Perimeter
To put a fence around a field, you measure all four sides and add the lengths. That total is the length of the fence, and it is also the perimeter of the field.
Add up every side of a figure and you get its perimeter. It is stated in length units such as cm and metres. Perimeter of a rectangle = 2 × (length + breadth) and perimeter of a square = 4 × side.
In the figure: The rectangle measures 6 cm by 4 cm. Adding all four sides: 6 + 4 + 6 + 4 = 20, which is 2 × (6 + 4) = 20. The side of the square is 3; four sides give 3 + 3 + 3 + 3 = 12, which is 4 × 3 = 12.
Rule 1: Get the perimeter from the formula — For a rectangle, add the length and breadth and multiply by 2. For a square, multiply the side by 4. For a triangle, just add the three sides.
- Easy: A square has side 25 cm. Perimeter 4 × 25 = 100 cm.
- Medium: A rectangle is 45 metres long and 30 metres wide. Perimeter 2 × (45 + 30) = 2 × 75 = 150 metres.
- Exam type: The sides of a triangle are 11 cm, 12 cm and 16 cm. Perimeter 11 + 12 + 16 = 39 cm.
Rule 2: Find a missing side from the perimeter — In a square, divide the perimeter by 4. In a rectangle, divide the perimeter by 2; that gives the sum of the length and breadth. Then subtract the side you know.
- Easy: The perimeter of a square is 36 cm. Side 36 ÷ 4 = 9 cm.
- Medium: A rectangle has perimeter 50 metres and length 16 metres. 50 ÷ 2 = 25, so the breadth is 25 − 16 = 9 metres.
- Exam type: A rectangle has perimeter 120 metres and breadth 20 metres. 120 ÷ 2 = 60, so the length is 60 − 20 = 40 metres.
In the figure: A 3 × 2 piece has been cut from the corner of an 8 × 6 rectangle. Now there are six sides: 5, 2, 3, 4, 8 and 6. Their sum is 5 + 2 + 3 + 4 + 8 + 6 = 28. Cutting the corner did not change the perimeter, so 2 × (8 + 6) = 28.
Rule 3: Perimeter of a bent figure — add every side — In a figure shaped like L, add the sides one by one. If a corner has been cut off, the perimeter equals that of the rectangle around it; use this to check.
- Easy: The sides are 7, 2, 3, 4, 10 and 6. 7 + 2 + 3 + 4 + 10 + 6 = 32 and for the rectangle around it 2 × (10 + 6) = 32.
- Medium: The sides are 9, 3, 4, 5, 13 and 8. 9 + 3 + 4 + 5 + 13 + 8 = 42 and 2 × (13 + 8) = 42.
- Exam type: A 5 metres × 4 metres piece is cut from the corner of a 20 metres × 12 metres rectangle. The perimeter stays the same: 2 × (20 + 12) = 64 metres.
Perimeter
In the app you can move this figure with your finger.
Open in the appPerimeter is the length of the boundary, so its unit is a length unit such as cm or metres. Do not use square units.
Writing only 6 + 4 = 10 for the perimeter of a rectangle. The figure has four sides, so the 10 must be multiplied by 2: 2 × 10 = 20.
If every side of a figure is made 2 times, the perimeter also becomes 2 times. Make a 6 × 4 rectangle 12 × 8 and 2 × (12 + 8) = 40, and 2 × 20 = 40.
Spot the difference
| Point | Rectangle | Square |
|---|---|---|
| Sides | two pairs of equal opposite sides | all four sides equal |
| Formula | 2 × (length + breadth) | 4 × side |
| Example in the figure | 20 for 6 and 4 | 12 for side 3 |
| Measures needed | 2 (length and breadth) | 1 (only the side) |
Practice questions
- 1The side of a square garden is 14 metres. What is its perimeter?
- 2A rectangular plot is 18 metres long and 7 metres wide. What is its perimeter?
- 3Find the perimeter of an equilateral triangle with side 4 cm.
- 4A rectangle has perimeter 70 metres and length 15 metres. Find its breadth.
- 5The side of a square ground is 20 metres. How many metres must you walk to go around it 3 times?
