Find the length of the arc of the following sectors, correct to one decimal place:
A sector of a circle of radius 27 \text{ cm} is formed from an angle of size \dfrac{2}{9} radians. Find the exact length of the arc.
Find the length of the arc of the following sectors, rounding your answers to two decimal places:
A sector of a circle of radius 4 \text{ cm} is formed from an angle of size \dfrac{5 \pi}{6} radians.
A sector of a circle of diameter 19 units is formed from an angle of size 3.5 radians.
The arc of a circle of radius 15 \text{ cm} subtends an angle of 150 \degree at the centre.
Convert 150 \degree to radians in exact form.
Hence find the exact length of the arc.
An arc AB subtends an angle of 90 \degree at the centre of a circle of radius 6 \sqrt{2} \text{ cm}.
Find the length of the chord AB.
Determine the exact difference in length between the minor arc AB and the chord AB.
A sector of a circle, radius 2 \text{ cm}, has an arc length of 3.9 \text{ cm}. Find the angle \theta formed at the centre of the circle in radians, correct to two decimal places.
The arc of a circle of radius 13 \text{ cm}, subtends an angle of \theta radians at the centre, and measures 11.7 \text{ cm} in length. Find the size of angle \theta.
A sector of a circle, radius 95 \text{ mm}, has an arc length of 159 \text{ mm}.
Find the size of angle \theta formed at the centre of the circle in radians, correct to two decimal places.
Find the size of angle \theta to the nearest degree.
Find the perimeter of the figure shown, correct to two decimal places.
Find the perimeter of the following sectors, correct to one decimal place:
Find the area of the following sectors, correct to two decimal places:
For each of the given sectors:
Find the perimeter of the sector, correct to one decimal place.
Find the area of the sector, correct to one decimal place.
Find the area of a sector of a circle with radius 5 \text{ cm}, and subtended by an angle of 4 radians at the centre.
Find the area of a sector of a circle with a radius of 24.1 \text{ m} and a central angle of \dfrac{4 \pi}{5} radians. Give your answer correct to two decimal places.
Find the area of the given sector, correct to one decimal place:
Consider the following sector:
Convert the central angle of 120 \degree to radians.
Find the exact area of the sector in terms of \pi.
Find the area of the shaded region, correct to one decimal place:
Find the area of the given figure shown, correct to one decimal place:
In the diagram, O is the centre of a circle with radius 6 \text{ cm}. Arc JK has a length measuring 2 \pi \text{ cm}.
Determine the exact area of sector OJK.
The given circle, centred at O, has radius 18\text{ cm} and arc AB has length 9\text{ cm}:
Find the area of the minor sector OAB.
The area of the sector below is 2960.92\text{ m}^2.
Find the length of the radius, correct to one decimal place.
Hence, find the perimeter of the sector, correct to one decimal place.
A formula to find the perimeter of a sector when the angle \theta is given in degrees is:
P = 2 r + \dfrac{\theta}{180} \pi rIf the perimeter of the following sector is 96.9 \text{ m}, find the size of the angle \theta, to the nearest degree.
Find the area of the sector. Round your answer to the nearest integer.
The diagram shows an arc JK of a circle, with centre O. The radius of the circle measures 15 \text{ cm} and the arc measures 12 \text{ cm} in length.
Calculate \angle JOK to the nearest degree.
Calculate the exact area of the sector.
A circle has a centre at O, and a radius of 20 \text{ m}. Sector AOC has an included angle of \theta radians and area of 160 \text{ m}^{2}.
Find the size of angle \theta, in radians.
Find the length of arc AC.
The diagram shows the sectors of two concentric circles with common centre O
where \angle{O} = \dfrac{\pi}{4} radians, OR = 8 \text{ cm}, and OQ = 12 \text{ cm}.
Calculate the exact area of the shaded region between the two circular arcs.
Calculate the exact perimeter of the shaded region.
In the diagram, OAB is a sector of a circle with radius 14\text{ cm}, where the central angle is \dfrac{\pi}{6} radians:
Find the exact value of the area of the triangle OAB.
Find the exact value of the area of the shaded segment.
The diagram shows a circle with radius 8 units, and chord AB subtending an angle of \dfrac{\pi}{3} radians at the centre.
Calculate the following:
Exact area of the minor segment cut off by chord AB.
Exact area of the major segment cut off by chord AB.
In the diagram, O is the centre of the circle, and the area of sector OAB is \dfrac{4}{9} of the area of the circle.
Find the area of the minor segment cut off by chord AB, correct to one decimal place.
The diagram shows a circle with radius 14 \text{ mm}, and chord AB subtending an angle of 2.3 radians at the centre.
Find the exact area of the minor segment cut off by chord AB.
A circle with centre O has an arc AB of length 24 \text{ cm} subtended by an angle of \theta radians at the centre. The radius of the circle measures 18 \text{ cm}.
Find the size of angle \theta, in radians.
Find the area of the minor segment cut off by chord AB, correct to one decimal place.
Consider a circle with centre O and a chord AB subtended by an angle of \theta radians at the centre. The radius is 30 \text{ cm} and the area of sector OAB is 75 \pi \text{ cm}^{2}.
Find the exact size of angle \theta, in radians.
Find the exact area of the minor segment cut off by chord AB.
Find the exact area of the major segment cut off by chord AB.
Determine the exact ratio of the area of the major segment to the area of the minor segment.
A chord AB of a circle with centre O has length measuring 6 \text{ cm}. The radius of the circle is 5 \text{ cm}.
Find the size of \angle AOB, in degrees correct to one decimal place.
