Area, surface area and volume

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What is Area?

Area is the amount of space inside a flat shape. We measure it in square units, like square centimetres (cm²) or square metres (m²).
The area of a rectangle is length × width. For a square, it's side × side.
You can find the area of a shape by counting how many unit squares fit inside it.
Area is additive: if you cut a shape into pieces, the total area is the sum of the areas of the pieces.

Area of Triangles

A triangle is half of a rectangle or parallelogram.
The base (b) is one side of the triangle, and the height (h) is the perpendicular distance from the base to the opposite corner.
Area of a triangle = \frac{1}{2} \times b \times h.
Example: If b = 6 cm and h = 4 cm, then area = \frac{1}{2} \times 6 \times 4 = 12 cm².

Area of Parallelograms

A parallelogram is a four-sided shape with opposite sides parallel.
Its area is base × height, where the height is the perpendicular distance between the base and its opposite side.
You can cut a parallelogram and rearrange it into a rectangle to see why the formula works.
Example: base = 5 cm, height = 3 cm, area = 5 × 3 = 15 cm².

Area of Trapeziums

A trapezium (or trapezoid) has one pair of parallel sides. Those are called the bases: b₁ and b₂.
The height (h) is the perpendicular distance between the two bases.
Area = \frac{1}{2} \times (b₁ + b₂) \times h.
You can split a trapezium into a rectangle and two triangles, or double it to make a parallelogram, to find the formula.

Compound Shapes

A compound shape is made of two or more simple shapes, like rectangles and triangles.
To find its area, decompose it into simple shapes, find each area, then add them together.
Sometimes you can also compose a shape by adding a piece to make a rectangle, then subtract that piece's area.
Always use the same units for all measurements before calculating.

Volume of Cuboids

Volume is the amount of space inside a 3-D shape. We measure it in cubic units, like cubic centimetres (cm³) or cubic metres (m³).
A cuboid is a box with six rectangular faces. A cube is a special cuboid with all edges equal.
Volume of a cuboid = length × width × height (V = l × w × h).
For fractional edge lengths, multiply the fractions. Example: \frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24} cm³.

Nets and Surface Area

A net is a flat pattern that folds up to make a 3-D shape.
The surface area (A) of a solid is the total area of all its faces. For a shape with flat faces, surface area = sum of the areas of the faces.
To find surface area using a net, calculate the area of each face on the net and add them up.
For a cuboid, there are 6 faces: opposite faces are equal. So surface area = 2 × (lw + lh + wh).
For a cube with side length s, surface area = 6 × s².

Choosing the Right Units

Use square units (like cm²) for area, and cubic units (like cm³) for volume.
Always match the units: if lengths are in metres, area is in m² and volume is in m³.
When solving problems, check if you need area (like painting a wall) or volume (like filling a box).
Real-world examples: wrapping a present uses surface area; filling a fish tank uses volume.

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1.What does the surface area of a solid object measure?
Easy
- AThe total area that the surface of the object occupies
- BThe space inside the object
- CThe length of the object's edges
- DThe weight of the object
2.Surface area is the sum of the areas of its faces for objects with flat polygonal faces.
Easy
True or false?
3.Which formula correctly gives the area of a triangle with base b and height h?
Easy
- AA = (b × h) / 2
- BA = b × h
- CA = 2 × b × h
- DA = b + h
4.A rectangle has length 8 cm and width 5 cm. What is its area?
Medium
- A40 cm²
- B13 cm²
- C26 cm²
- D45 cm²
5.A rectangular prism has length 5 cm, width 3 cm, and height 4 cm. What is its volume?
Medium
- A60 cm³
- B12 cm³
- C20 cm³
- D45 cm³
6.A triangular prism has a base area of 10 cm² and a height of 7 cm. What is its volume?
Medium
- A70 cm³
- B17 cm³
- C35 cm³
- D140 cm³
7.Which of the following shapes have flat faces? (select all that apply)
Medium
- ACube
- BSphere
- CCylinder
- DPyramid
- ECone
8.The surface area of a solid is the same regardless of its orientation in space.
Easy
True or false?

