GCSE Maths: Nets of solids (Higher)
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Nets of solids
A net is a flat pattern that folds up to make a solid. Imagine cutting along some edges of a cardboard box and unfolding it flat: the flat shape you get is a net. Nets show you every face of a solid at once, which makes them very useful for finding surface areas (Section 19.7).
Every net is made from the faces of the solid. A cuboid has six rectangular faces, so its net has six rectangles. A triangular prism has two triangles and three rectangles. A square-based pyramid has one square and four triangles. A cylinder has two circles and one rectangle. Counting the faces helps you decide which solid a net makes.
Nets of a cube. A cube has 6 identical square faces, so its net is 6 squares joined edge to edge. But not every arrangement of 6 squares folds into a cube. There are exactly 11 different nets of a cube. To test a net, imagine folding it: if two squares land on the same face, it fails. Squares in a straight row of four wrap round the cube like a belt, so a row of 5 or a 2 × 3 block never works.
Nets of cylinders. If you peel the label off a tin of soup and lay it flat, you get a rectangle. Its height is the height of the tin. Its width is the distance once round the tin, which is the circumference of the circular end, 2πr. So a cylinder’s net is a rectangle (2πr by h) with a circle of radius r touching the top edge and another touching the bottom edge.
Net of a cone. The curved surface of a cone opens out into a sector (a slice of a circle) whose radius is the cone’s slant height l. The arc of the sector is exactly as long as the circumference of the base circle. Its angle is 360 × r ÷ l degrees, where r is the base radius.
Key ideas
- A net is a flat pattern that folds to make a solid; it shows every face.
- There are 11 different nets of a cube. If two squares fold onto the same face, the net fails.
- Cuboid: 6 rectangles. Triangular prism: 2 triangles + 3 rectangles. Square-based pyramid: 1 square + 4 triangles. Cylinder: 2 circles + 1 rectangle.
- The rectangle in the net of a cylinder has width 2πr (the circumference) and height h.
- Higher: the sector in the net of a cone has radius l and angle 360 × r ÷ l.
Method. Mentally fold the net up. A net makes a cube when all six squares become six different faces.
- Pick one square to be the bottom of the cube. Imagine holding the first square flat on a table and folding the other squares up around it, like a paper box.
- Fold each neighbouring square up, one at a time. Every square must become a different face of the cube: bottom, top, front, back, left, right.
- Check that no two squares land on the same face. If all six squares go to six different faces, the net works.Pairs of opposite faces: A is opposite D; B is opposite E; C is opposite F
Check. There are exactly 11 different nets of a cube. This one is among them. ✓
Watch out. Rows of 4 squares with one square above and one below always work; a row of 5 or a 2 × 3 block never does.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.5 × 3 = 15 (and there are 2 of them, so 30)
- Front and back. Each is length × height.5 × 2 = 10 (2 of them: 20)
- Left and right. Each is width × height.3 × 2 = 6 (2 of them: 12)
- Add the three pairs. The surface area is every face added together.30 + 20 + 12 = 62
- Write the unit. Area, so square units.Surface area = 62 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (15 + 10 + 6) = 2 × 31 = 62. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Method. The rectangle in the net of a cylinder is as wide as the circumference of the circular end.
- Think about how the rectangle wraps round. When the net is folded up, the long edge of the rectangle goes right round the circular end once. So it is as long as the circumference of the circle.
- Use the circumference formula. Circumference = 2πr (or πd).C = 2 × π × 3 = 6π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Width = 6π cm
Check. Circumference is a bit more than 3 × the diameter: 3 × 6 = 18. ✓
Watch out. Use the circumference (2πr) for the width of the rectangle, not the area of the circle.
Method. The arc of the sector is as long as the circumference of the base. Find what fraction of the big circle (radius l) the arc is, then find that fraction of 360°.
