GCSE Maths: Surface area of cuboids, prisms and cylinders (Foundation)
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Surface area of cuboids, prisms and cylinders
The surface area of a solid is the total area of all of its faces. It tells you how much wrapping paper you need for a present, or how much paint covers a tank. Because it is an area, it is measured in square units such as cm2 or m2. Do not mix it up with volume, which is the space inside (cubic units).
The best way to find a surface area is to think of the net (Section 19.2). The net shows every face flat, so you can work out each area and add.
Cuboid. There are six rectangles in three matching pairs. Surface area = 2lw + 2lh + 2wh. For a cube with edge a all six faces are the same: surface area = 6a2.
Prisms. A prism has two identical end faces (the cross-section) and a rectangle along each side. Each rectangle is one edge of the cross-section multiplied by the length of the prism. So
surface area = 2 × (area of cross-section) + (perimeter of cross-section) × length
The second part works because all the rectangles have the same length, so together they make one big rectangle: the perimeter by the length.
Cylinders. The net of a cylinder is two circles and a rectangle. The rectangle wraps round the side, so its width is the circumference 2πr and its height is h.
surface area of a cylinder = 2πr2 + 2πrh
The first part is the two circles, the second part is the curved surface. If the cylinder has no lid (an open can) there is only one circle, so the surface area is πr2 + 2πrh.
Key ideas
- Surface area = total area of all the faces, in square units.
- Cuboid: 2lw + 2lh + 2wh. Cube: 6a2.
- Prism: 2 × (cross-section area) + (perimeter of cross-section) × length.
- Cylinder: 2πr2 + 2πrh (curved surface 2πrh). An open cylinder has only one circle.
- Use the net to check that you have included every face.
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 × 4 = 20 (and there are 2 of them, so 40)
- Front and back. Each is length × height.5 × 3 = 15 (2 of them: 30)
- 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.40 + 30 + 24 = 94
- Write the unit. Area, so square units.Surface area = 94 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (20 + 15 + 12) = 2 × 47 = 94. ✓
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. 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).
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 (L-shapes) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 8 by 2
Top rectangle: 3 by (6 − 2) = 3 by 4 - Find the area of each rectangle. Width × height each time.8 × 2 = 16 cm2
3 × 4 = 12 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 16 + 12 = 28 cm2
- Double it for the two ends. The two end faces are identical.2 × 28 = 56 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Bottom edge: 8 × 10 = 80
Right edge: 2 × 10 = 20
Inner horizontal edge: 5 × 10 = 50
Inner vertical edge: 4 × 10 = 40
Top edge: 3 × 10 = 30
Left edge: 6 × 10 = 60 - Add all the faces together. Surface area = the two ends + all the rectangles.56 + 80 + 20 + 50 + 40 + 30 + 60 = 336
- Write the unit. Area, so square units.Surface area = 336 cm2
Check. The rectangles together make (perimeter of an end) × length = 28 × 10 = 280. Then 56 + 280 = 336. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
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 of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 32 = 18π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 3 × 10 = 60π
- Add them together. Total surface area = circles + curved surface.18π + 60π = 78π
- 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.Surface area = 78π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 3 × 13 = 78π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. Work out the area of the five faces (base and four sides, no lid) and multiply by the cost per square metre.
- Area of the base. length × width.2 × 1.5 = 3 m2
- Area of the two long sides. Each is length × height.2 × 1 = 2 m2, so 2 of them: 4 m2
- Area of the two short sides. Each is width × height.1.5 × 1 = 1.5 m2, so 2 of them: 3 m2
- Add to get the area to paint. There is no lid so we do not count one.3 + 4 + 3 = 10 m2
- Multiply by the cost. £4 for each m2.10 × 4 = £40
Check. A closed tank would be 13 m2; removing the lid (3 m2) leaves 10 m2. ✓
Watch out. Check whether the question includes the top. An open tank has 5 faces.