- 6A square has side 4 cm. Stretch it to side 20 cm; how many times does the perimeter grow?
- 7The sides of an L shaped figure are 12, 4, 5, 6, 17 and 10 metres. Find the perimeter and check it against the perimeter of the rectangle around it.
- 8A rectangle is twice as long as it is wide, and its perimeter is 54 metres. Find its length and breadth.
Show answers (try them yourself first)
- 14 × 14 = 56 metres. 2) 2 × (18 + 7) = 2 × 25 = 50 metres. 3) 3 × 4 = 12 cm.
- 470 ÷ 2 = 35, so the breadth is 35 − 15 = 20 metres. 5) One round is 4 × 20 = 80 metres, so 3 × 80 = 240 metres. 6) 4 × 4 = 16 and 4 × 20 = 80; 80 ÷ 16 = 5 times.
- 712 + 4 + 5 + 6 + 17 + 10 = 54 metres. The rectangle around it is 17 × 10, so 2 × (17 + 10) = 54; both are the same.
- 854 ÷ 2 = 27 is the sum of the length and the breadth. That is 3 equal parts (1 of the breadth and 2 of the length), so the breadth is 27 ÷ 3 = 9 metres and the length is 9 × 2 = 18 metres.
Area
To lay tiles in a room, you first measure how much floor has to be covered. This flat space taken up by a figure is its area.
The flat space inside a figure is its area. It is given in square units (sq cm, sq metres). Area of a rectangle = length × breadth and area of a square = side × side.
In the figure: In the 6 × 4 grid there are 4 rows of 6 squares, so 6 × 4 = 24 squares in all. In the 3 × 3 grid there are 3 × 3 = 9 squares.
Rule 1: Get the area from the formula — In a rectangle, multiply the length and the breadth. In a square, multiply the side by itself. Write the answer in square units.
- Easy: A square has side 8 cm. Area 8 × 8 = 64 sq cm.
- Medium: A rectangle is 35 metres long and 20 metres wide. Area 35 × 20 = 700 sq metres.
- Exam type: A rectangle measures 120 cm by 90 cm. Area 120 × 90 = 10800 sq cm.
Rule 2: Find a missing side from the area — Divide the area by the side you know. In a square, look for the number that gives the area when multiplied by itself.
- Easy: A rectangle covers 96 sq cm and is 12 cm long. Breadth 96 ÷ 12 = 8 cm.
- Medium: The area of a square is 81 sq metres. 9 × 9 = 81, so the side is 9 metres.
- Exam type: A rectangle has area 600 sq metres and breadth 15 metres. Length 600 ÷ 15 = 40 metres, then perimeter 2 × (40 + 15) = 110 metres.
In the figure: Take a square of side 1 metre. 1 metre = 100 cm, so the area is 100 × 100 = 10000 sq cm. That is, 1 sq metre = 10000 sq cm.
Rule 3: When changing units, find the area first — To change sq metres to sq cm, multiply by 10000. To change sq cm to sq metres, divide by 10000. Do not multiply by 100 only.
- Easy: A rectangle is 200 cm × 50 cm. 200 × 50 = 10000 sq cm, which is 1 sq metre.
- Medium: A rectangle is 3 metres × 2 metres. 3 × 2 = 6 sq metres, then 6 × 10000 = 60000 sq cm.
- Exam type: A rectangle is 400 cm × 150 cm. 400 × 150 = 60000 sq cm, then 60000 ÷ 10000 = 6 sq metres.
Rule 4: Area × rate = cost — For the cost of tiles, paint or a lawn, first find the area and then multiply by the rate per sq metre.
- Easy: A 6 metres × 3 metres space costs 40 rupees per sq metre. 6 × 3 = 18, then 18 × 40 = 720 rupees.
- Medium: A 5 metres × 4 metres room costs 150 rupees per sq metre. 5 × 4 = 20, then 20 × 150 = 3000 rupees.
- Exam type: A 50 metres × 30 metres field costs 800 rupees per sq metre. 50 × 30 = 1500, then 1500 × 800 = 1200000 rupees.
Area
In the app you can move this figure with your finger.
