# Perimeter and area of shapes

Students need to be aware of the difference in meaning between perimeter and area and appreciate that shapes with the same perimeter can have different areas and vice-versa. Students also need to be aware of the relationship between perimeter and circumference. This resources package contains a variety of activities requiring students to solve problems involving perimeter and area of triangles, circles and composite shapes.

Visit the secondary mathematics webpage to access all lists.

### Area and Perimeter

This resource contains games, investigations, worksheets and practical activities.

Area and Perimeter pack one contains activities appropriate to this topic beginning with basic tasks such as **Which has the largest area?** requiring students to compare areas of shapes drawn on square grids. **What is the perimeter? **asks students to find areas of shapes drawn on to square grids and **Area 1** uses squared paper to draw shapes and find their area. **Eight squares **asks how many shapes can be found within an area of eight squares. More challenging activities include **Silver Earrings** in which students find the cost of a number of differently shapes earrings involving compound shapes and triangles.

Area and Perimeter pack two contains activities requiring students to make shapes of a given size using pentominoes, investigate the area and perimeter of rectangles, find the area of a right-angled triangle, calculate the area of polygons drawn on square dotted paper, investigate different ways of shading half a square, find the area of a triangle, find the area of compound shapes made from rectangles and find the area of a parallelogram.

Area and Perimeter pack three contains activities in which students investigate the connections between the area of a parallelogram and the area of a rectangle, the area of obtuse-angled triangles, further parallelogram problems, finding the area of a polygon, finding the area of a trapezium and calculating the area of irregular shapes.

### Areas and Perimeters

The textbook ‘**Areas and Perimeters’ **requires students to find the area of a shape by considering the number of squares the shape contains including estimating area where a shape does not contain an exact number of squares. Students explore the area and perimeter of squares and rectangles by attempting a number of searching questions investigating links between the area and perimeter before moving on to compound shapes comprising of squares and rectangles. The section concludes by students exploring how to find the area of a triangle.

The activities file contains a number of interesting activities. Students are asked to investigate the **Area of Hands** and **Areas and Points** explores areas of shapes drawn on dotted paper. Two further activities require students to investigate the area of a triangle. **Fence it Off** explores the maximum area that can be made with a fixed perimeter.

### Running Tracks

This activity gives the opportunity for a practical activity requiring accurate measurement, scale drawing and calculations involving pi. Students look at how a running track is marked out and distances measured accurately for runners in each of the lanes. Students could be asked to design their own stadium. This short video about building the Olympic park could be used as inspiration. Students could investigate where a track could be placed in the school grounds and how the track lanes can be marked accurately.

### Equal Areas Inquiry

This Maths Inquiry is designed to encourage students to explore relationships between the areas of different shapes.

‘**Equal areas’ **shows a diagram containing rectangles, triangles and circles and asks whether their areas can be the same. There are suggested questions that can be asked and directions that the inquiry could take.

**‘Lesson notes’** gives further information to help use the enquiry and suggests alternative inquiries.

### Shape, Space and Measures 3

**Area of simple shapes** page 21 (p.28 of the pdf) explains how to find the area of a rectangle, parallelogram, triangle, trapezium, kite, rhombus and a variety of compound shapes. There are a number of investigations and real-life problems mixed in with more standard questions.

**Circles: perimeter and area** page 71 (p. 78 of the pdf) begins with a practical task measuring the perimeter and diameter of circular objects to find an approximation for pi. The value of pi is then used to find the circumference of a circular object given the diameter. Students are asked to find how far a wheel will travel when passing through one revolution and how many revolutions are required when travelling a set distance.

There follows a nice explanation of the derivation of the formula for the area of a circle followed by a variety of questions finding the area of circles, halves and quarters of circles, the area of an annulus although this term is not referred to.

### Circle Measurement

This resource contains two packs of games, investigations, worksheets and practical activities.

**Circle measurement pack one** contains nine work cards with a wide variety of activities appropriate to this topic including **‘It’s not fair’** in which students find the length of a running track, **‘Circumference’** and ‘**Making Circles’** which explores how to find the circumference of a circle, a booklet ‘**All about circles’** which includes a variety of activities from naming different parts of a circle to calculating the distance travelled by the hand of a clock.

**Circle measurement pack two **contains activities such as **‘DIY earrings’** in which students find the cost of circular earrings and includes finding the area of compound shapes and ‘**Clover leaf’** in which distances are calculated when travelling through a motorway interchange system.

### Square and Circle Inquiry

This Maths Inquiry states "a square fits into a circle better than a circle fits into a square" leading to a rich task exploring areas, ratios and percentages.

A square fits better in a circle than a circle fits in a square describes the prompt and suggests a variety of ways in which it could be used in the classroom.

Lesson notes are included explaining how the inquiry can be generalised, extended and how conjectures can be proved. The notes cover the case of a circle and a square and the case of a circle and a hexagon.

### Circles

This resource contains six activities for use in the classroom when exploring the properties of circles. The section on Sectors begins by defining major and minor sector, and explaining how to calculate the arc length and the area of a sector of a circle. Students are then required to complete a table of values of properties of a circle and a sector of that circle given different properties from which to start with. It could be used as resource material for more able students working independently. Earlier sections could be referred to as a means of checking / consolidating any prior knowledge they may need.

### Four Crescents

A diagram is shown with a circle, centred on a rectangle with given dimensions. Semi-circular arcs are drawn with the sides of the rectangle as diameters. The areas between the semi-circles and the circles form crescents. The task is to calculate the area of the crescents.

When working on the numerical example it becomes apparent that the total area of the crescents is equal to the area of the rectangle. A formal proof of this relies on ideas of area of a circle, Pythagoras’ theorem, and algebraic manipulation.

### Polygon in Annulus

A polygon of side length two is shown with a circle circumscribing its vertices. A second circle is shown with the sides of the polygon tangential to the circle. The challenge is to calculate the area of the resulting annulus in terms of π. The result is always π, independent of the number of sides of the polygon.

The formal derivation of the result relies on calculating the area of each circle, factorising the result, and then substituting the length of the side into the result.

### Squaring the rectangle

On the left, a rectangle is given and the challenge is to construct a square to the right of it that has an equal area and sharing a common vertex with the rectangle. The construction must be made by using pencil, compass and straight edge.

When the left hand edge of the rectangle is fixed, and the height of the rectangle is increased, the top left hand vertex of the resulting squares traces out a circle.

### Two crescents and a triangle

This resource contains a diagram made from a circle, four semi-circular arcs and a rectangle with dimensions given in surd form. The challenge is to find the total area of a given shaded area.

The student worksheets each contain different dimensions for the diagram, but the solutions all have something in common.

The solution involves using the area of a circle and Pythagoras’ theorem. As an extension, students can be shown that Pythagoras’ theorem can be used with other similar shapes other than squares.