Compass and circles
Meena makes a round rangoli by tying a thread to a peg and turning a chalk round it. A gardener fixes a rope to a pole and walks round to mark a round flower bed. In both, every point on the edge is the same distance from one fixed point.
A circle is made of all the points that are the same distance from a fixed point called the centre. That distance is the radius, and a compass keeps it fixed when you open its two arms. A line through the centre that touches the circle at both ends is a diameter, and it is twice the radius.
In the figure: The radius goes from the centre O to the circle; the diameter goes right across through O.
Rule 1: Radius and diameter — All radii of one circle are equal. The diameter is twice the radius, so the radius is half the diameter. The compass opening is the radius, not the diameter.
Compass and circles
In the app you can move this figure with your finger.
Open in the app- Easy: Radius 6 cm: the diameter is 6 × 2 = 12 cm.
- Medium: Diameter 18 m: the radius is 18 ÷ 2 = 9 m.
- Exam-type: Compass opened to 5 cm: the farthest two points of the circle can be is 5 × 2 = 10 cm apart.
Rule 2: Inside, on or outside — Compare the distance of a point from the centre with the radius. Less than the radius means inside, equal means on the circle, more means outside.
- Easy: Radius 5 cm, point 3 cm from the centre: 3 is less than 5, so it is inside.
- Medium: Radius 5 cm, point 5 cm from the centre: equal, so it is on the circle.
- Exam-type: Radius 5 cm, point 8 cm from the centre: 8 is more than 5, so it is outside.
The diameter is the longest straight line you can draw across a circle, because it passes through the centre.
Taking the compass opening as the diameter, or finding the diameter by halving instead of doubling.
Diameter = radius doubled; radius = diameter halved. Memory clue: “Radius = Reach from the centre; Diameter = Double”.
Spot the difference
| Word | Meaning | Example |
|---|---|---|
| Centre | fixed point O | O |
| Radius | centre to the circle | 4 cm |
| Diameter | across through the centre | 8 cm |
| Compass opening | sets the radius | 4 cm |
Practice questions
- 1A circle has radius 7 cm. What is its diameter?
- 2A circular pond has diameter 20 m. What is its radius?
- 3A compass is opened to 9 cm to draw a circle. What is the diameter of the circle?
- 4A circle has diameter 26 cm. A point P is 12 cm from the centre. Is P inside, on or outside the circle?
- 5A compass is opened to 6 cm. What is the greatest distance between two points on the circle it draws?
- 6A circle has radius 8 cm. Points P, Q and R are 5 cm, 8 cm and 11 cm from the centre. Which point lies on the circle?
- 7A circle has radius 9 cm and a point lies 17 cm from its centre. How far is that point from the nearest point of the circle?
- 8A round pond has diameter 30 m. A 2 m wide path is built all round it, outside. What is the radius of the outer edge of the path?
Show answers (try them yourself first)
- 11) 14 cm
- 22) 10 m
- 33) 18 cm
- 44) Inside (12 is less than 13)
- 55) 12 cm
- 66) Q
- 77) 8 cm (17 − 9)
- 88) 17 m (15 + 2)
Properties of squares and rectangles
A carpenter checks a new window frame. He measures both pairs of opposite sides and fits a set square into each corner. If the opposite sides match and every corner fits, the frame is a true rectangle.
A rectangle has four right angles (90°), and its opposite sides are equal and parallel. A square is a rectangle whose four sides are all equal. So every square is a rectangle, but a rectangle with unequal sides is not a square.
In the figure: In both shapes the corners are right angles; in the square all four sides are also equal.
Rule 1: Sides and angles — In a rectangle the opposite sides are equal and all four angles are 90°. Turning the shape does not change either fact.
Properties of squares and rectangles
In the app you can move this figure with your finger.
Open in the app- Easy: Rectangle 9 cm by 5 cm: the other two sides are 9 cm and 5 cm, and the perimeter is 2 × 14 = 28 cm.
- Medium: Rectangle ABCD with AB = 12 cm and BC = 7 cm: CD = 12 cm and AD = 7 cm.
- Exam-type: Square with side 6 cm: all four sides are 6 cm, all four angles are 90°, perimeter 6 × 4 = 24 cm.