Open in the appThe unit of area is a square unit, such as sq cm and sq metres. Take the length and the breadth in the same unit before you multiply.
Swapping the perimeter 20 and the area 24 of a 6 × 4 rectangle. Perimeter is the sum of the sides; area is the number of squares in the grid.
Making a square's side 2 times as long makes its area 4 times. For side 3, 3 × 3 = 9; for side 6, 6 × 6 = 36, and 4 × 9 = 36.
Spot the difference
| Point | Perimeter | Area |
|---|---|---|
| What it measures | length of the boundary | flat space inside |
| Unit | cm, metres | sq cm, sq metres |
| Rectangle 6 × 4 | 20 | 24 |
| Square of side 3 | 12 | 9 |
Practice questions
- 1The side of a square plot is 12 metres. What is its area?
- 2Find the area of a rectangle that measures 15 cm by 8 cm.
- 3A rectangular piece of land is 20 metres long and 9 metres wide. What is its area?
- 4A rectangle has area 126 sq cm and length 18 cm. Find the breadth and the perimeter.
- 5A wall 8 metres long and 3 metres high is to be painted at 45 rupees per sq metre. What is the cost?
- 6A rectangle is 2 metres long and 75 cm wide. Give its area in sq cm.
- 7The side of a square is changed from 4 cm to 12 cm. How many times does the area grow?
- 8The perimeter of a rectangle is 28 metres and the length is 8 metres. Find its area.
Show answers (try them yourself first)
- 112 × 12 = 144 sq metres. 2) 15 × 8 = 120 sq cm. 3) 20 × 9 = 180 sq metres.
- 4Breadth 126 ÷ 18 = 7 cm. Perimeter 2 × (18 + 7) = 2 × 25 = 50 cm. 5) 8 × 3 = 24 sq metres, then 24 × 45 = 1080 rupees. 6) 2 metres = 200 cm, so 200 × 75 = 15000 sq cm.
- 74 × 4 = 16 and 12 × 12 = 144; 144 ÷ 16 = 9 times. The side became 3 times, so the area became 3 × 3 = 9 times.
- 828 ÷ 2 = 14, then the breadth is 14 − 8 = 6 metres. Area 8 × 6 = 48 sq metres.
Area of a triangle
If a rectangular sheet of paper is cut along a diagonal, you get two equal triangles. Each piece covers exactly half of the paper, so the area of a triangle is half that of the rectangle.
Area of a triangle = ½ × base × height. The height is the distance from the corner opposite the base to the base, measured at a right angle. It is not always a side of the triangle.
In the figure: A rectangle of 8 × 6 = 48 was cut along a diagonal. Two equal triangles were formed, each with area 48 ÷ 2 = 24. That is ½ × 8 × 6 = 24.
Rule 1: With a right angle, the two sides meeting there serve as base and height — Multiply those two sides and take half.
- Easy: The two sides at the right angle are 6 cm and 5 cm. Area ½ × 6 × 5 = 15 sq cm.
- Medium: The two sides at the right angle are 9 metres and 12 metres. Area ½ × 9 × 12 = 54 sq metres.
- Exam type: The two sides at the right angle are 1.6 cm and 2.5 cm. Area ½ × 1.6 × 2.5 = 2 sq cm.
In the figure: Both triangles have base 8 and height 6. In the first, the height falls inside the triangle. In the second, it falls outside the base, on the dotted line. Still both have the same area ½ × 8 × 6 = 24.
Rule 2: For any triangle, take a base and the height on it — Measure the height at a right angle to the base. Even if the height falls outside the triangle, the formula stays the same.
- Easy: The base is 10 cm and the height is 7 cm. Area ½ × 10 × 7 = 35 sq cm.
- Medium: The base is 14 metres and the height is 9 metres. Area ½ × 14 × 9 = 63 sq metres.
- Exam type: The base is 7.5 cm and the height is 4 cm. Area ½ × 7.5 × 4 = 15 sq cm.
Rule 3: Whichever side is taken as the base, the area is the same — When you change the base, the height on it changes too, but ½ × base × height stays the same.