Watch out
Do not forget faces! Use the net to count them. A cuboid has six faces, a triangular prism has five, and a closed cylinder has two circles and one curved surface. Give the answer in square units, and do not confuse surface area with volume.Exercise 19H Surface area (non-calculator)
Non-calculatorKey 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 × 4 = 20 (and there are 2 of them, so 40)
- Front and back. Each is length × height.5 × 3 = 15 (2 of them: 30)
- 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.40 + 30 + 24 = 94
- Write the unit. Area, so square units.Surface area = 94 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (20 + 15 + 12) = 2 × 47 = 94. ✓
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. 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 × 6 = 36 (and there are 2 of them, so 72)
- Front and back. Each is length × height.6 × 2 = 12 (2 of them: 24)
- Left and right. Each is width × height.6 × 2 = 12 (2 of them: 24)
- Add the three pairs. The surface area is every face added together.72 + 24 + 24 = 120
- Write the unit. Area, so square units.Surface area = 120 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (36 + 12 + 12) = 2 × 60 = 120. ✓
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. 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.10 × 3 = 30 (and there are 2 of them, so 60)
- Front and back. Each is length × height.10 × 2 = 20 (2 of them: 40)
- 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.60 + 40 + 12 = 112
- Write the unit. Area, so square units.Surface area = 112 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (30 + 20 + 6) = 2 × 56 = 112. ✓
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. 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.4 × 3 = 12 (and there are 2 of them, so 24)
- Front and back. Each is length × height.4 × 2.5 = 10 (2 of them: 20)
- Left and right. Each is width × height.3 × 2.5 = 7.5 (2 of them: 15)
- Add the three pairs. The surface area is every face added together.24 + 20 + 15 = 59
- Write the unit. Area, so square units.Surface area = 59 cm2
Check. Use the formula 2(lw + lh + wh) = 2 × (12 + 10 + 7.5) = 2 × 29.5 = 59. ✓
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.4 × 4 = 16
- Multiply by the number of faces. A cube has 6 faces.6 × 16 = 96
- Write the unit. Surface area = 96 cm2
Check. Formula 6a2 = 6 × 42 = 6 × 16 = 96. ✓
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. 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.7 × 7 = 49
- Multiply by the number of faces. A cube has 6 faces.6 × 49 = 294
- Write the unit. Surface area = 294 cm2
Check. Formula 6a2 = 6 × 72 = 6 × 49 = 294. ✓
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. 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.10 × 10 = 100
- Multiply by the number of faces. A cube has 6 faces.6 × 100 = 600
- Write the unit. Surface area = 600 mm2
Check. Formula 6a2 = 6 × 102 = 6 × 100 = 600. ✓
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 is the total area of all the faces. Work out the area of each face (or matching pair of faces) and add.
- Picture the five faces. A box with no lid has 5 faces: the base and four sides. The top is missing, so we do not count it.
- The base. Length × width, one face only.20 × 15 = 300
- The long sides (front and back). Each is length × height and there are 2.20 × 10 = 200, so 2 of them: 400
- The short sides (left and right). Each is width × height and there are 2.15 × 10 = 150, so 2 of them: 300
- Add them. 300 + 400 + 300 = 1000
- Write the unit. Area, so square units.Surface area = 1000 cm2
Check. A closed box would be 1300; taking away the lid (300) leaves 1000. ✓
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. 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).
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 (12 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 × 12 = 72
First sloping rectangle: 5 × 12 = 60
Second sloping rectangle: 5 × 12 = 60 - Add all the faces together. Surface area = the two ends + all the rectangles.24 + 72 + 60 + 60 = 216
- Write the unit. Area, so square units.Surface area = 216 cm2
Check. The rectangles together make (perimeter of an end) × length = 16 × 12 = 192. Then 24 + 192 = 216. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
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 (trapeziums) 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 trapezium end. Area of a trapezium = ½ × (sum of the parallel sides) × height.½ × (4 + 8) × 3 = 18 cm2
- Double it for the two ends. The two end faces are identical.2 × 18 = 36 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Rectangle on the top side: 4 × 10 = 40
Rectangle on the bottom side: 8 × 10 = 80
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.36 + 40 + 80 + 50 + 50 = 256
- Write the unit. Area, so square units.Surface area = 256 cm2
Check. The rectangles together make (perimeter of an end) × length = 22 × 10 = 220. Then 36 + 220 = 256. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
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 (L-shapes) and a rectangle for every edge of the end: each rectangle is the edge length × the length of the prism (10 cm).