Find the length of the minor arc AB.
Find the area of the minor segment formed by the chord AB, correct to one decimal place.
Consider a circle with centre O and a chord AB subtended by an angle of \dfrac{\pi}{3} radians at the centre. The exact area of the minor segment cut off by chord AB is (54 \pi - 81 \sqrt{3}) \text{ cm}^{2}.
Find the radius of the circle.
Find the exact length of arc AB.
A portable sprinker with adjustable spray is placed in one corner of a field of area 107 \text{ m}^{2} that is in the shape of an equilateral triangle.
Determine the length of the spray required so that the sprinker can water exactly one half of the area of the field. Give your answer correct to two decimal places.
A large 17 \text{ m} long sprinkler is placed in a crop field, with one end fixed and the other end free to move. As it turns, it waters everything underneath it.
If the sprinkler has turned an angle of 2.2 radians, find the area of the crop field it has watered.
Roald makes a lady beetle cake with a diameter of 24 \text{ cm} and a height of 10 \text{ cm} . Roald removes a slice of the cake with an angle measure of \dfrac{\pi}{4} radians.
Find the exact area that the black icing covers on the removed slice (both the top and side).
Find the exact area that the red icing covers on the remaining cake (both the top and side).
A sector formed from a central angle of \dfrac{10}{9} \pi radians, is cut from a circular plastic sheet of radius 63 \text{ cm}. The cut edges of the sector are brought together to form a plastic cone.
Find the exact circumference of the circular base of the funnel.
The diagram shows a piece of jewellery made out of gold:
Find the area of the piece. Round your answer to the nearest whole number.
If the gold costs \$4 per square millimetre, find the cost of the piece of jewellery.
A circular pizza of radius 12 \text{ cm} is cut into sectors where the angle at the centre of each sector is \theta, with 0 \lt \theta \lt \dfrac{\pi}{2}.
Each sector is to be placed on a circular plate of radius r \text{ cm} that is just large enough to contain that sector.
Find the radius of the plate that is needed for a slice that is \dfrac{1}{9} of the whole pizza. Round your answer to one decimal place.
The restaurant’s logo can be seen on the plate in one of the minor segments. Using the radius from part (a), find the area of the minor segment, correct to one decimal place.
A pendulum is 8\text{ cm} long and swings through an angle of 0.9 radians. The extreme points of the pendulum are indicated by points A and B in the diagram:
Find the length of the arc AB, correct to one decimal place.
Find x, the straight line distance between the points A and B, correct to one decimal place.
Find the area of the sector swept out by the pendulum, correct to one decimal place.
A circular metal plate is cut into two segments along a chord equal in length to the radius. Let \theta be the angle at the centre subtending the chord.
Find the exact value of angle \theta in radians.
Find the exact ratio of the area of the larger segment to the area of the smaller one, in fraction form.
The diagram shows three circles with radii \\r = 10 \text { cm}. The centres form the vertices of an equilateral triangle.
Find the exact area of the space formed between the three circles.
The radius, r, of the small circle is 5 \text{ cm} and the radius, R of the larger circle is 7 \text{ cm}. The distance between the centres, a + b is 10 \text{ cm}.
Use the cosine rule to find the size of angle x in radians, correct to two decimal places.
Hence, find the area of the green shaded segment A_1, correct to two decimal places.
Use the cosine rule to find the size of angle y in radians, correct to two decimal places.
Hence, find the area of the yellow shaded segment A_0, correct to two decimal places.
Use your answers from parts (b) and (d) to determine the area of the intersection of the circles, correct to two decimal places.
Two identical circles of radius length 9 \text{ cm} intersect at points A and B, and have their centres 14 \text{ cm} apart.
C is the midpoint of chord AB, and is x \text{ cm} away from each centre. Find x.
Solve for \theta, the angle (in radians) at the centre of each circle that subtends chord AB. Round your answer to two decimal places.
Hence, calculate the area of the intersecting region, correct to one decimal place.
Two circles of radius 6 \text{ cm} and 8 \text{ cm} are intersecting. Determine the area of the intersecting region, given that the distance between the centres of the circles is 12 \text{ cm}. Round your answer to two decimal places.
Two circles of radius 5 \text{ cm} and 8 \text{ cm} are intersecting. Determine the perimeter of the intersecting region, given that the distance between the centres of the circles is 11 \text{ cm}. Round your answer to two decimal places.
James wishes to find the length of the continuous belt required to enclosed the two circles as shown on the first diagram.
As shown on the second diagram, he first joins the radius of each circle to the point at which the belt attaches to each circle. This is the tangent and hence, he can form a rectangle with angles of 90 \degree.
Use Pythagoras’ theorem to determine the length of the longest side of the rectangle correct to two decimal places.
Use the cosine ratio for right angled triangles to determine the size of angle \theta. Give your answer in radians correct to two decimal places.
Determine the size of the angle subtended by the major arc in the larger circle \left( 2 \pi - 2\theta\right), correct to two decimal places.
Hence, find the length of the major arc created by the belt in the larger circle, correct to one decimal place.
Determine the size of the angle subtended by the minor arc in the smaller circle, correct to two decimal places.
Hence, find the length of the minor arc created by the belt in the smaller circle, correct to one decimal place.
Hence, determine the length of the belt that encloses the two circles, correct to one decimal place.
Determine the length of the continuous belt that encloses the two pulleys in the diagram below. Give your answer correct to the nearest metre.
Two pulleys touch each other as shown in the diagram below.
If R = 12 \text{ cm} and r = 5 \text{ cm}, determine the length of continuous belt required to enclose the two pulleys.