- Easy: In a right angled triangle with sides 6, 8, 10, the right angle is between 6 and 8. With base 8 the height is 6: ½ × 8 × 6 = 24. With base 10 the height is 4.8: ½ × 10 × 4.8 = 24.
- Medium: In one triangle the height is 8 when the base is 9: ½ × 9 × 8 = 36. With base 12 the height is 6: ½ × 12 × 6 = 36.
- Exam type: Base 20 with height 6 gives ½ × 20 × 6 = 60. Taking base 15, the height is 8: ½ × 15 × 8 = 60.
Rule 4: Find the base or the height from the area — Multiply the area by 2 and divide by the measure you know.
- Easy: The area is 40 sq cm and the base is 8 cm. Height 2 × 40 ÷ 8 = 10 cm.
- Medium: The area is 54 sq metres and the height is 9 metres. Base 2 × 54 ÷ 9 = 12 metres.
- Exam type: The area is 90 sq cm and the base is 15 cm. Height 2 × 90 ÷ 15 = 12 cm.
Area of a triangle
In the app you can move this figure with your finger.
Open in the appThe height is always measured at a right angle to the base. A slanting side is not the height; only in a right angled triangle is a side forming the right angle the height.
Forgetting to take half. 8 × 6 = 48 belongs to the rectangle; the triangle gets 48 ÷ 2 = 24.
First multiply the base and the height, then divide the answer by 2. Example: for base 14 and height 5, 14 × 5 = 70, then 70 ÷ 2 = 35.
Spot the difference
| Point | Right angled triangle | Any triangle |
|---|---|---|
| Where the height is | one side forming the right angle | a perpendicular segment drawn on the base |
| Formula | ½ × base × height | ½ × base × height |
| For base 8 and height 6 | 24 | 24 |
| Link with the rectangle | half of 8 × 6 = 48 | half of 8 × 6 = 48 |
Practice questions
- 1Find the area of a triangle with the two sides at the right angle 8 cm and 5 cm.
- 2A triangle has base 18 metres and height 6 metres. Find its area.
- 3A triangle has base 16 cm and height 5 cm. Find its area.
- 4The area of a triangle is 45 sq cm and the base is 15 cm. What is the height?
- 5A triangle has base 6.4 cm and height 5 cm. Find the area.
- 6In a right angled triangle with sides 12, 16, 20 cm, the right angle is between 12 and 16. Find the area. Then take the side 20 as the base and find the height on it.
- 7A triangular field has base 24 metres and height 18 metres. Grass is to be planted on it at 15 rupees per sq metre. What is the cost?
- 8One triangle has base 10 cm and height 6 cm. Another has base 15 cm and height 4 cm. Do both have the same area?
Show answers (try them yourself first)
- 1½ × 8 × 5 = 20 sq cm. 2) ½ × 18 × 6 = 54 sq metres. 3) ½ × 16 × 5 = 40 sq cm.
- 4Height 2 × 45 ÷ 15 = 6 cm. 5) ½ × 6.4 × 5 = 16 sq cm. 6) Area ½ × 12 × 16 = 96 sq cm. With base 20 the height is 2 × 96 ÷ 20 = 9.6 cm.
- 7Area ½ × 24 × 18 = 216 sq metres, then cost 216 × 15 = 3240 rupees.
- 8The first is ½ × 10 × 6 = 30 and the second is ½ × 15 × 4 = 30. Both are 30 sq cm, so the areas are the same.
Paths and tiles
A path for walking is built along the edge of a garden. Work out the whole space with the path and the space without it; the difference between the two is the path.
If a strip of the same width runs all around the inside or outside of a rectangle, then area of the strip = area of the bigger rectangle − area of the smaller rectangle. If the strip is inside, the inner rectangle loses 2 times the strip width in both its length and its breadth. If it is outside, both grow by that much.
In the figure: The garden is 14 × 10 and the path inside is 2 wide. The inner rectangle is (14 − 4) × (10 − 4) = 10 × 6 = 60. The whole garden is 14 × 10 = 140. So the path is 140 − 60 = 80.
Rule 1: Area of a path inside — Find the area of the whole space. Make the length and breadth each smaller by 2 times the width of the path to get the inner rectangle. Subtract the smaller from the bigger.
