Circumference, area of circles & sectors

Here is a plan of part of Macsen’s garden. There is a circle inside a triangle. The circle has a diameter of 7 m.

Macsen will cover the shaded area with gravel. Gravel is sold in bags. Each bag of gravel covers an area of 12.5 m²

Work out the number of bags of gravel Macsen will need.

Macsen finds that each bag of gravel only covers an area of 11 m²

How does this affect your answer to part (a)?

Here is a plan of part of Macsen’s garden. There is a circle inside a triangle. The circle has a diameter of 7 m.

Macsen will cover the shaded area with gravel. Gravel is sold in bags. Each bag of gravel covers an area of 12.5 m²

Work out the number of bags of gravel Macsen will need.

Macsen finds that each bag of gravel only covers an area of 11 m²

How does this affect your answer to part (a)?

The diagram shows trapezium ABCD.

AB = 12 cm AD = 8 cm The trapezium has an area of 112 cm² Work out the perimeter of the trapezium. Give your answer correct to 3 significant figures.

The diagram shows trapezium ABCD.

AB = 12 cm AD = 8 cm The trapezium has an area of 112 cm² Work out the perimeter of the trapezium. Give your answer correct to 3 significant figures.

The diagram shows a solid shape made from a cylinder and a cone.

The cone has a base radius of 8 cm. The cylinder has a radius of 8 cm. The vertical height of the cone is three times the height of the cylinder. The volume of the solid shape is 640πcm³ Work out the vertical height of the cone. You must show all your working.

Here is a square.

The square has a perimeter of 40 cm. Five of these squares are used to make the shape below.

Work out the perimeter of this shape.

OAB is a sector of a circle with centre O and radius 4.7 m.

The sector has a perimeter of 34.3 m. Find the size of the reflex angle AOB. Give your answer correct to the nearest degree.

The diagram shows a cone.

The radius of the base of the cone is of the height of the cone. The total surface area of the cone is 54πcm² Work out the height of the cone.

The diagram shows a triangle and a rectangle.

The perimeter of the rectangle is a quarter of the perimeter of the triangle. Work out the length of the rectangle.

The diagram shows a plan of a floor.

Petra is going to cover the floor with paint. Petra has 3 tins of paint. There are 2.5 litres of paint in each tin. Petra thinks 1 litre of paint will cover 10 m² of floor.

Assuming Petra is correct, does she have enough paint to cover the floor? You must show all your working.

Actually, 1 litre of paint will cover 11 m² of floor.

Does this affect your answer to part (a)? You must give a reason for your answer.

The diagram shows a plan of a floor.

Petra is going to cover the floor with paint. Petra has 3 tins of paint. There are 2.5 litres of paint in each tin. Petra thinks 1 litre of paint will cover 10 m² of floor.

Assuming Petra is correct, does she have enough paint to cover the floor? You must show all your working.

Actually, 1 litre of paint will cover 11 m² of floor.

Does this affect your answer to part (a)? You must give a reason for your answer.

OAB is a sector of a circle with centre O and radius 6 cm.

The length of the arc AB is 5πcm. Work out, in terms of π, the area of the sector. Give your answer in its simplest form.

The diagram shows a cube and a square-based pyramid.

The volume of the cube is equal to the volume of the pyramid. Work out the perpendicular height, hcm, of the pyramid.

The diagram shows a circle, radius r cm and two regular hexagons.

Each side of the larger hexagon ABCDEF is a tangent to the circle. Each side of the smaller hexagon PQRSTU is a chord of the circle. By considering perimeters, show that

The circumference of a circle is 10 m. Work out the area of the circle. Give your answer in terms of π.

A, B, C and D are points on a circle such that ABCD is a square.

The square ABCD has sides of length 3.5 cm. Calculate the circumference of the circle. Give your answer correct to 1 decimal place. You must show all your working.

The diagram shows a shaded sector POQ of a circle with centre O and radius 6.2 cm.

The area of the shaded sector is 82.6 cm² Calculate the size of angle x. Give your answer correct to 3 significant figures.