Thêm câu hỏi

9.Match each formula to the correct shape.
Medium
- A = l × w
- A = (b × h) / 2
- A = 6 × s²
- Rectangle
- Triangle
- Cube
10.Order the steps to find the surface area of a rectangular prism from a net.
Medium
- Find the area of each face.
- Add all the face areas together.
- Draw the net of the prism.
11.A rectangular prism has dimensions 2 cm by 3 cm by 4 cm. What is its surface area in square centimetres?
Hard
- A52
- B24
- C26
- D48
12.Match each shape to the correct formula for its area.
Easy
- Triangle
- Parallelogram
- Trapezium
- \A = \\frac{1}{2} \\times b \\times h\
- \A = b \\times h\
- \A = \\frac{1}{2} \\times (a + b) \\times h\
13.Put these steps in the correct order to find the surface area of a rectangular prism using a net.
Medium
- Draw the net of the prism.
- Find the area of each face.
- Add the areas of all the faces.
- Label the dimensions on the net.
14.Which of the following are correct ways to find the area of a compound shape? (select all that apply)
Medium
- ASplit it into rectangles and triangles, then add their areas.
- BMultiply all the side lengths together.
- CSubtract the area of a smaller shape from a larger shape.
- DCount the number of faces and multiply by 2.
15.The area of a triangle is always half the area of a rectangle with the same base and height.
Easy
True or false?
16.A rectangular prism has a length of 4 cm, a width of 3 cm, and a height of 2 cm. What is its surface area in square centimetres?
Medium
- A24
- B26
- C48
- D52
17.A cube has a side length of 5 cm. What is its volume in cubic centimetres?
Medium
- A25
- B50
- C125
- D150
18.Which of the following shapes have a volume? (select all that apply)
Medium
- AA cube
- BA triangle
- CA rectangular prism
- DA circle
19.What is the area of a rectangle with length 9 cm and width 4 cm?
Easy
- A36 cm²
- B13 cm²
- C26 cm²
- D40 cm²
20.What is the area of a triangle with base 8 cm and height 5 cm?
Medium
- A20 cm²
- B40 cm²
- C13 cm²
- D15 cm²
21.What is the area of a parallelogram with base 7 cm and height 6 cm?
Medium
- A42 cm²
- B13 cm²
- C26 cm²
- D48 cm²
22.A trapezium has parallel sides of 5 cm and 9 cm, and a height of 4 cm. What is its area?
Medium
- A28 cm²
- B36 cm²
- C56 cm²
- D14 cm²
23.What is the volume of a rectangular prism with length 5 cm, width 4 cm, and height 3 cm?
Medium
- A60 cm³
- B12 cm³
- C30 cm³
- D47 cm³
24.What is the surface area of a cube with side length 3 cm?
Medium
- A54 cm²
- B27 cm²
- C18 cm²
- D36 cm²
25.Which of the following formulas are correct? (select all that apply)
Medium
- AArea of a triangle = 1/2 × base × height
- BArea of a parallelogram = base × height
- CArea of a trapezium = 1/2 × (sum of parallel sides) × height
- DArea of a rectangle = length × width
- EVolume of a cube = side × side
26.To find the surface area of a rectangular prism, you multiply its length, width, and height.
Medium
True or false?
27.Put these steps in the correct order to find the surface area of a rectangular prism using its net.
Medium
- Find the area of each face.
- Add all the areas together.
- Draw the net of the prism.
- Label the dimensions on the net.
28.A rectangular prism has a length of 8 cm, a width of 5 cm, and a height of 3 cm. What is its volume?
Medium
- A120 cm³
- B120 cm²
- C40 cm³
- D16 cm³
29.A rectangular prism has a length of 7 cm, a width of 4 cm, and a height of 2 cm. What is its surface area?
Medium
- A100 cm²
- B56 cm²
- C88 cm²
- D28 cm²
30.The volume of a rectangular prism is found by multiplying its length, width, and height.
Easy
True or false?
31.Surface area is measured in cubic units.
Easy
True or false?
32.Match each 3-D shape to the correct formula for its volume.
Medium
- Rectangular prism
- Cube
- Triangular prism
- length × width × height
- side³
- base area × height
33.What does the surface area (symbol A) of a solid object measure?
Easy
- AThe total area that the surface of the object occupies
- BThe space inside the object
- CThe length of the object's edges
- DThe weight of the object
34.Surface area is invariant under rotations and translations, meaning it does not change if you move or rotate the object.
Easy
True or false?
35.Which of the following is the correct symbol for surface area?
Easy
- AA
- BV
- CP
- DS
36.Surface area is measured in what kind of units?
Easy
- ASquare units
- BCubic units
- CLinear units
- DWeight units
37.The surface area of a solid depends on its orientation in space.
Easy
True or false?
38.Which of the following is a correct statement about surface area?
Easy
- AThe area of the whole is the sum of the areas of its non-overlapping parts.
- BSurface area is measured in cubic units.
- CSurface area is the same as volume.
- DSurface area only applies to curved surfaces.
39.Match each term to its correct meaning.
Easy
- Surface area
- Volume
- Net
- Total area of the surface
- Space inside a solid
- Flat pattern that folds into a solid
40.Which of the following are properties of surface area? (select all that apply)
Medium
- AIt is additive: the area of the whole is the sum of the areas of the parts.
- BIt is measured in square units.
- CIt is measured in cubic units.
- DIt depends on the orientation of the object.
- EFor polyhedra, it is the sum of the areas of the faces.
41.A cube has 6 faces, each with an area of 4 square units. What is the total surface area of the cube?
Easy
42.Fill in the blanks with the correct units for area and volume.
Easy
Area is measured in ____ units, while volume is measured in ____ units.
43.What is the symbol commonly used to represent surface area?
Medium
44.A cube has a side length of 3 cm. What is its surface area in square centimetres?
Medium
45.Fill in the blank: The area of a parallelogram with base 10 cm and height 6 cm is ___ square centimetres.
Medium
The area is ____ cm².
46.What is the volume of the box in cubic centimetres?
Medium
47.What is the surface area of the box in square centimetres?
Medium
48.A parallelogram has a base of 12 cm and a height of 7 cm. What is its area in square centimetres?
Medium
49.A trapezium has parallel sides of 6 cm and 10 cm, and a height of 5 cm. What is its area in square centimetres?
Medium
50.Complete the sentence.
Easy
The volume of a cuboid is found by multiplying its length, width and ____.
51.A rectangle has a length of 12 cm and a width of 5 cm. What is its area? Give your answer in square centimetres.
Easy
52.A parallelogram has a base of 14 cm and a height of 9 cm. What is its area? Give your answer in square centimetres.
Medium
53.Explain why the surface area of a solid is the sum of the areas of its faces.
Hard
54.Explain how you would find the surface area of a triangular prism.
Medium
55.Explain how to find the surface area of a rectangular prism with length 4 cm, width 3 cm, and height 2 cm.
Medium
56.Explain how to find the surface area of a triangular prism.
Hard
57.A cube has 6 faces, each with an area of 4 square units. What is the total surface area of the cube?
Easy
- A24 square units
- B22 square units
- C28 square units
- D26 square units
58.A cube has a side length of 3 cm. What is its surface area in square centimetres?
Medium
- A64 cm²
- B54 cm²
- C59 cm²
- D49 cm²
59.What is the volume of the box in cubic centimetres?
Medium
- A86 cm³
- B72 cm³
- C79 cm³
- D65 cm³
60.What is the surface area of the box in square centimetres?
Medium
- A108 cm²
- B130 cm²
- C97 cm²
- D119 cm²
61.A parallelogram has a base of 12 cm and a height of 7 cm. What is its area in square centimetres?
Medium
- A92 cm²
- B100 cm²
- C84 cm²
- D76 cm²
62.A trapezium has parallel sides of 6 cm and 10 cm, and a height of 5 cm. What is its area in square centimetres?
Medium
- A36 cm²
- B48 cm²
- C44 cm²
- D40 cm²
63.A rectangle has a length of 12 cm and a width of 5 cm. What is its area? Give your answer in square centimetres.
Easy
- A60 cm²
- B66 cm²
- C54 cm²
- D72 cm²
64.A parallelogram has a base of 14 cm and a height of 9 cm. What is its area? Give your answer in square centimetres.
Medium
- A152 cm²
- B113 cm²
- C139 cm²
- D126 cm²

