Circumference, area of circles and sectors

A sphere has diameter 20 cm

Show that the radius of the sphere is 10 cm

The volume of a sphere is r³ where r is the radius. Work out the volume of a sphere with diameter 20 cm Give your answer in terms of

A rectangle has

• an area of 60 cm²

• a side length of 10 cm

Warren says, “The perimeter of the rectangle is 32 cm because 60 ÷ 10 = 6”

Is Warren correct? Tick a box.

Show working to support your answer.

Here is a circle, centre O.

The circle has a circumference of 20 cm Assume that angle x is 90 °

Work out the shaded area. Give your answer as a decimal.

In fact, angle x is smaller than 90 °

What does this mean about the shaded area? Tick one box.

Here is a circle, centre O.

The circle has a circumference of 20 cm Assume that angle x is 90 °

Work out the shaded area. Give your answer as a decimal.

In fact, angle x is smaller than 90 °

What does this mean about the shaded area? Tick one box.

Here is an equilateral triangle. All measurements are in centimetres.

Work out the value of the perimeter of the triangle.

Here is an equilateral triangle. All measurements are in centimetres.

Work out the value of the perimeter of the triangle.

Five identical spheres just fit inside a cylinder. Each sphere has radius r.

What fraction of the space inside the cylinder is filled by the spheres? You must show your working.

A large circle and a small circle are shown. The radius of the large circle is 12cm

radius of large circle: radius of small circle = 4: 1

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

A large circle and a small circle are shown. The radius of the large circle is 12cm

radius of large circle: radius of small circle = 4: 1

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

A circle has centre (0, 0) and circumference 36 π

Work out the equation of the circle.

Point J has coordinates (15, 0) and point K has coordinates (30, – 5)

Work out the equation of the straight line through J and K.

Ria works out the volume of this hemisphere in terms of π Here is her work.

Write down two mistakes she has made.

The metal used to make a sphere costs £4320 The metal costs £3.60 per gram. Each cubic centimetre of metal has a mass of 17.3 grams.

Work out the radius, r, of the sphere.

Here is a cone.

Beth tries to work out the curved surface area in terms of π

What mistake has she made?

Adam uses π = 3 to estimate the area of the base of the cone.

Work out his estimate.

Beth uses π = 3.14 to estimate the area of the base of the cone.

Is Beth’s estimate more than or less than Adam’s estimate? Tick a box.

Give a reason for your answer.

Here is a cone.

Beth tries to work out the curved surface area in terms of π

What mistake has she made?

Adam uses π = 3 to estimate the area of the base of the cone.

Work out his estimate.

Beth uses π = 3.14 to estimate the area of the base of the cone.

Is Beth’s estimate more than or less than Adam’s estimate? Tick a box.

Give a reason for your answer.

A sector has radius 12 cm and angle 60 °

Work out the length of the arc. Give your answer in terms of π

Shape ABCDE is drawn on a centimetre grid.

Complete this statement.

On this centimetre grid, draw a rectangle with the same area as shape ABCDE.

Use a ruler for this question. Here is an accurate drawing of an equilateral triangle.

By measuring, work out the perimeter of the triangle. State the units of your answer.

A cylinder, C, and a sphere, S, each have radius r C has height h

volume of C = volume of S

Work out the ratio r: h You must show your working.

A different cylinder has radius 3 r and height 2 h.

How many times bigger is the volume of this cylinder than the volume of C?

A regular pentagon has perimeter 100 mm

A shape is made by joining a square to the pentagon.

Work out the perimeter of the shape.

Shape B is a sector of a circle with radius 5cm

Which shape has the greater area, A or B? You must show your working.

In the diagrams, all lengths are in centimetres.

The two shapes have equal areas.

Work out k: m

Shape B is a sector of a circle with radius 5cm

Which shape has the greater area, A or B? You must show your working.

An open cardboard cylinder has radius 1.9 cm and height 10.2 cm

Harry assumes that the net of the cylinder is a rectangle with length l cm

Work out the area of this rectangle.