- Easy: A garden is 22 metres × 14 metres with a 1 metre wide path inside. 22 × 14 = 308, inner part 20 × 12 = 240, path 308 − 240 = 68 sq metres.
- Medium: A garden is 30 metres × 24 metres with a 3 metres wide path inside. 30 × 24 = 720, inner part 24 × 18 = 432, path 720 − 432 = 288 sq metres.
- Exam type: A garden is 17 metres × 12 metres with a 2 metres wide path inside. 17 × 12 = 204, inner part (17 − 4) × (12 − 4) = 13 × 8 = 104, path 204 − 104 = 100 sq metres.
In the figure: The ground is 9 × 5 = 45 and the path outside is 1 wide. The outer rectangle is (9 + 2) × (5 + 2) = 11 × 7 = 77. So the path is 77 − 45 = 32.
Rule 2: Area of a path outside — Make the length and breadth each larger by 2 times the width of the path to get the outer rectangle. Subtract the area of the original space from it.
- Easy: A ground is 40 metres × 25 metres with a 2 metres wide path outside. Ground 40 × 25 = 1000, outer rectangle 44 × 29 = 1276, path 1276 − 1000 = 276 sq metres.
- Medium: A ground is 18 metres × 12 metres with a 1 metre wide path outside. Ground 18 × 12 = 216, outer rectangle 20 × 14 = 280, path 280 − 216 = 64 sq metres.
- Exam type: The side of a square ground is 10 metres with a 3 metres wide path outside. Ground 10 × 10 = 100, outer square (10 + 6) × (10 + 6) = 16 × 16 = 256, path 256 − 100 = 156 sq metres.
Rule 3: Find the number of tiles — Bring the area of the room and the area of one tile to the same unit. Then divide the area of the room by the area of a tile.
- Easy: A room is 6 metres × 4 metres and the tile has a side of 20 cm. Room 600 × 400 = 240000 sq cm, tile 20 × 20 = 400 sq cm, tiles 240000 ÷ 400 = 600. Check: 30 tiles along the length and 20 along the breadth, which is 30 × 20 = 600.
- Medium: A room is 5 metres × 4 metres and the tile has a side of 50 cm. Room 500 × 400 = 200000, tile 50 × 50 = 2500, tiles 200000 ÷ 2500 = 80.
- Exam type: A hall is 8 metres × 6 metres and the tile has a side of 40 cm. Hall 800 × 600 = 480000, tile 40 × 40 = 1600, tiles 480000 ÷ 1600 = 300.
Rule 4: When the strips on the sides are different — From the length, subtract the left and right strips. From the breadth, subtract the top and bottom strips. Then find the area of the rectangle that is left.
- Easy: A page of 20 cm × 15 cm has a margin of 2 cm on all four sides. The inner part is (20 − 4) × (15 − 4) = 16 × 11 = 176 sq cm.
- Medium: In a picture of 24 cm × 18 cm the left and right strips are 3 cm each, the top is 2 cm and the bottom is 4 cm. Length 24 − 6 = 18, breadth 18 − 6 = 12, so 18 × 12 = 216 sq cm.
- Exam type: On a screen of 14 cm × 8 cm the left strip is 1, the right 2, the top 1 and the bottom 1 cm. Length 14 − 3 = 11, breadth 8 − 2 = 6, so 11 × 6 = 66 sq cm.
Paths and tiles
In the app you can move this figure with your finger.
Open in the appPaths and tiles
In the app you can move this figure with your finger.
Open in the appPaths and tiles
In the app you can move this figure with your finger.
Open in the appIf the strip is inside, the inner rectangle gets smaller; if it is outside, the outer rectangle gets bigger. Either way, the area of the strip is the bigger rectangle minus the smaller rectangle.
For an inside path, subtracting only the width of the path from the length, that is 14 − 2 = 12. The path lies along both edges, so 2 × 2 = 4 must be subtracted: 14 − 4 = 10.