The diagram above shows a frustum F of a cone. The frustum is made by removing a cone with height 10 cm from a solid cone with height 15 cm and base diameter 24 cm. The solid S is made by removing F from a solid hemisphere as shown in the diagram below.

The hemisphere has diameter 24 cm. Calculate the volume of solid S. Give your answer correct to 3 significant figures.

Here is a sphere.

of the surface area of this sphere is 75πcm² Find the diameter of the sphere.

The points A, B, C and D lie on a circle, centre O. ABCD is a rectangle.

AB = 8 cm BC = 10 cm Work out the circumference of the circle. Give your answer correct to 3 significant figures.

24The 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.

The diagram shows a shape on a centimetre grid.

Find the area of the shape.

Find the perimeter of the shape.

OAB is a triangle. OBC is a sector of a circle, centre O.

Calculate the area of OBC. Give your answer correct to 3 significant figures.

A solid frustum is made by removing a small cone from a large cone as shown in the diagram.

The slant height of the small cone is 6 cm. The slant height of the large cone is 10 cm. The radius of the base of the large cone is 3 cm. Calculate the total surface area of the frustum. Give your answer correct to 3 significant figures.

Here is a quadrilateral ABCD.

All the measurements are in centimetres. The perimeter of ABCD is 52 centimetres. Work out the length of DC.

The diagram shows a sector of a circle of radius 18 cm.

The length of the arc is 4πcm. Work out the value of x.

Here is a solid sphere and a solid cone.

Volume of sphere = πr³ Volume of cone = πr²h All measurements are in cm. The volume of the sphere is equal to the volume of the cone.

Find r:h Give your answer in its simplest form.

Here is a different solid sphere and a different solid cone.

Surface area of sphere = 4πr² Curved area of cone = πrl All measurements are in cm. The surface area of the sphere is equal to the total surface area of the cone.

Find r:h

The diagram shows a plan of Jason’s garden. ABCO and DEFO are rectangles. CDO is a right-angled triangle. AFO is a sector of a circle with centre O and angle AOF = 90°

Jason is going to cover his garden with grass seed. Each bag of grass seed covers 14 m² of garden. Each bag of grass seed costs £10.95 Work out how much it will cost Jason to buy all the bags of grass seed he needs.

The diagram shows a plan of Jason’s garden. ABCO and DEFO are rectangles. CDO is a right-angled triangle. AFO is a sector of a circle with centre O and angle AOF = 90°

Jason is going to cover his garden with grass seed. Each bag of grass seed covers 14 m² of garden. Each bag of grass seed costs £10.95 Work out how much it will cost Jason to buy all the bags of grass seed he needs.

A cube is placed on top of a cuboid, as shown in the diagram, to form a solid.

The cube has edges of length 4 cm. The cuboid has dimensions 7 cm by 6 cm by 5 cm. Work out the total surface area of the solid.

A right-angled triangle is formed by the diameters of three semicircular regions, A, B and C as shown in the diagram.

Show that area of region A = area of region B + area of region C

The diagram shows three circles, each of radius 4 cm. The centres of the circles are A, B and C such that ABC is a straight line and AB = BC = 4 cm.

Work out the total area of the two shaded regions. Give your answer in terms of π

A solid cone is joined to a solid hemisphere to make the solid T shown below.

The diameter of the base of the cone is 7 cm. The diameter of the hemisphere is 7 cm. The total volume of T is 120πcm³ The total height of T is ycm.

Calculate the value of y. Give your answer correct to 3 significant figures.

The diameter of the base of the cone and the diameter of the hemisphere are both increased by the same amount. Assuming the total volume of T does not change,

explain the effect this would have on your answer to part (a).

26The diagram shows a cube with edges of length xcm and a sphere of radius 3 cm.

The surface area of the cube is equal to the surface area of the sphere.

The diagram shows a cube with edges of length xcm and a sphere of radius 3 cm.

The surface area of the cube is equal to the surface area of the sphere.

Here is a shape with all its measurements in centimetres.