Ghi chú ôn tập

## Area, Surface Area and Volume – Quick Revision ### Area of 2-D Shapes - **Rectangle**: Area = length × width. Units: square units (e.g., cm², m²). - **Triangle**: Area = ½ × base × height. The height is the perpendicular distance from the base to the opposite vertex. - **Parallelogram**: Area = base × height. The height is the perpendicular distance between the two bases. - **Trapezium**: Area = ½ × (a + b) × h, where a and b are the lengths of the parallel sides, and h is the perpendicular height between them. - **Compound shapes**: Split the shape into rectangles and triangles, find each area, then add them together. ### Volume of a Cuboid - **Volume** = length × width × height. Units: cubic units (e.g., cm³, m³). - Works for fractional edge lengths too – just multiply the fractions. ### Surface Area of 3-D Solids - **Surface area** is the total area of all the faces of a solid. It is measured in square units. - **Net**: A flat pattern that folds into a 3-D shape. To find surface area, draw the net, find the area of each face, then add them all up. - For a **cube** with side length s, surface area = 6 × s² (because it has 6 equal square faces). - For a **cuboid** with length l, width w, height h, surface area = 2(lw + lh + wh). - For **prisms** and **pyramids**, use the net: add the areas of all faces. ### Choosing Units - Use **square units** (like cm²) for area and surface area. - Use **cubic units** (like cm³) for volume. - Always match the unit to the problem: if lengths are in metres, area is in m², volume in m³. ### Real-World Problems - **Packaging**: Find surface area to know how much cardboard is needed. - **Painting**: Find surface area to know how much paint to buy. - **Filling containers**: Find volume to know how much liquid fits. Remember: For any shape, the area of the whole is the sum of the areas of its parts. This is called **additivity**.

