What is the perimeter of the sector to the nearest centimetre? (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(78 cm)\ \ \ \ text((nearest cm))`

Show Worked Solution

♦ Mean mark 39%
MARKER’S COMMENT: The formula of the length of an arc is given in the formula sheet.

`text(Perimeter)`	`= 2 xx text(radius) + text(arc length)`
`text(Arc length)`	`= theta/360 xx 2 xx pi xx r`
	`= 230/360 xx 2 xx pi xx 13`
	`= 52.1853…`

`:.\ text(Perimeter)`	`= 2 xx 13 + 52.1853…`
	`= 78.1853…`
	`=text(78 cm)\ \ \ \ text((nearest cm))`