The area of the shape is Acm² Show that A = 2x² + 24x + 46

Here is a right-angled triangle.

The shaded shape below is made from two of these triangles.

Work out the perimeter of the shaded shape. Give your answer correct to 3 significant figures.

Here is a right‑angled triangle.

The shaded shape below is made from two of these triangles.

Work out the perimeter of the shaded shape. Give your answer correct to 3 significant figures.

This diagram shows two squares.

Work out the area of the square shown shaded in the diagram.

The diagram shows a sector OPQR of a circle, centre O and radius 8 cm.

OPR is a triangle. Work out the area of the shaded segment PQR. Give your answer correct to 3 significant figures.

The diagram shows a sector OPQR of a circle, centre O and radius 8 cm.

OPR is a triangle. Work out the area of the shaded segment PQR. Give your answer correct to 3 significant figures.

The diagram shows two shaded shapes, A and B. Shape A is formed by removing a sector of a circle with radius (3x – 1) cm from a sector of the circle with radius (5 x – 1) cm. Shape B is a circle of diameter (2 – 2 x) cm.

The area of shape A is equal to the area of shape B. Find the value of x. You must show all your working.

The diagram shows a right-angled triangle and a quarter circle.

The right-angled triangle ABC has angle ABC = 90 The quarter circle has centre C and radius CB. Work out the area of the quarter circle. Give your answer correct to 3 significant figures. You must show all your working.

The diagram shows a right-angled triangle and a quarter circle.

The right-angled triangle ABC has angle ABC = 90 The quarter circle has centre C and radius CB. Work out the area of the quarter circle. Give your answer correct to 3 significant figures. You must show all your working.

The diagram represents a solid cone.

The cone has a base diameter of 20 cm and a slant height of 25 cm. A circle is drawn around the surface of the cone at a slant height of 10 cm above the base. The curved surface of the cone above the circle is painted grey. Work out the area of the curved surface of the cone that is not painted grey. Give your answer as a multiple of π You must show all your working.

Work out the perimeter of this shape.

The diagram shows a cube and a cuboid.

The total surface area of the cube is equal to the total surface area of the cuboid.

Janet says,

“The volume of the cube is equal to the volume of the cuboid.”

Is Janet correct? You must show how you get your answer.

The diagram shows a hemisphere with diameter 8.4 cm.

Work out the volume of the hemisphere. Give your answer correct to 3 significant figures.

Here is a rectangle.

Coby has to find the perimeter of this rectangle.

He writes,

What mistake has Coby made?

Here is a triangle.

Iram solves a problem about this triangle to find the value of x.

Her answer is

Explain why Iram’s answer must be wrong.

A is in the shape of a quarter circle of radius 15 cm. B is in the shape of a circle.

The area of A is 9 times the area of B.

Show that the radius of B is 2.5 cm.

The cone has volume 56.8 cm³ and height 3.6 cm.

Calculate the size of angle AOB of sector OACB. Give your answer correct to 3 significant figures. You must show all your working.

Here are two rectangles.

QR = 10 cm BC = PQ The perimeter of ABCD is 26 cm The area of PQRS is 45 cm² Find the length of AB.

Here are two rectangles.

QR = 10 cm BC = PQ The perimeter of ABCD is 26 cm The area of PQRS is 45 cm² Find the length of AB.

The diagram shows a solid shape. The shape is a cone on top of a hemisphere.

The height of the cone is 10 cm. The base of the cone has a diameter of 6 cm. The hemisphere has a diameter of 6 cm. The total volume of the shape is cm³, where k is an integer. Work out the value of k.

OAB is a sector of a circle with centre O and radius 7 cm.

The area of the sector is 40 cm² Calculate the perimeter of the sector. Give your answer correct to 3 significant figures.

Here is a rectangle.

The 6-sided shape below is made from two of these rectangles.

Work out the perimeter of this 6-sided shape.

The diagram shows a square ABCD with sides of length 20 cm. It also shows a semicircle and an arc of a circle.

AB is the diameter of the semicircle. AC is an arc of a circle with centre B.

