Year 6: Shapes, angles, circles and coordinates
Part of the Year 6 sats maths revision chapter. Chapter overview · whole chapter on one page
Shapes, angles, circles and coordinates
Angles are measured in degrees. A straight line is 180 degrees, a full turn around a point is 360 degrees, and the angles in a triangle add up to 180 degrees. A quadrilateral has angles that add up to 360 degrees.
Points on the coordinate grid are given as (across, up). Points on the left of the vertical axis have a negative across number. A reflection flips a shape over an axis, and every point stays the same distance from the mirror line. A translation slides a shape without turning it.
A circle has a radius from the centre to the edge, and a diameter across the middle. The diameter is twice the radius. The circumference is pi times the diameter, and we use 3.14 for pi.
Key ideas
- Angles on a straight line add up to 180 degrees; angles at a point add up to 360 degrees.
- The angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
- Opposite angles made by two crossing lines are equal.
- Reflection: one number changes sign. Translation: add the movement to each number.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Use the fact about the total, add the angles you know, then take them away from the total.
- Write the fact. Angles on a straight line add up to 180 degrees.the total is 180°
- Add the angles you know. Add the angles that are given.112 = 112°
- Take the total away. The missing angle b is what is left over.180 − 112 = 68
Check. 68 + 112 = 180, so the missing angle is right. ✓
Watch out. Use the total from the fact, not 90 or 360 by mistake.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Find the total of the interior angles, then share it equally between the angles of the regular shape.
- Write the number of sides. A regular shape has the same number of sides and angles.n = 8 sides
- Find the total of the interior angles. A shape with 8 sides can be split into 6 triangles, and each triangle has 180 degrees.(8 − 2) × 180 = 1080°
- Share the total equally. A regular shape has 8 angles that are all the same.1080 ÷ 8 = 135
Check. 135 × 8 = 1080, so the answer is right. ✓
Watch out. Use the number of sides minus 2, not the number of sides, when you find the total.
Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.
Method. Reflect the point across the axis: the distance from the axis stays the same, and the point is on the other side.
- Write the point. Read the across number first, then the up number.(2, -5)
- Reflect the point. Reflecting in the x-axis keeps the across number and changes the sign of the up number.(2, 5)
- Write the answer. The reflected point is the same distance from the mirror line on the other side.(2, 5)
Check. The reflected point is the same distance from the axis, on the opposite side. ✓
Watch out. Only one number changes sign. For the x-axis, the up number changes sign.
Watch out
Common mistake: reflecting in the x-axis changes the sign of the up number, not the across number. Also, an angle at a point is 360 degrees, not 180.
Exercise 13F Timed set: angles, circles and coordinates (about 12 minutes)
Non-calculatorKey words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Use the fact about the total, add the angles you know, then take them away from the total.
- Write the fact. Angles on a straight line add up to 180 degrees.the total is 180°
- Add the angles you know. Add the angles that are given.67 = 67°
- Take the total away. The missing angle a is what is left over.180 − 67 = 113
Check. 113 + 67 = 180, so the missing angle is right. ✓
Watch out. Use the total from the fact, not 90 or 360 by mistake.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Use the fact about the total, add the angles you know, then take them away from the total.
- Write the fact. Angles at a point add up to 360 degrees.the total is 360°
- Add the angles you know. Add the angles that are given.90 + 128 + 75 = 293°
- Take the total away. The missing angle x is what is left over.360 − 293 = 67
Check. 67 + 293 = 360, so the missing angle is right. ✓
Watch out. Use the total from the fact, not 90 or 360 by mistake.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Use the fact about the total, add the angles you know, then take them away from the total.
- Write the fact. Angles in a triangle add up to 180 degrees.the total is 180°
- Add the angles you know. Add the angles that are given.48 + 67 = 115°
- Take the total away. The missing angle x is what is left over.180 − 115 = 65
Check. 65 + 115 = 180, so the missing angle is right. ✓
Watch out. Use the total from the fact, not 90 or 360 by mistake.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Use the fact about the total, add the angles you know, then take them away from the total.
- Write the fact. Angles in a quadrilateral add up to 360 degrees.the total is 360°
- Add the angles you know. Add the angles that are given.110 + 75 + 80 = 265°
- Take the total away. The missing angle x is what is left over.360 − 265 = 95
Check. 95 + 265 = 360, so the missing angle is right. ✓
Watch out. Use the total from the fact, not 90 or 360 by mistake.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Vertically opposite angles are equal.
