1.A Rectangle Has Length 7.1 Cm and Width 3.6 Cm

The Robert Smyth SchoolTopic 3

Mathematics FacultyProbabilityPerimeter, Area and Volume

Problem solving with perimeter and area

1.A rectangle has length 7.1 cm and width 3.6 cm.

(a)Calculate the area of the rectangle.
Give your answer to 1 decimal place.

......

Answer ...... cm2

(3)

(b)The diagram shows a parallelogram.

Explain why the area of the parallelogram is equal to the area of the rectangle.

......

(1)

(c)This diagram shows a different parallelogram of length 11.5 cm, height 4.9 cm and slant height 5.3 cm.

Calculate the area of this parallelogram.

......

Answer ...... cm2

(2)

(Total 6 marks)

2.(a)Two squares of side 4 cm are removed from a square

of side 12 cm as shown. Work out the shaded area.

......

Answer ………………………………………

(3)

(b)Two squares of side x cm are removed from a square of side 3x cm as shown.

Work out the fraction of the large square which remains.

Give your answer in its simplest form.

......

Answer ………………………………………

(3)

(Total 6 marks)

3.What percentage of this shape is shaded?

...... ……………………………………………………………………………..

Answer ...... %

(Total 4 marks)

4.The length of a rectangle is 9 cm.
The perimeter of the rectangle is 28 cm.

Calculate the width of the rectangle.

......

Answer...... cm

(Total 3 marks)

5.The length of a rectangle is 10.8 cm.
The perimeter of the rectangle is 28.8 cm.

Calculate the width of the rectangle.

...... ………......

Answer ...... cm

(Total 3 marks)

6.A rectangle has an area of 40 cm2 and a perimeter of 26 cm.
Find the length and width of the rectangle.
You may use the grid to help you.

...... ……………………………………………………………………………..

Answer Length ...... cm Width ………………….cm (Total 2 marks)

7.A shop sells square carpet tiles in two different sizes.

(a)What is the area of a small carpet tile?

……………………………………………….....…………………………………….

Answer ...... cm2

(2)

(b)What is the length of a side of a large carpet tile?

……………………………………………….....…………………………………….

Answer ...... cm

(1)

(c)The floor of a rectangular room is 300 cm long and 180 cm wide.
How many small tiles are needed to carpet the floor?

Answer ……………………………………..

(3)

(Total 6 marks)

Problem solving using area and volume

1.In the diagram below, PQ = 10.8 cm, QR = 11.6 cm, RS = 17.5 cm and PS = 9.5 cm. The angles at P and S are 90°

Calculate the area of PQRS.

...... ………......

Answer ...... cm2

(Total 3 marks)

2.A shape has dimensions as shown.

Calculate the shaded area.

...... ……...... …...

Answer ...... cm2

(Total 3 marks)

3.The diagram shows a silver bar.

The cross-section of the silver bar is a trapezium.

(a)Calculate the area of the cross-section.

...... ……………………………………………………………………………..

Answer ...... cm2

(2)

(b)The silver bar is 15 cm long.
The bar is melted and the silver is then made into a cuboid.
The base of the cuboid is 10 cm by 10 cm.

What is the height, h, of the cuboid?

...... ……………………………………………………………………………..

Answer ...... cm

(3)

(Total 5 marks)

4.A cuboid is shown below.
The cuboid has volume 60 cm3.
The base is 6.2 cm long and 3.7 cm wide.

(a)Calculate the height of the cuboid.
Give your Answer to a sensible degree of accuracy.

......

Answer ...... cm

(3)

(b)A tile is shown below.

Find the area of the tile.
Give your answer in m2.

......

Answer ...... m2

(2)

(Total 5 marks)

5.The diagram shows a beam of uniform cross-section and length 4 metres.

Calculate the volume of the beam.

Give your answer in cubic centimetres.

......

Answer ...... cm3

(Total 5 marks)

6.Calculate the area of the shape.

......

Answer...... cm2

(Total 3 marks)

7.(a)This L-shape is made of rectangles. Calculate the area of the shape.

......

Answer ……………………...... cm2

(3)

(b)This T-shape is also made of rectangles. The perimeter is 29cm. Work out the value of x.

......

Answer ……………………...... cm

(4) (Total 7 marks)

Pythagoras and 3D objects

1.(a)ABC is a right-angled triangle.
AC = 19 cm and AB = 9 cm.

Calculate the length of BC.

………...…...... ……...... …………….....…………......

Answer ...... cm

(3)

(Total 3 marks)

2.(a)The diagram shows a right-angled triangle ABC.
AB = 10 cm and AC = 15 cm

Calculate the length of BC. Leave your answer as a square root.

......

Answer ...... cm

(3)

(Total 3 marks)

3.A rectangular field ABCD is shown.
The length of the field, AB = 160 m.
The width of the field, BC = 75 m.