- Circumference of the base. The base circle has radius 3.2 × π × 3 = 6π cm
- Circumference of the full circle of radius l = 5. If the sector were a whole circle, its arc would be this long.2 × π × 5 = 10π cm
- Find the fraction of the full circle. The arc is 6π out of 10π.6π ÷ 10π = 6 ÷ 10 = 3 ÷ 5
- Find that fraction of 360°. 3 ÷ 5 × 360 = 216°
Check. Formula: 360 × r ÷ l = 360 × 3 ÷ 5 = 216. ✓
Watch out. The radius of the sector is the slant height l, not the base radius r.
Watch out
Squares that touch in a net do not always touch on the cube, and squares that look far apart can be opposite. Always fold the net in your head one square at a time. For a cylinder the width of the rectangle is the circumference (2πr), not the radius and not the area of the circle.Exercise 19B Nets of solids
Non-calculatorMethod. Mentally fold the net up. A net makes a cube when all six squares become six different faces.
- Pick one square to be the bottom of the cube. Imagine holding the first square flat on a table and folding the other squares up around it, like a paper box.
- Fold each neighbouring square up, one at a time. Every square must become a different face of the cube: bottom, top, front, back, left, right.
- Check that no two squares land on the same face. If all six squares go to six different faces, the net works.Pairs of opposite faces: A is opposite F; C is opposite E; D is opposite B
Check. There are exactly 11 different nets of a cube. This one is among them. ✓
Watch out. Rows of 4 squares with one square above and one below always work; a row of 5 or a 2 × 3 block never does.
Method. Mentally fold the net up. If two squares land on the same face, the net does not make a cube.
- Pick one square to be the bottom of the cube. Imagine holding it flat on a table and folding the other squares up around it.
- Fold the other squares round, one at a time. Each square becomes a face of the cube. We follow where each one lands.
- Look for squares that land on top of each other. A cube has 6 different faces. If two squares fold onto the same face, one face is covered twice and another face is left open.Squares C and D fold onto the same face, so there is a hole in the cube.
Check. Count the faces round the net: a cube needs each of its 6 faces exactly once. ✓
Watch out. A net needs more than six squares joined together: they must also fold into six different faces.
Method. Mentally fold the net up. A net makes a cube when all six squares become six different faces.
- Pick one square to be the bottom of the cube. Imagine holding the first square flat on a table and folding the other squares up around it, like a paper box.
- Fold each neighbouring square up, one at a time. Every square must become a different face of the cube: bottom, top, front, back, left, right.
- Check that no two squares land on the same face. If all six squares go to six different faces, the net works.Pairs of opposite faces: A is opposite D; B is opposite E; C is opposite F
Check. There are exactly 11 different nets of a cube. This one is among them. ✓
Watch out. Rows of 4 squares with one square above and one below always work; a row of 5 or a 2 × 3 block never does.
Method. Mentally fold the net up. If two squares land on the same face, the net does not make a cube.
- Pick one square to be the bottom of the cube. Imagine holding it flat on a table and folding the other squares up around it.
- Fold the other squares round, one at a time. Each square becomes a face of the cube. We follow where each one lands.
- Look for squares that land on top of each other. A cube has 6 different faces. If two squares fold onto the same face, one face is covered twice and another face is left open.Squares A and E fold onto the same face, so there is a hole in the cube.
Check. Count the faces round the net: a cube needs each of its 6 faces exactly once. ✓
Watch out. A net needs more than six squares joined together: they must also fold into six different faces.
Method. Fold the net into a cube in your head. Two faces are opposite if there is exactly one face between them going round the cube, so they can never touch.
- Choose one face to be the bottom. Fold the rest up around it.
- Find the face that folds up to be directly opposite face A. The opposite face is the one that is not next to it. In the net, squares that are two apart in a straight row are opposite.Opposite pairs: A–F, B–D, C–E
- Read off the answer. The face opposite A is F.
Check. Squares in a straight line with exactly one square between them are opposite. ✓
Watch out. Touching in the net does not always mean touching on the cube, and being far apart in the net does not mean opposite.