- Look at the two lines. The two lines cross at one point, making four angles.one angle is 58°
- Find the angle opposite. Angles that are opposite each other when two lines cross are equal.opposite angle = 58°
- Write the answer. 58
Check. The angle next to it is 122 degrees, and 58 + 122 = 180, which is a straight line. ✓
Watch out. Opposite angles are equal. Angles next to each other add up to 180 degrees.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Double the radius to find the diameter.
- Write the rule. The diameter goes right across the circle through the centre, so it is twice the radius.diameter = 2 × radius
- Multiply the radius by 2. Double the radius to get the diameter.2 × 6.5 = 13
Check. The radius is half the diameter: 13 ÷ 2 = 6.5. ✓
Watch out. The radius is half the diameter. Do not halve the radius to find the diameter.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Find the total of the interior angles, then share it equally between the angles of the regular shape.
- Write the number of sides. A regular shape has the same number of sides and angles.n = 6 sides
- Find the total of the interior angles. A shape with 6 sides can be split into 4 triangles, and each triangle has 180 degrees.(6 − 2) × 180 = 720°
- Share the total equally. A regular shape has 6 angles that are all the same.720 ÷ 6 = 120
Check. 120 × 6 = 720, so the answer is right. ✓
Watch out. Use the number of sides minus 2, not the number of sides, when you find the total.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Find the total of the interior angles, then share it equally between the angles of the regular shape.
- Write the number of sides. A regular shape has the same number of sides and angles.n = 5 sides
- Find the total of the interior angles. A shape with 5 sides can be split into 3 triangles, and each triangle has 180 degrees.(5 − 2) × 180 = 540°
- Share the total equally. A regular shape has 5 angles that are all the same.540 ÷ 5 = 108
Check. 108 × 5 = 540, so the answer is right. ✓
Watch out. Use the number of sides minus 2, not the number of sides, when you find the total.
Key words. Angles on a straight line add up to 180 degrees. Angles at a point add up to 360 degrees. Angles in a triangle add up to 180 degrees, and in a quadrilateral they add up to 360 degrees.
Method. Multiply pi, which is 3.14, by the diameter.
- Write the formula. The circumference is pi times the diameter. Use 3.14 for pi.C = 3.14 × 14
- Multiply. Multiply 3.14 by the diameter.3.14 × 14 = 43.96
Check. The circumference is a little more than 3 times the diameter, and 43.96 is about 3 × 14. ✓
Watch out. Use the diameter, not the radius. If you are given the radius, double it first.
Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.
Method. Reflect the point across the axis: the distance from the axis stays the same, and the point is on the other side.
- Write the point. Read the across number first, then the up number.(-3, 4)
- Reflect the point. Reflecting in the y-axis changes the sign of the across number and keeps the up number.(3, 4)
- Write the answer. The reflected point is the same distance from the mirror line on the other side.(3, 4)
Check. The reflected point is the same distance from the axis, on the opposite side. ✓
Watch out. Only one number changes sign. For the x-axis, the up number changes sign.
Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.
Method. Reflect the point across the axis: the distance from the axis stays the same, and the point is on the other side.
- Write the point. Read the across number first, then the up number.(5, -2)
- Reflect the point. Reflecting in the x-axis keeps the across number and changes the sign of the up number.(5, 2)
- Write the answer. The reflected point is the same distance from the mirror line on the other side.(5, 2)
Check. The reflected point is the same distance from the axis, on the opposite side. ✓
Watch out. Only one number changes sign. For the x-axis, the up number changes sign.
Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.
Method. Move across by the first number, then up or down by the second number.
- Move across. A positive number means right, a negative number means left.-2 + 4 = 2
- Move up or down. A positive number means up, a negative number means down.3 − 5 = -2
- Write the answer. (2, -2)
Check. Reversing both moves takes (2, -2) back to (-2, 3). ✓
Watch out. Move across first, then up or down. Keep the order of the numbers in the bracket.
Key words. A coordinate gives a point as (across, up). A reflection flips a shape over a line, and a translation slides it.
Method. Find the x-value and the y-value that appear only once among the three corners.
- Look at the x-values. Two corners share the same across number.x-values: 2, 6, 6
- Look at the y-values. Two corners share the same up number.y-values: 2, 2, 6
- Find the missing corner. The fourth corner uses the number that appears only once for x and for y.x = 2 and y = 6
- Write the answer. (2, 6)
Check. The side from (2, 6) to each of the other corners is parallel to an axis. ✓
Watch out. Find the number that only appears once in each list. That is the missing corner.