- Split the L-shape into two rectangles. We can only use the area formula on simple shapes, so cut the L into two rectangles.Bottom rectangle: 8 by 2
Top rectangle: 3 by (6 − 2) = 3 by 4 - Find the area of each rectangle. Width × height each time.8 × 2 = 16 cm2
3 × 4 = 12 cm2 - Add the two areas. The whole cross-section is the two parts together.A = 16 + 12 = 28 cm2
- Double it for the two ends. The two end faces are identical.2 × 28 = 56 cm2
- Find each rectangular face. Each rectangle is an edge of the end multiplied by the length of the prism.Bottom edge: 8 × 10 = 80
Right edge: 2 × 10 = 20
Inner horizontal edge: 5 × 10 = 50
Inner vertical edge: 4 × 10 = 40
Top edge: 3 × 10 = 30
Left edge: 6 × 10 = 60 - Add all the faces together. Surface area = the two ends + all the rectangles.56 + 80 + 20 + 50 + 40 + 30 + 60 = 336
- Write the unit. Area, so square units.Surface area = 336 cm2
Check. The rectangles together make (perimeter of an end) × length = 28 × 10 = 280. Then 56 + 280 = 336. ✓
Watch out. Do not forget the two ends, and make sure each rectangle uses the edge length (not the height of the triangle).
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 of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 32 = 18π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 3 × 10 = 60π
- Add them together. Total surface area = circles + curved surface.18π + 60π = 78π
- 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.Surface area = 78π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 3 × 13 = 78π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
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 of a cylinder = two circles + the curved surface (a rectangle).
- Find the radius. The formulas use the radius, which is half the diameter.r = 8 ÷ 2 = 4
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 42 = 32π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 4 × 5 = 40π
- Add them together. Total surface area = circles + curved surface.32π + 40π = 72π
- 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.Surface area = 72π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 4 × 9 = 72π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
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 of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 52 = 50π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 5 × 2 = 20π
- Add them together. Total surface area = circles + curved surface.50π + 20π = 70π
- 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.Surface area = 70π cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 5 × 7 = 70π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. The curved surface of a cylinder opens out into a rectangle of width 2πr and height h.
- Think of the curved surface as a rectangle. If you cut a label off a can and flatten it, you get a rectangle. Its width is the circumference of the circle (2πr) and its height is h.
- Multiply width by height. Area of a rectangle = width × height.2πr × h = 2 × π × 4 × 10 = 80π
- 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.Curved surface area = 80π cm2
Check. Circumference = 2πr = 8π; times the height 10 = 80π. ✓
Watch out. The curved surface does not include the circular ends.
Method. The curved surface of a cylinder opens out into a rectangle of width 2πr and height h.
- Think of the curved surface as a rectangle. If you cut a label off a can and flatten it, you get a rectangle. Its width is the circumference of the circle (2πr) and its height is h.
- Multiply width by height. Area of a rectangle = width × height.2πr × h = 2 × π × 3 × 7 = 42π
- 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.Curved surface area = 42π cm2
Check. Circumference = 2πr = 6π; times the height 7 = 42π. ✓
Watch out. The curved surface does not include the circular ends.
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 of an open cylinder = one circle + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into one circle (the base) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the base circle. Area of a circle = πr2. Only one circle because there is no lid.π × 52 = 25π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 5 × 12 = 120π
- Add them together. Total surface area = circles + curved surface.25π + 120π = 145π
- 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.Surface area = 145π cm2
Check. Using πr(r + 2h) = π × 5 × 29 = 145π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. Find the surface area of the box and the area of one sheet, then divide and round up.