The diagram shows a square ABCD with sides of length 20 cm. It also shows a semicircle and an arc of a circle.

AB is the diameter of the semicircle. AC is an arc of a circle with centre B.

A square has an area of 81 cm²

Find the perimeter of the square.

The diagram shows a right-angled triangle and a parallelogram.

The area of the parallelogram is 5 times the area of the triangle. The perpendicular height of the parallelogram is h cm.

Find the value of h.

Here is a frustum of a cone.

The diagram shows that the frustum is made by removing a cone with height 3.2 cm from a solid cone with height 6.4 cm and base diameter 7.2 cm.

The frustum is joined to a solid hemisphere of diameter 7.2 cm to form the solid S shown

The density of the frustum is 2.4 g/cm³ The density of the hemisphere is 4.8 g/cm³

Calculate the average density of solid S.

Here is a shaded shape ABCD.

The shape is made from a triangle and a sector of a circle, centre O and radius 6 cm. OCD is a straight line.

AD = 14 cm Angle AOD = 140° Angle OAD = 24°

Calculate the perimeter of the shape. Give your answer correct to 3 significant figures.

Here is a solid square-based pyramid, VABCD.

The base of the pyramid is a square of side 6 cm. The height of the pyramid is 4 cm. M is the midpoint of BC and VM = 5 cm.

Draw an accurate front elevation of the pyramid from the direction of the arrow.

Work out the total surface area of the pyramid.

Here is a solid square-based pyramid, VABCD.

The base of the pyramid is a square of side 6 cm. The height of the pyramid is 4 cm. M is the midpoint of BC and VM = 5 cm.

Draw an accurate front elevation of the pyramid from the direction of the arrow.

Work out the total surface area of the pyramid.

A farmer has a field in the shape of a semicircle of diameter 50 m.

The farmer asks Jim to build a fence around the edge of the field. Jim tells him how much it will cost.

Jim takes three days to build the fence.

Work out the total cost.

The diagram shows a logo made from three circles.

Each circle has centre O. Daisy says that exactly of the logo is shaded.

Is Daisy correct? You must show all your working.

The diagram shows a logo made from three circles.

Each circle has centre O. Daisy says that exactly of the logo is shaded.

Is Daisy correct? You must show all your working.

Hasmeet walks once round a circle with diameter 80 metres.

There are 8 points equally spaced on the circumference of the circle.

Find the distance Hasmeet walks between one point and the next point.

Four of the points are moved, as shown in the diagram below.

Hasmeet walks once round the circle again.

Has the mean distance that Hasmeet walks between one point and the next point changed? You must give a reason for your answer.

Hasmeet walks once round a circle with diameter 80 metres.

There are 8 points equally spaced on the circumference of the circle.

Find the distance Hasmeet walks between one point and the next point.

Four of the points are moved, as shown in the diagram below.

Hasmeet walks once round the circle again.

Has the mean distance that Hasmeet walks between one point and the next point changed? You must give a reason for your answer.

OAC is a sector of a circle, centre O, radius 10 m.

BA is the tangent to the circle at point A. BC is the tangent to the circle at point C.

Angle AOC

Calculate the area of the shaded region. Give your answer correct to 3 significant figures.

Work out an estimate for the number of boxes of grass seed Balena needs. You must show your working.

Is your estimate for part (a) an underestimate or an overestimate? Give a reason for your answer.

ONQ is a sector of a circle with centre O and radius 11 cm.

A is the point on ON and B is the point on OQ such that AOB is an equilateral triangle of side 7 cm.

Calculate the area of the shaded region as a percentage of the area of the sector ONQ. Give your answer correct to 1 decimal place.

A square, with sides of length x cm, is inside a circle. Each vertex of the square is on the circumference of the circle.

The area of the circle is 49 cm².

Work out the value of x. Give your answer correct to 3 significant figures.

A, B and C are points on a circle of radius 5 cm, centre O. DA and DC are tangents to the circle. DO = 9 cm

Work out the length of arc ABC. Give your answer correct to 3 significant figures.