OCR GCSE Maths
Topic questions Geometry and measures Angle facts
59 questions · 298 marks · about 4.5 hours
The diagram shows a triangle with one side extended.
Work out the value of a.
Complete the statement below using the correct symbol and value from this list.
< 90 ° = 90° > 90° < 18 0 ° = 18 0 ° > 18 0 °
When the two smallest angles in a triangle are added the answer is always........
T he diagram shows a regular pentagon and part of another regular polygon meeting at point P. One of the angles at point P is 92°.
Show that the size of one interior angle of a regular pentagon is 108°.
Find the number of sides of the other regular polygon.
The diagram shows a triangle, ABC. The angle at A is 30°.
What does the symbol at angle B mean? Choose from this list.
An acute
The biggest An obtuse A reflex
A right angle angle angle angle angle
Work out the angle at C. Give a geometrical reason for your answer.
The angle at C is........ ° because........
A student works out the size of angle x in this diagram.
The student says the angle is 240°.
Explain why the student’s answer cannot be correct.
Work out the size of an exterior angle of a regular hexagon.
Use your answer to part (a) to write down the size of an interior angle of a regular hexagon.
A school is deciding on a charity to support. Each student at the school votes for one of four charities, A, B, C or D. The results are to be shown in a pie chart.
This pie chart shows the sector for charity A. Twice as many students voted for charity C than charity B. Twice as many students voted for charity D than charity C.
Show that the sector for charity B will have an angle of 44°.
Complete the pie chart.
39 students voted for charity A.
Calculate the total number of students at the school.
The school asks 80 of the students to choose a new logo from three designs G, H and J. The same results are shown in a pie chart and in a bar chart.
Give one advantage of using the pie chart rather than the bar chart.
Give one disadvantage of using the pie chart rather than the bar chart.
The diagram shows a straight line crossing two parallel lines.
Find the value of y. You must show your working.
The diagram shows two straight lines AB and DC that intersect at P. DA is parallel to BC.
Complete these statements to show that triangle PAD is similar to triangle PBC.
ADP BCP alternate Angle........ = angle........ because they are........ angles
APD Angle........ = angle........ because they are........ angles
Angle........ = angle........ because they are........ angles
Triangle PAD is similar to triangle PBC because........
Also involves Rearranging formulae
The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.
Show that the area of the pentagon is 247.75 cm2, correct to 2 decimal places.
The regular pentagon is the base of a pyramid.
The pyramid has volume 450 cm3. The perpendicular height of the pyramid is h cm.
Calculate the value of h. [The volume of a pyramid is 
× area of base × perpendicular height.]
A school is deciding on a charity to support. Each student at the school votes for one of four charities, A, B, C or D. The results are to be shown in a pie chart.
This pie chart shows the sector for charity A. Twice as many students voted for charity C than charity B. Twice as many students voted for charity D than charity C.
Show that the sector for charity B will have an angle of 44°.
Complete the pie chart.
39 students voted for charity A.
Calculate the total number of students at the school.
The school asks 80 of the students to choose a new logo from three designs G, H and J. The same results are shown in a pie chart and in a bar chart.
Give one advantage of using the pie chart rather than the bar chart.
Give one disadvantage of using the pie chart rather than the bar chart.
Also involves Linear equations
A regular polygon has n sides. The interior angle of the polygon is 15 times its exterior angle.
Find the value of n.
The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.
Show that the area of the pentagon is 247.75 cm2, correct to 2 decimal places.
The regular pentagon is the base of a pyramid.
The pyramid has volume 450 cm3. The perpendicular height of the pyramid is h cm.
Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]
The diagram shows two triangles ABC and ADC.
Angle ABC = 60 °, angle ACB = 20 ° and angle ACD = 55 °. AB = 8 cm and DC = 6 cm.
Calculate length AD. You must show your working.
Four lines meet at a point.
Work out the size of angle a.
The diagram shows two straight lines crossing a pair of parallel lines.
Find the value of x. Give a geometrical reason for your answer.
Find the value of y. Give a geometrical reason for your answer.
Find the value of z.
A regular polygon is being constructed inside a circle, centre O. Part of the construction is shown in this diagram.
Give a reason why OA = OB.
Write down the mathematical name of triangle OAB.
The regular polygon is completed.
Work out the sum of the interior angles of the regular polygon.
The diagram shows the positions of three towns A, B and C.
The bearing of town B from town A is 053°. The bearing of town C from town B is 108°. AB = 35 km and BC = 61 km.
Calculate AC. You must show your working.
Also involves Geometric reasoning and proof
The diagram shows four triangles that are joined together.
• Points A, B, C and D lie on a straight line.