- Surface area of the box. Three pairs of faces: 30 × 20, 30 × 10, 20 × 10.2 × (600 + 300 + 200) = 2 × 1100 = 2200 cm2
- Area of one sheet. 50 × 40.2000 cm2
- Divide. See how many sheets are needed.2200 ÷ 2000 = 1.1
- Round up. 1.1 sheets means more than one sheet is needed, so 2 whole sheets.2 sheets
Check. 1 sheet gives 2000 cm2, which is less than 2200. 2 sheets give 4000, which is enough. ✓
Watch out. Round up when you need whole sheets.
Method. Find the area of the five faces and multiply by the cost per m2.
- Area of the five faces. Use the open-top cuboid surface area: base + 2 long sides + 2 short sides.3 + 2 × 2 + 2 × 1.5 = 3 + 4 + 3 = 10 m2
- Multiply by the cost. £4 for each m2.10 × 4 = £40
Check. 10 m2 at £4 is £40. ✓
Method. Work backwards: divide the total by 6 to get one face, then find the square root.
- Area of one face. A cube has 6 identical faces.216 ÷ 6 = 36 cm2
- Find the edge. Each face is a square, so edge × edge = 36. Since 6 × 6 = 36 the edge is 6 cm.Edge = 6 cm
Check. 6 × 62 = 6 × 36 = 216. ✓
Watch out. Square root (not divide by 2) to get the edge from the area of a face.
Method. Work out the volume and surface area of both solids and compare.
- Volume of the cube. edge3.6 × 6 × 6 = 216 cm3
- Volume of the cuboid. length × width × height.9 × 6 = 54, then 54 × 4 = 216 cm3
- Surface area of the cube. 6 faces of 36.6 × 36 = 216 cm2
- Surface area of the cuboid. 2(lw + lh + wh).2 × (54 + 36 + 24) = 2 × 114 = 228 cm2
- Compare. Same volume. The cube has the smaller surface area: 216 < 228.
Check. A cube is the most compact cuboid, so its surface area is the smallest for that volume. ✓
Method. Work out the curved surface and the two circles separately and compare with 60π.
- Curved surface. 2πrh with r = 3 and h = 10.2 × π × 3 × 10 = 60π
- Tom’s mistake. His answer is only the curved surface. He left out the two circular ends.Two circles: 2 × π × 32 = 18π
- Add all the parts. 60π + 18π = 78π cm2
Check. 2πr(r + h) = 2 × π × 3 × 13 = 78π. ✓
Method. Work out the surface area before, then of both halves after, and subtract.
- Surface area of the original. 2(lw + lh + wh).2 × (24 + 18 + 12) = 2 × 54 = 108 cm2
- Surface area of one half. Use 6 by 4 by 1.5.2 × (24 + 9 + 6) = 2 × 39 = 78 cm2
- Both halves. There are two identical halves.2 × 78 = 156 cm2
- Find the increase. Subtract.156 − 108 = 48 cm2. ✓
Check. The cut makes two new faces of 6 × 4 = 24 each, and 2 × 24 = 48. ✓
Method. Find the surface area of the cuboid and of the two cubes and compare.
- Surface area of the cuboid. 2(lw + lh + wh) with 4, 2 and 2.2 × (8 + 8 + 4) = 2 × 20 = 40 cm2
- Surface area of two separate cubes. Each cube is 6 × 2 × 2 = 24 cm2.2 × 24 = 48 cm2
- Explain the difference. The faces that touch are no longer on the outside. Each is 2 × 2 = 4 cm2 and there are 2 of them.48 − 40 = 8 = 2 × 4 ✓
Check. The hidden faces total 8 cm2. ✓
Method. Each face of a cube is a square.
- Write the area formula for a square. Area = side × side.Area = 3 × 3
- Multiply. 9 cm2
Check. A 3 by 3 grid has 9 squares. ✓
Method. A cuboid has 6 faces in 3 matching pairs. Work out one face from each pair.