Inside path = 2 × width of path × (length + breadth − 2 × width of path). Example: 2 × 2 × (14 + 10 − 4) = 4 × 20 = 80. Outside path = 2 × width of path × (length + breadth + 2 × width of path). Example: 2 × 1 × (9 + 5 + 2) = 2 × 16 = 32.
Spot the difference
| Point | Path inside | Path outside |
|---|---|---|
| Smaller rectangle | the inner part | the original space |
| Measures of the rectangle | length and breadth each smaller by 2 times the width | length and breadth each larger by 2 times the width |
| Calculation in the figure | 140 − 60 = 80 | 77 − 45 = 32 |
| Area of the path | 80 | 32 |
Practice questions
- 1A garden is 18 metres × 10 metres with a 1 metre wide path inside. How much area does the path cover?
- 2A garden is 16 metres × 12 metres with a 2 metres wide path inside. How much area does the path cover?
- 3A room is 6 metres × 3 metres and the tile has a side of 30 cm. How many tiles are needed?
- 4A ground is 21 metres × 11 metres with a 2 metres wide path outside. How much area does the path cover?
- 5The side of a square ground is 20 metres with a 2 metres wide path outside. How much area does the path cover?
- 6The area of a room is 144 sq metres. How many tiles of 30 cm × 40 cm are needed?
- 7A 30 metres × 20 metres plot has a 2 metres wide path inside. The path costs 120 rupees per sq metre. What is the total cost?
- 8A picture of 30 cm × 22 cm has a margin of 2 cm on all four sides inside it. How much area does the margin cover?
Show answers (try them yourself first)
- 1Garden 18 × 10 = 180, inner part 16 × 8 = 128, path 180 − 128 = 52 sq metres. 2) Garden 16 × 12 = 192, inner part 12 × 8 = 96, path 192 − 96 = 96 sq metres. 3) Room 600 × 300 = 180000, tile 30 × 30 = 900, tiles 180000 ÷ 900 = 200.
- 4Ground 21 × 11 = 231, outer rectangle 25 × 15 = 375, path 375 − 231 = 144 sq metres. 5) Ground 20 × 20 = 400, outer square 24 × 24 = 576, path 576 − 400 = 176 sq metres. 6) 144 × 10000 = 1440000 sq cm, tile 30 × 40 = 1200, tiles 1440000 ÷ 1200 = 1200.
- 730 × 20 = 600, inner part 26 × 16 = 416, path 600 − 416 = 184 sq metres. Cost 184 × 120 = 22080 rupees.
- 8Picture 30 × 22 = 660, inner part (30 − 4) × (22 − 4) = 26 × 18 = 468, margin 660 − 468 = 192 sq cm.
Surface area
To paint a box on the outside, every face of it needs paint. The area of all those faces together is the surface area of the box.
Adding the areas of every face of a cuboid or a cube gives its surface area. A cuboid has 6 faces; opposite faces are equal in pairs. Surface area of a cuboid = 2 × (length × breadth + breadth × height + height × length). Surface area of a cube = 6 × side × side.
In the figure: The cuboid has length 6, breadth 4 and height 3. The faces in three pairs: 6 × 4 = 24, 4 × 3 = 12 and 6 × 3 = 18. So the surface area is 2 × (24 + 12 + 18) = 2 × 54 = 108.
Rule 1: Surface area of a cuboid — Find the area of the three different faces, add them and multiply by 2.
- Easy: A cuboid measures 7 cm, 3 cm and 2 cm. 7 × 3 = 21, 3 × 2 = 6, 7 × 2 = 14, so 2 × (21 + 6 + 14) = 2 × 41 = 82 sq cm.
- Medium: A cuboid measures 40 cm, 30 cm and 20 cm. 40 × 30 = 1200, 30 × 20 = 600, 40 × 20 = 800, so 2 × (1200 + 600 + 800) = 2 × 2600 = 5200 sq cm.
- Exam type: A cuboid measures 2 metres, 1.5 metres and 1 metre. 2 × 1.5 = 3, 1.5 × 1 = 1.5, 2 × 1 = 2, so 2 × (3 + 1.5 + 2) = 2 × 6.5 = 13 sq metres.