• Triangles ABF and CDE are equilateral triangles.
• Triangle BCF is an isosceles triangle with BF = BC.
Show that triangle CEF is a right-angled triangle. Give a reason for each stage of your working. Use the template below to help present your work. You may not need all of the lines.
Angle........ =........ ° because........
Angle........ =........ ° because........
Angle........ =........ ° because........
Angle........ =........ ° because........
Angle........ =........ ° because........
Angle........ =........ ° because........
Angle........ =........ ° because........
Also involves Ratio
In the diagram, line AB is parallel to line ED. The points A, C and D lie on a straight line. The points B, C and E lie on a straight line.
Angle BCA = 45°, angle ABC = x ° and angle CDE = y °. The ratio x: y is 3: 2.
Work out the value of x.
A student draws two different regular polygons. The exterior angle of one polygon is p °. The exterior angle of the other polygon is q °.
The sum of p and q is 112°. The difference between p and q is 32°.
Find the number of sides of each polygon. You must show your working.
........ sides and........ sides
The diagram shows two straight lines crossing a pair of parallel lines.
Write down the value of a. Give a reason for your answer.
Work out the value of b.
A student draws two different regular polygons. The exterior angle of one polygon is p °. The exterior angle of the other polygon is q °.
The sum of p and q is 112°. The difference between p and q is 32°.
Find the number of sides of each polygon. You must show your working.
........ sides and........ sides
Taylor designs a logo using isosceles triangles joined at a central point, P.
This is the start of Taylor’s design.
The completed design will have rotational symmetry, order 60 about point P.
Each triangle has base, b, and height, h, measured in mm.
Calculate h when b = 40 mm.
Give your answer correct to 1 decimal place.
B is 12 km due east of A. C is south-east of A and on a bearing of 225° from B.
Complete the diagram to show the positions of A, B and C. Show clearly the values of all three angles in triangle ABC.
Scale: 1 cm represents 2 km
The diagram shows two intersecting straight lines.
Find the value of a. Give a reason for your answer.
Find the value of b. Give a reason for your answer.
The diagram shows part of a regular 12-sided polygon.
For this polygon, find the ratio of the size of one exterior angle to the size of one interior angle. Give your answer in its simplest form. You must show your working.
B is 12 km due east of A. C is south-east of A and on a bearing of 225° from B.
Complete the diagram to show the positions of A, B and C. Show clearly the values of all three angles in triangle ABC.
Scale: 1 cm represents 2 km
The diagram shows a straight line crossing a pair of parallel lines.
Find the value of y. You must show your working.
Points A, B, C and D lie on the circumference of a circle. Line AC intersects line BD at point E.
Prove that triangle AED is similar to triangle BEC.
The diagram shows a pair of parallel lines.
Write down the value of a.
Write down the value of b.
Also involves Fractions
Some students were asked how they travel to school. Each student gave one answer. The pie chart shows the proportion who travel by bus, by train or walk.
All of the remaining students travel to school either by bike or by car. The ratio of the number who travel by bike to the number who travel by car is 2: 3.
Complete the pie chart.
You must show your working.
Which way of travelling to school is the mode?
Three regular polygons meet at a point.
Two of the polygons are pentagons.
Find the number of sides of the third polygon. You must show your working.
Points A, B and C lie on the circumference of a circle. EAF is a tangent to the circle.
Write down the value of angle BCA giving a reason for your answer.
Angle BCA = ……………….° because........
Points G, H and J lie on the circumference of a circle, centre O.
Angle GOJ = 52° and angle GJH = x °. Lines JO and GH are parallel.
Find the value of x. You must show your working.
The diagram shows a straight line crossing two parallel lines.
Find the value of y. You must show your working.
The diagram shows triangles ABC and ADE.
B lies on AD and C lies on AE. BC is parallel to DE.
Complete these statements to show that triangles ABC and ADE are similar.
Angle ABC = angle ADE because they are corresponding angles.
Angle ACB = angle........ because........
Triangles ABC and ADE are similar because........
An equilateral triangle, a regular 10-sided polygon and another regular polygon meet at a point.
Show that angle A is 156°.
Work out the number of sides of the other regular polygon.
Work out the size of the exterior angle of a regular 12-sided polygon.
Use your answer to part (a) to write down the size of the interior angle of a regular 12-sided polygon.
Also involves Proof
In the diagram, AB and DC are parallel lines of equal length.
Prove that angle DAB = angle BCD.
Work out the size of the exterior angle of a regular 12-sided polygon.
Use your answer to part (a) to write down the size of the interior angle of a regular 12-sided polygon.
Three lines meet at a point.
Work out the size of angle a.
XY and CW are parallel lines. AB = CB. Angle CAX = 130°.