Rule 2: Rectangle or square? — If the angles are 90° and only the opposite sides are equal, it is a rectangle. If all four sides are equal as well, it is a square. Four equal sides alone are not enough, the angles must be 90°.
- Easy: Sides 8 cm, 8 cm, 8 cm, 8 cm with right angles: a square.
- Medium: Sides 8 cm and 5 cm with right angles: a rectangle, not a square.
- Exam-type: Four sides of 5 cm but angles 70° and 110°: not a square (it is a rhombus).
A square has every property of a rectangle, plus all its sides are equal.
Saying a square is not a rectangle, or thinking a shape with four equal sides must be a square.
Square = rectangle + all four sides equal. Memory clue: “Rectangle = 4 right angles; Square = 4 right angles + 4 equal sides”.
Spot the difference
| Property | Rectangle | Square |
|---|---|---|
| Opposite sides | equal | equal |
| All four sides | usually unequal | always equal |
| Angles | four of 90° | four of 90° |
| Example sides | 9 cm and 5 cm | 6 cm and 6 cm |
Practice questions
- 1Rectangle ABCD has AB = 11 cm and BC = 6 cm. Write the lengths of CD and AD.
- 2How many right angles does a square have?
- 3A square has side 9 cm. What is its perimeter?
- 4A rectangular door is 2 m wide and its perimeter is 10 m. How tall is the door?
- 5A rectangle has two neighbouring sides of 8 cm and 8 cm. What special name does it have?
- 6Four rods of 5 cm each are joined, and one corner measures 80°. Is it a square?
- 7A square and a rectangle both have perimeter 40 cm. The rectangle is 14 cm long. Find the rectangle’s width and the square’s side.
- 8A rectangle has perimeter 34 cm and its length is 5 cm more than its width. Find the width and the length.
Show answers (try them yourself first)
- 11) CD = 11 cm, AD = 6 cm
- 22) 4
- 33) 36 cm
- 44) 3 m
- 55) A square
- 66) No, a corner of 80° is not a right angle
- 77) Width 6 cm, square side 10 cm
- 88) Width 6 cm, length 11 cm
Constructing squares and rectangles
A tailor marks a rectangular piece of cloth before cutting. She draws one straight edge, makes a square corner at each end with a set square, and measures the same height up both sides. Careful drawing gives a true rectangle.
To construct a rectangle means to draw it exactly from given measurements. Draw the base AB equal to the length, make a right angle at A and at B, measure the width along both, mark D and C, and join D and C. A square is built the same way with the width equal to the length.
In the figure: The four steps: base, right angles, mark the height, join.
Rule 1: A rectangle in four steps — Step 1: draw AB. Step 2: make right angles at A and B. Step 3: measure the width up both lines to mark D and C. Step 4: join D and C. The side DC then measures the same as AB.
Constructing squares and rectangles
In the app you can move this figure with your finger.
Open in the app- Easy: Rectangle 7 cm by 4 cm: AB = 7 cm, then AD = BC = 4 cm; DC will also be 7 cm.
- Medium: Rectangle 5 cm by 3 cm: base 5 cm, height 3 cm on both sides, then join the tops.
- Exam-type: Check: if DC measures 7.4 cm instead of 7 cm, the angles were not exactly 90°, so redraw.
Rule 2: How many measurements? — A rectangle needs two measurements, its length and its width. A square needs only one, its side, because all sides are equal. If you are given the perimeter, first find the missing measurement.
- Easy: Rectangle 8 cm by 3 cm: two measurements are needed.
- Medium: Square of side 5 cm: one measurement is enough, draw AB = 5 cm and raise 5 cm on both ends.
- Exam-type: Rectangle with perimeter 20 cm and length 6 cm: half of 20 is 10 and 10 − 6 = 4, so draw 6 cm by 4 cm.
Draw thin pencil lines first and check the angles; a small slip at the corner makes the last side the wrong length.
Measuring the height from the wrong point, or drawing the two side lines at an angle that is not exactly 90°.
Base, two right angles, two equal heights, join the tops. Memory clue: “Base – Right angles – Height – Join”.
