# Angles, Polygons and Geometrical Proof Stage 4 - Short Problems

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Angles, Polygons and Geometrical Proof - Stage 4.

Printable worksheets containing selections of these problems are available here:

 Stage 4 ★★ Sheet 1 Solutions Sheet 2 Solutions Stage 4 ★★★ Sheet 1 Solutions

### Centred

##### KS 4 Short Challenge Level:

Weekly Problem 13 - 2012
The diagram shows contains some equal lengths. Can you work out one of the angles?

### Eulerian

##### KS 4 Short Challenge Level:

Weekly Problem 37 - 2014
Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

### Octagonal Ratio

##### KS 4 Short Challenge Level:

Can you find the ratio of the area shaded in this regular octagon to the unshaded area?

### Internal - External

##### KS 4 Short Challenge Level:

Weekly Problem 12 - 2016
The diagram shows a square PQRS and two equilateral triangles RSU and PST. PQ has length 1. What is the length of TU?

### Block Challenge

##### KS 4 Short Challenge Level:

Can you work out the shaded area in this shape?

##### KS 4 Short Challenge Level:

Weekly Problem 41 - 2016
The diagram shows a square, with lines drawn from its centre. What is the shaded area?

### Isosceles Reduction

##### KS 4 Short Challenge Level:

Weekly Problem 29 - 2010
An isosceles triangle is drawn inside another triangle. Can you work out the length of its base?

### Diagonal Touch

##### KS 4 Short Challenge Level:

Weekly Problem 21 - 2012
Two rectangles are drawn in a rectangle. What fraction of the rectangle is shaded?

### Inscribed Semicircle

##### KS 4 Short Challenge Level:

Weekly Problem 43 - 2017
The diagram shows a semicircle inscribed in a right angled triangle. What is the radius of the semicircle?

### Tunnel Vision

##### KS 4 Short Challenge Level:

How wide is this tunnel?

### Incentre Angle

##### KS 4 Short Challenge Level:

Weekly Problem 1 - 2011
Use facts about the angle bisectors of this triangle to work out another internal angle.

### Two Right Angles

##### KS 4 Short Challenge Level:

Weekly Problem 4 - 2008
In the figure given in the problem, calculate the length of an edge.

### Quarters

##### KS 4 Short Challenge Level:

Weekly Problem 27 - 2014
Four congruent isosceles trapezia are placed in a square. What fraction of the square is shaded?

### 3-4-5 Circle

##### KS 4 Short Challenge Level:

Can you find the radius of the circle inscribed inside a '3-4-5 triangle'?

### Angle to Chord

##### KS 4 Short Challenge Level:

Weekly Problem 23 - 2008
A triangle has been drawn inside this circle. Can you find the length of the chord it forms?

### Slide Height

##### KS 4 Short Challenge Level:

How high is the top of the slide?

### Overlapping Semicircles

##### KS 4 Short Challenge Level:

Two semicircles overlap, can you find this length?

### Cut-up Square

##### KS 4 Short Challenge Level:

Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?

### Two Equilateral Triangles

##### KS 4 Short Challenge Level:

Prove that these two lengths are equal.

### Garden Fence

##### KS 4 Short Challenge Level:

Weekly Problem 44 - 2009
A garden has the shape of a right-angled triangle. A fence goes from the corner with the right-angle to a point on the opposite side. How long is the fence?

### Semicircle in a Triangle

##### KS 4 Short Challenge Level:

A semicircle is drawn inside a right-angled triangle. Find the distance marked on the diagram.

### Altitude Inequalities

##### KS 4 Short Challenge Level:

Weekly Problem 8 - 2010
Are you able to find triangles such that these five statements are true?

### Folded Square

##### KS 4 Short Challenge Level:

This square piece of paper has been folded and creased. Where does the crease meet the side AD?

### Height and Sides

##### KS 4 Short Challenge Level:

Can you find the area of the triangle from its height and two sides?

### Circular Inscription

##### KS 4 Short Challenge Level:

In the diagram, the radius of the circle is equal to the length AB. Can you find the size of angle ACB?

### Overlapping Annuli

##### KS 4 Short Challenge Level:

Just from the diagram, can you work out the radius of the smaller circles?

### Circled Square

##### KS 4 Short Challenge Level:

Can you find the area of this square inside a circle?

### Griddy Region

##### KS 4 Short Challenge Level:

Weekly Problem 34 - 2008
What is the area of the region common to this triangle and square?