Complete this sentence.
Angle CAB = 50° because........
Work out angle BCW. Give a reason for each angle you work out.
The diagram shows a kite, ABCD. AFE and CEF are equilateral triangles.
Write down a mathematical name for quadrilateral AFCE.
The ratio of angle DAE: angle EAF = 1: 4.
Work out angle x. Write on the diagram the values of any other angles you use in your working.
AB and CD are parallel lines. EF and FB are straight lines.
Complete the following statements.
The diagram shows two intersecting straight lines.
Find the value of y.
Also involves Geometric reasoning and proof
The diagram shows triangle ABC. CD is parallel to AB. A, C and E lie in a straight line. Angles of size a °, b ° and c ° are shown.
Insert a °, b ° or c ° to make this statement true. Give a reason for your answer.
Angle DCE =........ because........
Use the diagram and the answer to part (a) to show that the angles of a triangle add up to 180°.
Give a reason for each statement you make.
Also involves Forming and solving equations
An interior angle of an isosceles triangle is p ° and an exterior angle is q °.
It is given that q = 5 p.
Write the ratio p: q in its simplest form.
Work out the two different possible sets of angles for the isosceles triangle.
(b) Triangle 1:........ °,........ °,........ °
Triangle 2:........ °,........ °,........ °
In the diagram AB is parallel to CD. AED and BEC are straight lines.
Prove that triangle ABE is similar to triangle CDE.
Also involves Properties of triangles and quadrilaterals
The diagram shows a square, a regular hexagon and part of another regular polygon meeting at
Show that the size of one interior angle of a regular hexagon is 120°.
Find the number of sides of the other regular polygon.
Also involves Forming and solving equations
A regular polygon has n sides. The polygon’s interior angle is 5 times the size of its exterior angle.
Find n.
In the diagram, AB is parallel to DC.
Work out angle BDC. Give a reason for each angle you work out.
Two groups of students go on a water sport holiday. Each student chooses one activity.
Students in Group A choose from Diving, Swimming, Paddleboarding and Kayaking. Their choices are to be shown in a pie chart.
Complete this table for Group A.
| Activity | Number of students | Angle of sector |
| Diving | 5 | 60 |
| Swimming | 120 | |
| Paddleboarding | ||
| Kayaking | 9 | 108 |
Complete the pie chart for Group A.
One student in Group A changes activity. There is now a new modal activity for Group A.
Write down the student’s original activity and new activity.
original activity........
The choices made by Group B are shown in this pie chart.
A teacher thinks more students chose Diving in Group B than in Group A.
Give a reason why the teacher may be wrong.
The diagram below shows two triangles.
Prove that triangle ABC is congruent to triangle ACD.
Also involves Trigonometric ratios
The angles in a triangle are in the ratio 1: 2: 3.
Show that the triangle is a right-angled triangle.
The hypotenuse of the triangle is 15 cm long.
Calculate the length of the shortest side in the triangle.
Also involves Symmetry
Angle BAE is part of a regular 18-sided polygon. Angle CAD is part of a regular 10-sided polygon. The dashed line through A is a line of symmetry of both polygons.
Work out angle BAC.
The angles in a triangle are in the ratio 1: 2: 3.
Show that the triangle is a right-angled triangle.
The hypotenuse of the triangle is 15 cm long.
Calculate the length of the shortest side in the triangle.
Also involves Forming and solving equations
The diagram shows the positions of two towns, Amton and Bisham.
The bearing of Bisham from Amton is b °. The bearing of Amton from Bisham is 6 b °.
Calculate the 3-figure bearing of Amton from Bisham.
Measure angle a.
Write down the mathematical name of this type of angle.
Choose one of these words to complete the following sentence.
These are........ lines.
Also involves Fractions
Jorge recorded the scorers of 120 goals. He started to draw a pie chart to show the results.
How many goals did Simon score?
The table shows the other players who scored goals.
| Name of scorer | Number of goals | Angle of sector |
| Wayne | 48 | 144° |
| Harry | 5 | |
| Obi | 72° | |
| Antony |
Complete the table.
Complete the pie chart.
A, B, C and D are four towns.
Work out the bearing of B from C.
Calculate the bearing of D from B.
XY and BD are parallel lines. X is a point on AB and C is a point on BD.
Complete this sentence.
Angle XBC = 65° because........
Work out angle BXC. Give a reason for each angle you work out.
Imran joins two tiles together as shown below. One tile is a regular hexagon and the other tile is a regular pentagon.
Show that angle a is 132°.
Imran thinks that another tile in the shape of a regular polygon will fit exactly into angle a
Is Imran correct? Show your reasoning.
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