In fact, the net is a parallelogram, not a rectangle. Not drawn

What does this mean about the area of the net? Tick one box.

What does this mean about the perimeter of the net? Tick one box.

A solid trophy consists of a stand and a player.

The stand is made by removing a small pyramid from a large pyramid.

Large pyramid Square base, edge 8 cm Perpendicular height 16 cm Small pyramid Square base, edge 5 cm Perpendicular height 10 cm

Show that the volume of the stand is 258 cm³

The trophy is made from a metal of density 7.5 grams per cm³ The total mass of the trophy is 2340 grams.

Work out the volume of the player.

Here is a square.

Use a ruler to measure a side length of the square. Give your answer in millimetres.

Here is a quadrilateral.

Use a protractor to measure the size of the smallest angle.

A different quadrilateral is drawn on a centimetre grid.

By counting squares, work out the area of the quadrilateral.

One side of a rectangle is drawn on this centimetre grid.

Complete the rectangle so that it has a perimeter of 20 cm

A large rectangle has a rectangular hole cut out.

Work out the percentage of the large rectangle that is shaded.

A solid hemisphere has radius x. A solid cylinder has radius 3 x and height x.

Surface area of a sphere = 4πr² where r is the radius

Work out the ratio total surface area of the hemisphere: total surface area of the cylinder Give your answer in its simplest form. You must show your working.

A circle, centre O, has circumference 20 π cm Q is a point on the circle.

Work out the value of a. You must show your working.

Here are some shapes. Each shape has an area of six square centimetres.

Which has the bigger perimeter, shape A or shape B ? You must show the lengths of both perimeters.

Which shape is congruent to shape A ?

Which two shapes fit together to make a rectangle?

On this grid draw a reflection of shape C. Show your mirror line.

A bowl is a hemisphere with radius 12 cm

Water is poured into the bowl at a rate of 325 cm³ per second for 8 seconds. Does the water fill more than 70% of the bowl? You must show your working.

A bowl is a hemisphere with radius 12 cm

Water is poured into the bowl at a rate of 325 cm³ per second for 8 seconds. Does the water fill more than 70% of the bowl? You must show your working.

Here is cuboid A.

Cuboid B is made from two of cuboid A.

volume of A: volume of B = 1: 2 Matthew says, “surface area of A: surface area of B must be 1: 2 because B is made of 2 of A.” Is Matthew correct? Tick one box.

Give a reason for your answer.

Here is cuboid A.

Cuboid B is made from two of cuboid A.

volume of A: volume of B = 1: 2 Matthew says, “surface area of A: surface area of B must be 1: 2 because B is made of 2 of A.” Is Matthew correct? Tick one box.

Give a reason for your answer.

The length of this rectangle is 6 times the width.

Two of these rectangles are joined, with no overlap, to make this L-shape.

The perimeter of the L-shape is 98.8 cm Work out the value of the perimeter of one of the rectangles.

The length of this rectangle is 6 times the width.

Two of these rectangles are joined, with no overlap, to make this L-shape.

The perimeter of the L-shape is 98.8 cm Work out the value of the perimeter of one of the rectangles.

ABCDE is a pentagon.

Work out the area of the pentagon.

Part of a running track is the arc of a semicircle joined to a straight line. The semicircle has diameter 45 metres. The straight line has length 75 metres.

Abby runs once along this part of the track in 18 seconds. Work out her average speed. Give your answer to 2 significant figures.

ABCDE is a pentagon.

Work out the area of the pentagon.

Part of a running track is the arc of a semicircle joined to a straight line. The semicircle has diameter 45 metres. The straight line has length 75 metres.

Abby runs once along this part of the track in 18 seconds.

Work out her average speed. Give your answer to 2 significant figures.

Alec makes a bowl for dog food from a solid wooden cone. The sketches show how the bowl is made.

The cone has radius 9 cm and perpendicular height 30 cm A smaller cone, with radius 6 cm, is removed.

A hemisphere with radius 6 cm is then removed.