Not to scale

(a)Calculate the length of the diagonal BD.

Give your answer to a suitable degree of accuracy.

......

Answer ...... m

(4)

(Total 4 marks)

4.The diagram shows a square and its diagonals.
Each diagonal is 8 centimetres long.

Calculate the area of the square.

......

Answer ...... cm2

(3)

5.PQRS is a quadrilateral.
Angles RQS and QSP are right angles.
PS = 4 cm, QR = 12 cm and RS = 13 cm.

Not to scale

Show that the length of PQ is

......

(Total 4 marks)

6.The diagram shows a cuboid.
AB = 3 cm, AE = 4 cm, BC = 12 cm.

Not drawn accurately

(a)Find the length of BH.

......

Answer ...... cm

(2)

(Total 2 marks)

7.(a)Multiply out and simplify (x +)2.

......

Answer ......

(2)

(b)Triangle ABC has a right angle at B.

Find the value of x.

You must explain clearly how you obtain your answer.

......

Answer x = ......

(5)

(Total 7 marks)

8.VABCD is a right pyramid on a rectangular base.

VA = VB = VC = VD = 16 cm.

AB = 20 cm and BC = 14 cm.

(a) Prove that the perpendicular height, h, of the pyramid is 15.6 cm (1dp.)

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(5)

(b) Work out the volume of the pyramid

......

Answer ......

(2)

(Total 7 marks)

Using½ ab Sin C and finding the volume of pyramids

1.(a)The diagram shows a right-angled triangle ABC.
AB = 10 cm and AC = 15 cm

Calculate the length of BC. Leave your answer as a square root.

......

Answer ...... cm

(3)

(Total 3 marks)

2.The diagram shows a triangle ABC.
AB = 6 cm, BC = 5 cm and angle B = 75°

You are given that sin 75° = 0.966 to 3 significant figures.

Calculate the area of the triangle.
Give your answer to a suitable degree of accuracy.

...... ………......

Answer ...... cm2

(Total 3 marks)

3.In triangle ABC, AB = 11 cm, BC = 9 cm and CA = 10 cm.

Angle CAB = 50.5°

Find the area of triangle ABC.

Give your answer to a suitable degree of accuracy.

...... ……………………………………………………………………………..

Answer ......

(Total 4 marks)

4.In triangle ABC, AB = 5 cm, BC = 8 cm and AC = 9 cm.

Angle BCA = 32°

Find the area of triangle ABC.

Give your answer to a suitable degree of accuracy.

......

Answer ......

(Total 4 marks)

5.ABCD is a quadrilateral.
AB = 7 cm, AD = 6 cm and BC = 9 cm, AC = 10cm
Angle ABC = 75° and angle ADC = 90°

Calculate the area of ABCD.

...... ….

Answer ...... cm

(Total 6 marks)

6.VABCD is a right pyramid on a rectangular base.
VA = VB = VC = VD = 11 cm. 0AB= 12 cm and BC = 7 cm.

VO is the perpendicular height.

Calculate the volume of the pyramid.

Hints: Use Pythagoras to first find the length AC, then the perpendicular height VO.

......

Answer...... cm3

(Total 5 marks)

7.A square-based pyramid A is divided into two parts:
a square-based pyramid B and a frustum C, as shown.

Pyramid A is similar to pyramid B.

The base of pyramid A is a square of side 10 cm.
The base of pyramid B is a square of side 5 cm.

The vertical height of pyramid A is 12 cm.

(a)You are given the formula

Volume of a pyramid = area of base × vertical height

Calculate the volume of the frustum C.

......

Answer ...... cm3

(4)

(b)Express the volume of the frustum C as a fraction of the volume of the larger pyramid A. Give your answer in its simplest form.

......

Answer ......

(2) (Total 6 marks)

8.A square-based pyramid has a base of edge 5 cm.
The vertex of the pyramid is directly over the midpoint of the base.
The volume of the pyramid is 100cm3.

Find the length of the slant edge of the pyramid (marked x in the diagram).

...... ……………………………………………………………………………..

Answer ...... cm

(Total 5 marks)

Extension Questions

9.(a)Show that(√12 + √3)2 = 27

......

(2)

(b)In the diagramPQ = (√12 + √3) cm

QS = (√8 + √2) cm

SR = √2cm

Not drawn accurately

(i)Show that PS = 3 cm

......

(4)

(ii)Find the area of triangle PQR giving your answer in the form a√2, where a is a positive integer.

......

Answer ...... cm2

(3)

(Total 9 marks)

10.A trapezium has parallel sides of length (x + 1) cm and (x + 2) cm.
The perpendicular distance between the parallel sides is x cm.
The area of the trapezium is 10 cm2.

Not drawn accurately

Find the value of x.

......

Answer x = ...... cm

(Total 5 marks)

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