Method. Fold the net into a cube in your head. Two faces are opposite if there is exactly one face between them going round the cube, so they can never touch.
- Choose one face to be the bottom. Fold the rest up around it.
- Find the face that folds up to be directly opposite face B. The opposite face is the one that is not next to it. In the net, squares that are two apart in a straight row are opposite.Opposite pairs: A–F, B–D, C–E
- Read off the answer. The face opposite B is D.
Check. Squares in a straight line with exactly one square between them are opposite. ✓
Watch out. Touching in the net does not always mean touching on the cube, and being far apart in the net does not mean opposite.
Method. Fold the net into a cube in your head. Two faces are opposite if there is exactly one face between them going round the cube, so they can never touch.
- Choose one face to be the bottom. Fold the rest up around it.
- Find the face that folds up to be directly opposite face F. The opposite face is the one that is not next to it. In the net, squares that are two apart in a straight row are opposite.Opposite pairs: A–F, B–D, C–E
- Read off the answer. The face opposite F is A.
Check. Squares in a straight line with exactly one square between them are opposite. ✓
Watch out. Touching in the net does not always mean touching on the cube, and being far apart in the net does not mean opposite.
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has six rectangles.
- Name the solid. The net makes a cuboid.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has two triangles and three rectangles.
- Name the solid. The net makes a triangular prism.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has one square and four triangles.
- Name the solid. The net makes a square-based pyramid.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. Count the shapes in the net and match them to a solid.
- Describe the net. It has two circles and one rectangle.
- Name the solid. The net makes a cylinder.
Check. Compare the number of each kind of face with the list in the Key ideas box. ✓
Method. A prism has two end faces and a rectangle for each side of the end.
- Count the ends. The ends are triangles.2 triangles
- Count the rectangles. A triangle has 3 sides, so there are 3 rectangles.3 rectangles
Check. 2 + 3 = 5 faces. ✓
Method. A pyramid has one base and a triangle on each side of the base.
- Count the base. The base is a square.1 square
- Count the triangles. A square has 4 sides.4 triangles
Check. 1 + 4 = 5 faces. ✓
Method. A prism has two end faces and a rectangle for each side of the end.
- Count the ends. The ends are pentagons.2 pentagons
- Count the rectangles. A pentagon has 5 sides.5 rectangles
Check. 2 + 5 = 7 faces; Euler gives 10 − 15 + 7 = 2. ✓
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the six faces. A cuboid has 6 rectangular faces that come in 3 matching pairs: top and bottom, front and back, left and right.
- Top and bottom. Each is length × width.6 × 4 = 24 (and there are 2 of them, so 48)
- Front and back. Each is length × height.6 × 3 = 18 (2 of them: 36)
- Left and right. Each is width × height.4 × 3 = 12 (2 of them: 24)
- Add the three pairs. The surface area is every face added together.48 + 36 + 24 = 108
- Write the unit. Area, so square units.Surface area = 108 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (24 + 18 + 12) = 2 × 54 = 108. ✓
Watch out. Count every face: forgetting one pair of faces is the most common error. Give the answer in cm2, not cm3.
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. A cube has 6 identical square faces. Find the area of one face and multiply by 6.
- Area of one face. Each face is a square with side equal to the edge length.5 × 5 = 25
- Multiply by the number of faces. A cube has 6 faces.6 × 25 = 150
- Write the unit. Surface area = 150 cm2
Check. Formula 6a2 = 6 × 52 = 6 × 25 = 150. ✓
Watch out. Do not confuse with the volume: volume cubes the side (a3), surface area squares it (6a2).
