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There are **28** NRICH Mathematical resources connected to **Area - circles, sectors and segments**, you may find related items under Measuring and calculating with units.

Problem
Primary curriculum
Secondary curriculum
### Curvy Areas

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Partly Circles

What is the same and what is different about these circle questions? What connections can you make?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gutter

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Spokes

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Salinon

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles and Petals

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Approximating Pi

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Blue and White

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Compare Areas

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadarc

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circular Area

How could you find out the area of a circle? Take a look at these ways.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Rational Search

Investigate constructible images which contain rational areas.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Shapes & Printer Ink

If I print this page which shape will require the more yellow ink?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bound to Be

Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squaring the Circle

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Crescents and Triangles

Can you find a relationship between the area of the crescents and the area of the triangle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bull's Eye

What fractions of the largest circle are the two shaded regions?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

Age 14 to 16

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### The Pillar of Chios

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Floored

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round and Round

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Pegs

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Circles

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Get Cross

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Giant Holly Leaf

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).

Age 14 to 16

Challenge Level