Similarity, lengths, areas and volumes

These two shapes are similar.

Write down the value of y.

Work out the value of x.

x = cm

Triangles ABC and DEF are similar.

Work out the area of DEF.

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

These two triangles are congruent.

Write down the value of x.

x = cm

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.

Here are two square-based paving stones. The stones are similar solids.

The price per cm³ is the same for both stones. The price of the larger stone is £17.50

Work out the price of the smaller stone.

ABCD is a trapezium. All four sides are different lengths. AB is parallel to CD. The diagonals intersect at X.

For each statement, tick the correct box.

Show that these two rectangles are similar.

Show that these two rectangles are similar.

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.

Trapezium DEFG is formed by joining triangle DEH

ABC is similar to DEH. Work out the area of DEFG.

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.

Here are two shapes, P and Q.

How many times bigger is the area of P than the area of Q? You must show your working.

Here are two shapes, P and Q.

How many times bigger is the area of P than the area of Q? You must show your working.

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?

A dolphin and a whale are drawn to scale.

The actual length of the dolphin is 3 metres. Estimate the actual length of the whale. You must show your working.

Here are two rectangles.

Show that the rectangles are similar.

Here are two similar cones.

The surface area of cone A is 2 m² The surface area of cone B is 4.5 m² Work out the ratio radius of cone A: radius of cone B Give your answer in the form 1: n

A and B are similar solid cylinders. base area of A: base area of B = 9: 25 Complete these ratios.

These two triangles are similar.

Work out the value of a.

These two triangles are similar.

Work out the value of a.

Here are two triangles, P and Q.

Here is a statement.

Make one criticism of the statement.

Here are two shapes, C and D.

Here is a statement.

Make one criticism of the statement.

A map of an island is shown on a centimetre grid. A, B and C are houses.

The actual distance between A and B is 1500 metres. Show that the scale on the map is 1: 30 000

Work out the actual distance between A and C. Give your answer in kilometres.

Here are three similar cuboids, A, B and C. A has length 5 cm, width 2 cm and height 3 cm B has length 10 cm

The total surface area of A is 62 cm 2 Tim wants to work out the total surface area of B. Here is his working.

Make one criticism of Tim’s method.

Work out the value of x.

Here is quadrilateral PQRS.

X is a point on PS where PX: XS = 1: 2 Y is a point on RQ where RY: YQ = 2: 1

Is XY parallel to PQ ? Show working to support your answer.

A drawing has a scale of 1: 40 On the drawing, a bedroom is a rectangle measuring 10 cm by 18 cm A kitchen has an actual area of 300 000 cm² Which has the bigger actual area, the kitchen or the bedroom? You must show your working.

Here are two similar shapes, A and B.

length of edges in A: length of edges in B = 2: 5 The perimeter of A is 210 mm Work out the perimeter of B.

A and B are similar cuboids. surface area of A: surface area of B = 16: 25 Work out volume of A: volume of B Circle your answer.

4: 5 16: 25 64: 125 256: 625

Prisms A and B are similar. The cross sections are shaded. Prism A Prism B

area of the cross section of A: area of the cross section of B = 4: 9 Work out the area of the cross section of B.

Each side of a square is made 3 times as long. What happens to the perimeter? Circle your answer.

× 3 × 6 × 9 × 12

A small square has length x cm A large square has length 15 cm

The area of the small square is of the area of the large square. Work out the value of x.

The diagram shows the side view of a step ladder with a horizontal strut of length 48 cm The strut is one third of the way up the ladder. The symmetrical cross section of the ladder shows two similar triangles.

Work out the vertical height, h cm, of the ladder.

Solids X and Y are similar.

Work out the surface area of Y.

A and B are similar shapes. B is an enlargement of A with scale factor 1.5 Not drawn

Work out the values of x, h and w.

A and B are similar shapes. B is an enlargement of A with scale factor 1.5 Not drawn

Work out the values of x, h and w.

x = degrees h = cm w = cm

A and B are similar solids.

‘‘The volume of B is double the volume of A because the length of B is double the length of A.’’ Is he correct?

Give a reason for your answer.

PRT and QRS are similar triangles.

Circle your answer.

Triangles ABC and DEF are similar.

Work out the value of x.

Write down the size of angle y.