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. Use the perpendicular height of the triangle, not the sloping side.½ × 6 × 4 = 12 cm2
- Double it for the two ends. The two end faces are identical.2 × 12 = 24 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle under the base: 6 × 10 = 60
First sloping rectangle: 5 × 10 = 50
Second sloping rectangle: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.24 + 60 + 50 + 50 = 184
- Write the unit. Area, so square units.Surface area = 184 cm2
Check. The rectangles together make (perimeter of an end) × length = 16 × 10 = 160. Then 24 + 160 = 184. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Method. The rectangle in the net of a cylinder is as wide as the circumference of the circular end.
- Think about how the rectangle wraps round. When the net is folded up, the long edge of the rectangle goes right round the circular end once. So it is as long as the circumference of the circle.
- Use the circumference formula. Circumference = 2πr (or πd).C = 2 × π × 4 = 8π
- Write the answer in terms of π. Leave the answer in terms of π: write the number in front of π and do not multiply by 3.14.Width = 8π cm
Check. Circumference is a bit more than 3 × the diameter: 3 × 8 = 24. ✓
Watch out. Use the circumference (2πr) for the width of the rectangle, not the area of the circle.
Method. Replace π in 8π by 3.14 and multiply.
- Write the exact width from part (a). The width is the circumference.8π cm
- Replace π by 3.14. This gives an estimate of the width.8 × 3.14 = 25.12
Check. 8 × 3 = 24, and 25.12 is just over 24. ✓
Method. The label wraps once round the tin, so its length is the circumference.
- Write the circumference formula. C = 2πr.C = 2 × 22/7 × 3.5
- Work it out. 2 × 3.5 = 7, and 7 × 22/7 = 22.C = 7 × 22/7 = 22 cm
Check. The diameter is 7 cm and 7 × 22/7 = 22. ✓
Method. The label is a rectangle: length (the circumference) × height of the tin.
- Write the sizes. The length is 22 cm from part (a) and the height is 10 cm.22 cm by 10 cm
- Multiply. Area of a rectangle = length × width.22 × 10 = 220 cm2
Check. Area of curved surface = 2πrh = 2 × 22/7 × 3.5 × 10 = 220. ✓
Key words. Volume is the space inside a 3D shape. A cubic centimetre (cm3) is the volume of a cube with sides of 1 cm.
Method. Volume is the amount of space inside a solid. For a cuboid, multiply length × width × height.
- Write down the formula. A cuboid is built from layers of 1-unit cubes. One layer holds length × width cubes and the height tells us how many layers there are.Volume = length × width × height
- Substitute the lengths. All three lengths are in cm, so the units match.V = 8 × 5 × 3
- Multiply the length and the width first. This is the area of the base: the number of cubes in one layer.8 × 5 = 40
- Multiply by the height. This counts all the layers.40 × 3 = 120
- Write the answer with its unit. Three lengths multiplied together give a cubic unit.V = 120 cm3
Check. Multiply in a different order: 5 × 3 = 15, then 15 × 8 = 120. ✓
Watch out. Volume is in cm3, not cm or cm2. Do not add the lengths.
Method. Fold the net up in your head, starting from one end of the row.
- Fold the row of five. Four squares in a row wrap round to make the four side faces of the cube, like a belt.Squares 1, 2, 3 and 4 make the sides.
- Fold in the fifth square. It keeps going round the same belt, so it lands on top of square 1.Square 5 lands on square 1.
- State the conclusion. Two squares fold onto the same face, so one face of the cube is missing and the net fails.It is not a net of a cube. ✓
Check. The same test (fold each square once) shows it fails. ✓
Method. The three rectangles are 3, 4 and 5 cm wide and each is 10 cm long. Put them side by side.
- Add the widths. The widths are the three sides of the triangle.3 + 4 + 5 = 12 cm
- State the size. The rectangles share the same 10 cm length, so together they form one big rectangle.One rectangle 12 cm by 10 cm
Check. 12 × 10 = 120 = 30 + 40 + 50, the three separate areas. ✓
Key words. The surface area of a solid is the total area of all its faces added together. It is measured in square units (cm2), because it is an area.