Volume of a hemisphere = πr³ where r is the radius

Work out the volume of the remaining wood that forms the bowl.

Outside a cafe there is a large plastic ice cream cornet. The cornet is a hemisphere on top of a cone.

The cone and the hemisphere each have radius 24 cm The cone has perpendicular height 117 cm

r is the radius r is the radius h is the perpendicular height

Work out the total volume of the cornet.

The actual cornets that the cafe sells are similar to the plastic one. For the actual cornets, the cone and the hemisphere each have radius 2 cm How many times greater is the volume of the plastic cornet than an actual cornet?

An empty container is a cylinder of radius 3.5 cm and height 40 cm A tennis ball is a sphere of radius 3.5 cm Will six of the tennis balls fit in the container? Tick a box.

Show working to support your answer.

The L-shape is made from rectangles.

Work out the perimeter.

The vertices of a regular hexagon lie on a circle with centre O and radius 5 cm

Work out the shaded area.

A ball contains 5000 cm³ of air. More air is pumped into the ball at a rate of 160 cm³ per second. The ball is full of air when it becomes a sphere with radius 15 cm

Does it take less than 1 minute to fill the ball? You must show your working.

A ball contains 5000 cm³ of air. More air is pumped into the ball at a rate of 160 cm³ per second. The ball is full of air when it becomes a sphere with radius 15 cm

Does it take less than 1 minute to fill the ball? You must show your working.

Work out the area of this triangle.

A circle has a radius of 11.5 cm

Work out the area of the circle.

Here are a triangle and a rectangle.

a and b are positive numbers. Which shape has the larger perimeter? You must work out expressions for both perimeters.

Tick a box.

Two wire shapes make an earring. The shapes are a circle with radius 21 mm and a quarter circle.

radius of circle: radius of quarter circle = 7: 2

Show that the radius of the quarter circle is 6 mm

Work out the total length of the wire in the earring. Give your answer in the form a π + b where a and b are integers.

The shorter side of a parallelogram has length 6.5 cm

The length of the shorter side is of the perimeter. Work out the length of the longer side.

Two wire shapes make an earring. The shapes are a circle with radius 21 mm and a quarter circle.

radius of circle: radius of quarter circle = 7: 2

Show that the radius of the quarter circle is 6 mm

Work out the total length of the wire in the earring. Give your answer in the form a π + b where a and b are integers.

A shape is made by joining triangle ABC to a semicircle with diameter AC.

Work out the total area of the shape.

Here are two solids. Cylinder

Hemisphere

Which solid has the greater volume? You must show your working.

Here are two solids. Cylinder

Hemisphere

Which solid has the greater volume? You must show your working.

Here are a circle and a sector of the circle. They each have radius r.

circumference of circle = perimeter of sector Work out the size of angle x. Give your answer in terms of π

A shaded semicircle is inside a circle as shown.

The radius of the circle is 10 cm The diameter of the semicircle is 8 cm How many times bigger is the unshaded area than the shaded area?

Work out the arc length, in metres, of a semicircle of radius 6 metres. Circle your answer.

3 π 6 π 12 π 18 π

A shaded semicircle is inside a circle as shown.

The radius of the circle is 10 cm The diameter of the semicircle is 8 cm How many times bigger is the unshaded area than the shaded area?

A solid shape is made by joining two cones. Each cone has the same radius.

One cone has slant height = 2 × radius The other cone has slant height = 3 × radius The total surface area of the shape is 57.8π cm² Curved surface area of a cone = π rl where r is the radius and l is the slant height

Work out the radius.

Here are two sectors from different circles.

Which sector has the bigger area?

Show working to support your answer.

In this question, all lengths are in centimetres. A is a point on a circle, centre O. B is a point on a different circle, centre O.

The equation of the larger circle is x² + y² = 144 radius of smaller circle: radius of larger circle = 4: 5

Work out the size of angle A O B.

Here is a rectangle.

Work out the perimeter. Circle your answer.

12 cm 24 cm 35 cm 70 cm

A circle has circumference C and diameter d.

What value does the constant k represent?

