Area & Volume of Similar Shapes (AQA GCSE Maths: Higher): Exam Questions

Fred is making two rectangular flower beds.

The dimensions of the larger rectangle will be three times the dimensions of the smaller rectangle.

There is going to be the same depth of soil in each flower bed. Fred needs 180 kg of soil for the smaller flower bed.

Work out how much soil Fred needs for the larger flower bed.

and are two triangular prisms that are mathematically similar.

Prism has triangle  as its cross section.
Prism  has triangle  as its cross section.

 = 6 cm = 12 cm

The area of the cross section of prism is 10 cm².
The length of prism is 15 cm.

Work out the volume of prism .

and  are mathematically similar trapeziums.

Work out the length of  .

Trapezium  has an area of 36 cm².

Work out the area of the shaded region.

Here are two similar solid shapes.

surface area of shape A : surface area of shape B = 3:4

The volume of shape B is 10 cm³

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

Solid A and solid B are mathematically similar.
The ratio of the surface area of solid A to the surface area of solid B is 4: 9

The volume of solid B is 405 cm³.

Show that the volume of solid A is 120cm³.

Mark has made a clay model.
He will now make a clay statue that is mathematically similar to the clay model.

The model has a base area of 6cm²
The statue will have a base area of 253.5 cm²

Mark used 2kg of clay to make the model.

Clay is sold in 10kg bags.
Mark has to buy all the clay he needs to make the statue.

How many bags of clay will Mark need to buy?

Cone A and cone B are mathematically similar.
The ratio of the volume of cone A to the volume of cone B is 27:8

The surface area of cone A is 297 cm²

Show that the surface area of cone B is 132 cm²

The circumference of circle B is 90% of the circumference of circle A.

Find the ratio of the area of circle A to the area of circle B.

Square E has sides of length cm.
Square F has sides of length cm.

The area of square E is 44% greater than the area of square F.

Work out the ratio

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
C has length cm

The total surface area of A is 62 cm²

Tim wants to work out the total surface area of B.
Here is his working.

Make one criticism of Tim’s method.

Volume of  Volume of C

Work out the value of .

A pattern is made from two similar trapeziums.

Show that the shaded area is 294 cm².

The three solids and C are similar such that

the surface area of the surface area of

and

the volume of the volume of

Work out the ratio

the height of the height of

Give your ratio in its simplest form.

and are regular hexagons each with centre .

is an enlargement of , with centre and scale factor 2

The triangle has an area of 5 cm²

Calculate the area of the shaded region.

....................................................... cm²

and  are regular pentagons.

and are straight lines.

The area of is 

Calculate the area of the shaded region.

.............................

The diagram consists of three mathematically similar shapes. The heights of the shapes are in the ratio 1 : 4 : 5.

Find the ratio

total shaded area : total unshaded area.

Give your answer in its simplest form.

total shaded area : total unshaded area ............ : ..........

Prism P and prism Q are similar.
The ratio of the surface area of prism P to the surface area of prism Q is 1 : 3.

i) Jay says

The height of prism P is one third of the height of prism Q.

Explain why he is wrong.

[1]

ii) The volume of prism Q is 86 cm³.

Calculate the volume of prism P.

.................................................... cm³ [3]

 is a right-angled triangle.

Angle = 90°
is parallel to .

The area of triangle is 8 cm²
The area of triangle  is 16 cm²

Work out the area of triangle .

The diagram shows two similar vases,  and .

The height of vase  is 9cm and the height of vase  is 13 cm.

Given that

surface area of vase  + surface area of vase  = 1800

calculate the surface area of vase .

........................

A sphere of radius has a surface area of  and a volume of . A cylinder of radius  and height  has a curved surface area of  and a volume of .

The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder.

The radius of the hemisphere is  cm.
The radius of the cylinder is twice the radius of the hemisphere.

Given that

find the value of .

A frustum is made by removing a small cone from a large cone.
The cones are mathematically similar.

The large cone has base radius cm and height cm.
Given that

find an expression, in terms of , for the height of the frustum.

....................................................... cm

andare two similar vases.

Vase has height 10 cm.
Vase has height 15 cm.

The difference between the volume of vase and the volume of vase is 1197 cm³

Calculate the volume of vase .

...................................................... cm³

A standard tin and a large tin are mathematically similar.
The volume of the large tin is 50% more than the volume of the standard tin.
Both tins are cylinders.
The radius of the standard tin is 10cm.

Calculate the radius of the large tin.

........................... cm

The diagram shows a company logo made from a rectangle and a major sector of a circle.
The circle has centre O and radius OA.
OA = OD = 0.5 cm and AB = 1.5 cm.
E is a point on OC such that OE = 0.25 cm and angle OED = 90°.

Calculate the perimeter of the logo.  

............................................. cm

Calculate the area of the logo.

............................................ cm²

A mathematically similar logo is drawn.
The area of this logo is 77.44 cm².

Calculate the radius of the major sector in this logo.

............................................. cm