Spot the difference
| Shape | Measurements needed | Example |
|---|---|---|
| Rectangle | length and width | 7 cm and 4 cm |
| Square | one side | 6 cm |
| Check DC | equals AB | 7 cm = 7 cm |
| Check corners | each 90° | set square fits |
Practice questions
- 1How many measurements are needed to draw a square?
- 2You draw a rectangle with AB = 6 cm and BC = 3 cm. How long should side DC be?
- 3In rectangle ABCD, AB = 9 cm and BC = 5 cm. How long should AD be?
- 4A rectangle has perimeter 24 cm and width 5 cm. What length of AB should you draw?
- 5A square has perimeter 28 cm. What side should you draw?
- 6You draw a rectangle with AB = 8 cm and BC = 4 cm, but DC measures 8.3 cm. Is something wrong?
- 7A rectangle must have area 30 square cm and length 6 cm. What width do you draw?
- 8A rectangle has sides 10 cm and 6 cm. A square has the same perimeter. What is the side of the square?
Show answers (try them yourself first)
- 11) 1
- 22) 6 cm
- 33) 5 cm
- 44) 7 cm (12 − 5)
- 55) 7 cm (28 ÷ 4)
- 66) Yes, DC should equal AB, so the angles were not exactly 90°
- 77) 5 cm (30 ÷ 6)
- 88) 8 cm (perimeter 32, 32 ÷ 4)
Diagonals of rectangles and squares
A carpenter nails a slanting plank across a wooden gate so that the gate does not sag. The plank joins two opposite corners, and a second plank from the other two corners would cross it exactly in the middle.
A diagonal is a line segment joining two opposite corners. A rectangle has two diagonals, and they are equal in length and cut each other at the middle point O. In a square the diagonals are equal too, and they also cross at a right angle.
In the figure: Rectangle ABCD: the diagonals AC and BD are equal and meet at O, the middle of both.
Rule 1: Diagonals of a rectangle — Both diagonals of a rectangle are equal, and each is cut into two equal halves at O. So if you know one half, double it to get the whole diagonal.
Diagonals of rectangles and squares
In the app you can move this figure with your finger.
Open in the app- Easy: One diagonal of a rectangle is 10 cm, so the other diagonal is also 10 cm.
- Medium: AO = 6 cm, so AC = 6 × 2 = 12 cm, BD = 12 cm and BO = 6 cm.
- Exam-type: Diagonal 18 cm: each half is 18 ÷ 2 = 9 cm.
Rule 2: Squares, and a line across a rectangle — The diagonals of a square are equal and cross at 90°. Now join a point on one side of a rectangle to a point on the opposite side: this line is shortest when it goes straight across and longest when it is a diagonal.
- Easy: Square with diagonal 14 cm: the diagonals meet at O at 90°, and each half is 7 cm.
- Medium: Rectangle 9 cm by 5 cm: a line from one 5 cm side to the opposite 5 cm side is never shorter than 9 cm.
- Exam-type: Rectangle 12 cm by 5 cm with diagonal 13 cm: such a line is always between 12 cm and 13 cm.
Only the line between opposite corners is a diagonal; the sides of the shape are never diagonals.
Thinking the diagonals of every rectangle cross at a right angle; that happens only in a square.
Rectangle: diagonals equal and halve each other. Square: also at 90°. Memory clue: “Diagonals: Duplicates (equal) and Divided in the middle”.
Spot the difference
| Figure | Diagonals equal? | Cross at 90°? |
|---|---|---|
| Rectangle | yes | no |
| Square | yes | yes |
| Parallelogram | no | no |
| Kite | no | yes |
Practice questions
- 1A rectangle has diagonals AC and BD. If AC = 15 cm, what is BD?
- 2The diagonals of a rectangle meet at O. If AC = 16 cm, how long is AO?
- 3How many diagonals does a rectangle have?
- 4In a rectangle the diagonals meet at O and BO = 9 cm. Find the length of AC.
- 5At what angle do the diagonals of a square cross?
- 6A rectangle is 8 cm long and 6 cm wide and its diagonal is 10 cm. X is on one 6 cm side and Y is on the opposite 6 cm side. Between which two lengths does XY lie?