- Top and bottom. 4 × 3 = 12, and 2 of them.12 + 12 = 24
- Front and back. 4 × 2 = 8, and 2 of them.8 + 8 = 16
- Left and right. 3 × 2 = 6, and 2 of them.6 + 6 = 12
- Add them. 24 + 16 + 12 = 52 cm2
Check. 2(12 + 8 + 6) = 2 × 26 = 52. ✓
Exercise 19I Surface area of cylinders (calculator)
Calculator allowedKey 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 of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 42 = 32π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 4 × 10 = 80π
- Add them together. Total surface area = circles + curved surface.32π + 80π = 112π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.112π = 351.8584…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.351.8584… = 352 cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 4 × 14 = 112π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
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 of a cylinder = two circles + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 2.52 = 12.5π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 2.5 × 7.2 = 36π
- Add them together. Total surface area = circles + curved surface.12.5π + 36π = 48.5π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.48.5π = 152.3672…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.152.3672… = 152 cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 2.5 × 9.7 = 48.5π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
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 of a cylinder = two circles + the curved surface (a rectangle).
- Find the radius. The formulas use the radius, which is half the diameter.r = 12 ÷ 2 = 6
- Think of the net. A cylinder opens out into two circles (top and bottom) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the two circles. Area of a circle = πr2. There are two of them.2 × π × 62 = 72π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 6 × 15 = 180π
- Add them together. Total surface area = circles + curved surface.72π + 180π = 252π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.252π = 791.6813…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.791.6813… = 792 cm2
Check. Using the factorised formula: 2πr(r + h) = 2 × π × 6 × 21 = 252π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
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 of an open cylinder = one circle + the curved surface (a rectangle).
- Think of the net. A cylinder opens out into one circle (the base) and a rectangle. The rectangle wraps round the curved side, so its width is the circumference of the circle, 2πr, and its height is h.
- Area of the base circle. Area of a circle = πr2. Only one circle because there is no lid.π × 62 = 36π
- Area of the curved surface. This is the rectangle: circumference × height = 2πr × h.2 × π × 6 × 14 = 168π
- Add them together. Total surface area = circles + curved surface.36π + 168π = 204π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.204π = 640.8849…
- Round the answer. The question wants 1 decimal place. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.640.8849… = 640.9 cm2
Check. Using πr(r + 2h) = π × 6 × 34 = 204π. ✓
Watch out. The curved surface is 2πr × h. Do not use the diameter for the circles, and remember both circular ends (unless the question says one end is open).
Method. The curved surface of a cylinder opens out into a rectangle of width 2πr and height h.
- Think of the curved surface as a rectangle. If you cut a label off a can and flatten it, you get a rectangle. Its width is the circumference of the circle (2πr) and its height is h.
- Multiply width by height. Area of a rectangle = width × height.2πr × h = 2 × π × 3.7 × 10.8 = 79.92π
- Use your calculator. Type the calculation in using the π button on your calculator (not 3.14). Write down the whole display before you round.2 × π × 3.7 × 10.8 = 251.0761…
- Round the answer. The question wants 3 significant figures. Look at the digit just after the last one you are keeping: 5 or more rounds up, 4 or less leaves it.251.0761… = 251 cm2
Check. Circumference = 2πr = 7.4π; times the height 10.8 = 79.92π. ✓
Watch out. The curved surface does not include the circular ends.
Method. Find the surface area of the base and curved surface, then divide by the area one litre covers. Round up because you cannot buy part of a litre.
- Area of the base circle. Only one circle because the tank is open.π × 0.82 = 0.64π
- Area of the curved surface. 2πrh.2 × π × 0.8 × 1.5 = 2.4π
- Add and use the calculator. 0.64π + 2.4π = 3.04π.3.04π = 9.550… m2
- Divide by 4 m2 per litre. This gives the litres needed.9.550… ÷ 4 = 2.388…
- Round up. Part of a litre must be a whole extra litre.3 litres
Check. 2 litres would cover only 8 m2, which is less than the area. ✓
Watch out. Round up when you need to buy whole units.
Method. Work out both areas in terms of π and write them as a ratio. The π cancels.
- Curved surface area. 2πrh.2 × π × 3 × 8 = 48π
- Total surface area. 2πr2 + 2πrh.2 × π × 9 + 48π = 18π + 48π = 66π
- Write the ratio. The π is in both so it cancels.48 : 66
- Simplify. Divide both parts by 6.48 : 66 = 8 : 11
Check. 8 × 6 = 48 and 11 × 6 = 66. ✓