Area: The amount of space inside a flat shape. Measured in square units (like cm²).
Surface area: The total area of all the faces of a 3-D solid. Measured in square units.
Volume: The amount of space inside a 3-D solid. Measured in cubic units (like cm³).
Compound shape: A shape made by joining two or more simple shapes (like rectangles and triangles).
Net: A flat pattern that can be folded to make a 3-D shape.
Cuboid: A 3-D shape with six rectangular faces. A cube is a special cuboid with all square faces.
Prism: A 3-D shape with two identical ends and flat sides. The ends are polygons.
Pyramid: A 3-D shape with a polygon base and triangular faces that meet at a point (apex).
Additivity: The property that the area of a whole shape is the sum of the areas of its parts.

Thẻ ghi nhớ

What is the area of a triangle with base 6 cm and height 4 cm?

A = 1/2 × b × h = 1/2 × 6 × 4 = 12 cm².

What is the area of a parallelogram with base 5 cm and height 3 cm?

A = b × h = 5 × 3 = 15 cm².

What is the area of a trapezium with parallel sides 4 cm and 6 cm, and height 5 cm?

A = (a + b)/2 × h = (4 + 6)/2 × 5 = 25 cm².

How do you find the area of a compound shape?

Split it into rectangles and triangles, find each area, then add them together.

What is the volume of a cuboid with length 3 cm, width 2 cm, and height 5 cm?

V = l × w × h = 3 × 2 × 5 = 30 cm³.

What is a net of a 3-D shape?

A flat shape that folds to make the 3-D shape.

How do you find the surface area of a cuboid?

Add the areas of all six faces: 2(lw + lh + wh).

What is the surface area of a cube with side length 4 cm?

Each face is 4×4=16 cm², and there are 6 faces, so total = 6×16 = 96 cm².

What is the surface area of a cuboid with length 5 cm, width 3 cm, and height 2 cm?

A = 2(lw + lh + wh) = 2(5×3 + 5×2 + 3×2) = 2(15+10+6) = 62 cm².

What is the volume of a cuboid with length 2.5 cm, width 2 cm, and height 1.5 cm?

V = l × w × h = 2.5 × 2 × 1.5 = 7.5 cm³.

What is the surface area of a triangular prism?

Find the area of the two triangles and the three rectangles, then add them all together.

What is the surface area of a square pyramid?

Add the area of the square base and the areas of the four triangular faces.

What is the volume of a cuboid with fractional edge lengths 1/2 cm, 2 cm, and 3 cm?

V = 1/2 × 2 × 3 = 3 cm³.

What is the volume of a cube with side length 1/2 cm?

V = (1/2)³ = 1/8 cm³.

What is the surface area of a cube with side length 1/2 cm?

Each face is (1/2)² = 1/4 cm², so total = 6 × 1/4 = 1.5 cm².

What is the area of a triangle with base 3/4 cm and height 2/3 cm?

A = 1/2 × 3/4 × 2/3 = 1/4 cm².

What is the area of a parallelogram with base 2.5 cm and height 1.2 cm?

A = 2.5 × 1.2 = 3 cm².

What is the area of a trapezium with parallel sides 3 cm and 5 cm, and height 4 cm?

A = (3+5)/2 × 4 = 16 cm².

What is the volume of a cuboid with length 4 cm, width 2.5 cm, and height 3 cm?

V = 4 × 2.5 × 3 = 30 cm³.

What is the surface area of a cuboid with length 2 cm, width 2 cm, and height 2 cm?

It's a cube: A = 6 × 2² = 24 cm².

What is the volume of a cuboid with length 1.5 cm, width 1 cm, and height 2 cm?

V = 1.5 × 1 × 2 = 3 cm³.

What is the surface area of a triangular prism with triangle base 3 cm, height 4 cm, and prism length 5 cm?

Area of each triangle = 1/2 × 3 × 4 = 6 cm², so two triangles = 12 cm². Rectangles: 3×5=15, 4×5=20, and 5×5=25, sum = 60. Total = 12+60 = 72 cm².

What is the surface area of a square pyramid with base side 4 cm and triangular face height 6 cm?

Base area = 4×4=16 cm². Each triangle area = 1/2 × 4 × 6 = 12 cm², four triangles = 48 cm². Total = 16+48 = 64 cm².

What is the volume of a cuboid with length 2 cm, width 1/2 cm, and height 3 cm?

V = 2 × 1/2 × 3 = 3 cm³.

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