In the figure: The side of the cube is 4. Each face is 4 × 4 = 16. The six faces are equal, so the surface area is 6 × 16 = 96.
Rule 2: Surface area of a cube — Find the area of one face and multiply it by 6.
- Easy: The side of a cube is 3 cm. Surface area 6 × 3² = 6 × 9 = 54 sq cm.
- Medium: The side of a cube is 10 cm. Surface area 6 × 10² = 6 × 100 = 600 sq cm.
- Exam type: The side of a cube shaped box is 0.5 metres. Surface area 6 × 0.5² = 6 × 0.25 = 1.5 sq metres. If the paint costs 80 rupees per sq metre, the cost is 1.5 × 80 = 120 rupees.
Rule 3: When some faces are missing, subtract their area — For an open box, an open tank or only the four walls of a room, subtract the faces that are not needed from the full surface area.
- Easy: A box of 6 cm × 4 cm × 3 cm is open at the top. The full surface area is 108 and the lid is 6 × 4 = 24. So 108 − 24 = 84 sq cm.
- Medium: A room is 5 metres × 4 metres and 3 metres high. The area of only the four walls is 2 × (5 × 3 + 4 × 3) = 2 × (15 + 12) = 54 sq metres.
- Exam type: An open tank is 3 metres long, 2 metres wide and 2 metres high. The full surface area is 2 × (6 + 4 + 6) = 32, the top is 3 × 2 = 6, so 32 − 6 = 26 sq metres. If painting the inside costs 25 rupees per sq metre, the cost is 26 × 25 = 650 rupees.
Surface area
In the app you can move this figure with your finger.
Open in the appThe unit of surface area is a square unit. A cuboid has 6 faces in 3 pairs. Leave out the faces that are not to be painted.
Forgetting to multiply by 2. 24 + 12 + 18 = 54 is the area of only three faces; the real surface area is 2 × 54 = 108.
In a cube all six faces are equal, so find one and multiply by 6. For side 4, 6 × 4² = 6 × 16 = 96.
Spot the difference
| Point | Cuboid | Cube |
|---|---|---|
| Number of faces | 6 | 6 |
| Equal faces | opposite ones in pairs | all six |
| Formula | 2 × (sum of three different faces) | 6 × side × side |
| Example in the figure | 108 | 96 |
Practice questions
- 1The side of a cube is 7 cm. What is its surface area?
- 2A cuboid measures 8 cm by 5 cm by 2 cm. Find its surface area.
- 3The side of a cube is 9 cm. What is its surface area?
- 4The areas of the three different faces of a cuboid are 60, 24 and 40 sq cm. What is its surface area?
- 52 cubes of side 20 cm are to be painted. How many sq cm must be painted in all?
- 6A box of 5 cm × 3 cm × 2 cm is open at the top. What is the surface area outside it?
- 7A room is 9 metres × 6 metres and 3 metres high. Only the four walls are to be painted at 30 rupees per sq metre. What is the cost?
- 8The side of a cube is changed from 3 cm to 6 cm. How many times does the surface area grow?
Show answers (try them yourself first)
- 16 × 7² = 6 × 49 = 294 sq cm. 2) 8 × 5 = 40, 5 × 2 = 10, 8 × 2 = 16, so 2 × (40 + 10 + 16) = 2 × 66 = 132 sq cm. 3) 6 × 9² = 6 × 81 = 486 sq cm.
- 42 × (60 + 24 + 40) = 2 × 124 = 248 sq cm. 5) One cube is 6 × 20² = 6 × 400 = 2400, so 2 cubes are 2 × 2400 = 4800 sq cm. 6) The base is 5 × 3 = 15 and the four sides are 2 × (3 × 2 + 5 × 2) = 2 × 16 = 32, so 15 + 32 = 47 sq cm.
- 7Four walls 2 × (9 × 3 + 6 × 3) = 2 × (27 + 18) = 90 sq metres. Cost 90 × 30 = 2700 rupees.
- 8For side 3, 6 × 3² = 6 × 9 = 54 and for side 6, 6 × 6² = 6 × 36 = 216; 216 ÷ 54 = 4 times. The side became 2 times, so the surface area became 2 × 2 = 4 times.