Two identical quarter circles are cut from a rectangle as shown.

Work out the shaded area.

This triangle is equilateral.

Is the perimeter of the triangle greater than one metre? You must show your working.

A circle has circumference C and diameter d.

What value does the constant k represent?

Two identical quarter circles are cut from a rectangle as shown.

Work out the shaded area.

A motor racing circuit consists of two parallel straight sections, each of length 0.75 km a semicircle of diameter 0.9 km three equal, smaller semicircles.

The length of a motor race must be greater than 305 km What is the lowest number of full laps needed at this circuit? You must show your working.

A motor racing circuit consists of two parallel straight sections, each of length 0.75 km a semicircle of diameter 0.9 km three equal, smaller semicircles.

The length of a motor race must be greater than 305 km What is the lowest number of full laps needed at this circuit? You must show your working.

A company makes two types of lampshade using fabric on wire frames. Lampshade A Fabric is used to make the curved surface of a cylinder. The cylinder has radius 8 cm and height 22 cm

Lampshade B Fabric is used to make the four triangular faces of a pyramid.

Each triangular face has base 15 cm and perpendicular height 24 cm

Work out the ratio cost of one lampshade A: cost of one lampshade B Give your answer in the form n: 1

24

A sphere has radius 2 x cm A cone has radius 3 x cm perpendicular height h cm The sphere and the cone have the same volume. Work out radius of cone: perpendicular height of cone Give your answer in the form a: b where a and b are integers.

Here is a tunnel for a toy train.

The diagram below shows the cross section of the tunnel.

AD is a semicircular arc of radius 10 cm BC is a semicircular arc of radius 7 cm The length of the tunnel is 30 cm Work out the total area of all six faces of the tunnel. Give your answer in terms of π.

A circle has radius 4.2 cm Work out the length of the circumference. Give your answer to 1 decimal place.

The circle below has centre O. Draw a sector on the circle.

A container is a hemisphere of radius 30 cm

Sand fills the container at a rate of 4000 cm³ per minute. Does it take less than a quarter of an hour to fill the container? You must show your working.

A container is a hemisphere of radius 30 cm

Sand fills the container at a rate of 4000 cm³ per minute. Does it take less than a quarter of an hour to fill the container? You must show your working.

A pentagon is made from a square and an isosceles triangle.

Work out the perimeter of the pentagon.

Here is a quarter circle of radius 6 cm

Work out the area of the quarter circle. Give your answer in terms of π.

Here is a quarter circle of radius 6 cm

Work out the area of the quarter circle. Give your answer in terms of π.

A wheel is made of a circular rim and 8 spokes as shown.

The length of each spoke is 37 cm

Work out the total length of the rim and spokes.

The cross section of an earring is a semicircle, centre C, radius 25 mm The earring is black and white. The shaded area is black.

Sector BCD is white and has radius 12 mm

Is more than 20% of the semicircle white?

You must show your working.

A cube has edge length 0.9 metres.

Work out the total surface area of the cube. Give your answer in square centimetres.

VABCD is a square-based pyramid. The horizontal base ABCD has side length 10 cm and centre M. Angle VMA is 90° Angle VAM is 68°

× area of base × perpendicular height

Work out the volume of the pyramid.

A triangle has perimeter 32 cm

Two sides of the shapes are put together to make a pentagon.

Work out the perimeter of the pentagon.

Here is a circle touching a square.

Work out the area of the circle. Give your answer in terms of .

Here is a circle touching a square.

Work out the area of the circle. Give your answer in terms of .

A cone has a horizontal base of radius 5 cm height of 15 cm The cone contains water to a depth of 9 cm

Work out the volume of the water, in cm³ Give your answer in terms of π.

The diagram shows a logo. ABE and DCE are congruent triangles. BCE is a sector of a circle, centre E.

Show that the area of the logo is 510 cm² to 2 significant figures.

A circle is drawn on a centimetre grid.

Draw a tangent to the circle.

Grace works out that the area of the circle is more than 9 cm² Why must this be wrong?