Method. Surface area = the area of the two ends + the area of all the rectangular faces. Using the net helps you not to miss a face.
- Picture the faces. A prism has two identical ends (triangles) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Area of one triangular end. It is a right-angled triangle, so use the two shorter sides as base and height: ½ × base × height.½ × 3 × 4 = 6 cm2
- Double it for the two ends. The two end faces are identical.2 × 6 = 12 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the side of length 3: 3 × 10 = 30
Rectangle on the side of length 4: 4 × 10 = 40
Rectangle on the side of length 5: 5 × 10 = 50 - Add all the faces together. Surface area = the two ends + all the rectangles.12 + 30 + 40 + 50 = 132
- Write the unit. Area, so square units.Surface area = 132 cm2
Check. The rectangles together make (perimeter of an end) × length = 12 × 10 = 120. Then 12 + 120 = 132. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
Method. Use the area to find one face, then the edge length, then the volume.
- Find the area of one face. A cube has 6 identical faces.150 ÷ 6 = 25 cm2
- Find the edge length. Each face is a square, so edge × edge = 25. Since 5 × 5 = 25, the edge is 5 cm.Edge = 5 cm
- Find the volume. Volume of a cube = edge × edge × edge.5 × 5 × 5 = 125 cm3
Check. Surface area of a cube with edge 5 is 6 × 25 = 150. ✓
Watch out. Divide by 6 (faces), not by 4 or by 2.
Method. The arc of the sector equals the circumference of the base. Find what fraction of the circle of radius l it is, then take that fraction of 360°.
- Circumference of the base. 2π × 4 = 8π
- Circumference of the full circle of radius 10. The sector is part of this circle.2π × 10 = 20π
- Find the fraction. Divide the base circumference by the full circumference.8π ÷ 20π = 4 ÷ 10
- Find the angle. Take that fraction of a full turn.4 ÷ 10 × 360 = 144°
Check. Formula: 360 × r ÷ l = 360 × 4 ÷ 10 = 144. ✓
Watch out. Use the slant height l for the radius of the sector.
Method. The arc of the sector equals the circumference of the base. Find what fraction of the circle of radius l it is, then take that fraction of 360°.
- Circumference of the base. 2π × 6 = 12π
- Circumference of the full circle of radius 9. The sector is part of this circle.2π × 9 = 18π
- Find the fraction. Divide the base circumference by the full circumference.12π ÷ 18π = 6 ÷ 9
- Find the angle. Take that fraction of a full turn.6 ÷ 9 × 360 = 240°
Check. Formula: 360 × r ÷ l = 360 × 6 ÷ 9 = 240. ✓
Watch out. Use the slant height l for the radius of the sector.
Method. The arc of the sector equals the circumference of the base. Find what fraction of the circle of radius l it is, then take that fraction of 360°.
- Circumference of the base. 2π × 5 = 10π
- Circumference of the full circle of radius 12. The sector is part of this circle.2π × 12 = 24π
- Find the fraction. Divide the base circumference by the full circumference.10π ÷ 24π = 5 ÷ 12
- Find the angle. Take that fraction of a full turn.5 ÷ 12 × 360 = 150°
Check. Formula: 360 × r ÷ l = 360 × 5 ÷ 12 = 150. ✓
Watch out. Use the slant height l for the radius of the sector.
Method. The curved edge (arc) of the sector wraps round the base of the cone, so the arc length equals the circumference of the base.
- Arc length of the sector. A sector with angle x° is x/360 of a circle of radius l.Arc = (x ÷ 360) × 2πl
- Circumference of the base. C = 2πr
- Put them equal. The arc wraps exactly once round the base.(x ÷ 360) × 2πl = 2πr
- Solve for x. Divide both sides by 2πl, then multiply by 360.x ÷ 360 = r ÷ l, so x = 360r ÷ l ✓
Check. With r = 3 and l = 5: 360 × 3 ÷ 5 = 216, as in the worked example. ✓