- 7The two diagonals of a rectangle together measure 36 cm. How long is AO, where O is the meeting point?
- 8Can a rectangle have diagonals of 10 cm and 12 cm? Give a reason.
Show answers (try them yourself first)
- 11) 15 cm
- 22) 8 cm
- 33) 2
- 44) 18 cm
- 55) 90°
- 66) Between 8 cm and 10 cm
- 77) 9 cm (each diagonal is 18 cm)
- 88) No, the diagonals of a rectangle are always equal
Points equidistant from two points
Two villages want one common well that is the same distance from both. A bus stop may be placed so that two schools are equally far from it. Such a place is found by measuring with a compass from both ends.
A point is equidistant from B and C when it is the same distance from both. To find such points, draw an arc of the same radius from B and from C. Where the arcs cross, the point is the same distance from B and from C. The arcs meet only if the radius is more than half of BC.
In the figure: Arcs from B and C with equal radius meet at A and A’, and AB = AC.
Rule 1: Arcs from both ends — Keep the compass opening the same for both arcs. If the radius is less than half of BC the arcs do not meet; if it is exactly half they touch at the middle of BC; if it is more they meet at two points, one on each side of BC.
Points equidistant from two points
In the app you can move this figure with your finger.
Open in the app- Easy: BC = 8 cm, radius 5 cm: more than half (4), so the arcs meet at two points, each 5 cm from B and C.
- Medium: BC = 8 cm, radius 4 cm: exactly half, so the arcs just touch at the middle of BC.
- Exam-type: BC = 8 cm, radius 3 cm: less than half, so 3 + 3 = 6 is shorter than 8 and the arcs never meet.
Rule 2: A triangle from three sides — Draw the base BC. Draw an arc of radius AB from B and an arc of radius AC from C. They meet at A. This works only when the two shorter sides together are longer than the third side.
- Easy: Sides 6 cm, 5 cm and 4 cm: base 6 cm, arcs of 5 cm and 4 cm meet, since 5 + 4 = 9 is more than 6.
- Medium: Sides 3 cm, 4 cm and 8 cm: 3 + 4 = 7 is less than 8, so no triangle can be made.
- Exam-type: Sides 5 cm, 5 cm and 5 cm: base 5 cm and two arcs of 5 cm give a triangle with all sides equal.
For two arcs to meet, the radii added together must be more than the distance BC.
Changing the compass opening between the two arcs, or forgetting that there is a second point on the other side of BC.
Same radius from B and from C: where the arcs cross is equally far from both. Memory clue: “Equal radius from both ends gives equal distance”.
Spot the difference
| Radius (BC = 10 cm) | Compared with half of BC | Result |
|---|---|---|
| 3 cm | less than 5 | arcs do not meet |
| 5 cm | equal to 5 | touch at the middle |
| 6 cm | more than 5 | meet at two points |
| 8 cm | more than 5 | two points, farther apart |
Practice questions
- 1B and C are 12 cm apart. Arcs of radius 6 cm are drawn from both. What happens?
- 2A point A is equidistant from B and C and AB = 7 cm. What is AC?
- 3BC = 10 cm. Can arcs of radius 4 cm drawn from B and C meet?
- 4BC = 6 cm. What is the smallest whole-number radius for which arcs from B and C meet at two points?
- 5To draw a triangle with sides 7 cm, 5 cm and 4 cm, you take 7 cm as the base. What radii do you use for the two arcs?
- 6Can a triangle have sides 2 cm, 3 cm and 6 cm?
- 7B and C are 8 cm apart. A and A’ are the two points that are 5 cm from both. The line AA’ cuts BC at its middle M, and AM = 3 cm. How long is AA’?
- 8Two sides of a triangle are 9 cm and 5 cm. What are the smallest and largest whole-number lengths the third side can have?
Show answers (try them yourself first)
- 11) The arcs just touch at the middle of BC
- 22) 7 cm
- 33) No (4 + 4 = 8 is less than 10)
- 44) 4 cm
- 55) 5 cm and 4 cm
- 66) No (2 + 3 = 5 is less than 6)
- 77) 6 cm (3 + 3)
- 88) 5 cm and 13 cm