Area, Surface Area, and Volume: practice solutions, Grade 6

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Practice solutions Grade 6 : Area, Surface Area, and Volume — Zyro the alien explorer of Planète Maths

Written solutions to the chapter problems. Check each step, then correct yourself.

1 Area of a triangle ★★★

Use $ A = \dfrac{1}{2} b h $.

$ A = \dfrac{1}{2} \times 12 \times 7 = 6 \times 7 = 42 $.

The area of the sail is 42 cm².

2 Area of a parallelogram ★★★

The area is base times perpendicular height: $ A = 15 \times 6 = 90 $. The slanted side (8 in) is not needed.

The area is 90 in².

3 True or false: triangle area ★★★

He is wrong. He found the area of a rectangle $ 10 \times 6 = 60 $. A triangle is half of that rectangle: $ \dfrac{1}{2} \times 10 \times 6 = 30 $.

The correct area is 30 m².

4 Area of a trapezoid ★★★

$ A = \dfrac{5 + 9}{2} \times 4 = 7 \times 4 = 28 $.

The area is 28 cm².

5 Rectangle on a grid ★★★

The sides are horizontal and vertical, so the shape is a rectangle. Length: $ 8 - 2 = 6 $. Width: $ 5 - 1 = 4 $.

$ A = 6 \times 4 = 24 $. The area is 24 square units.

6 Name the solid from its net ★★★

A cube.
A square pyramid (the square is the base, the triangles are the lateral faces).
A triangular prism (two triangular bases and three rectangular lateral faces).

7 Volume of a box ★★★

$ V = \ell \times w \times h = 8 \times 5 \times 3 = 120 $.

The volume is 120 cm³.

8 Finding a missing height ★★★

$ A = \dfrac{1}{2} b h $, so $ 54 = \dfrac{1}{2} \times 12 \times h = 6h $.

$ h = 54 \div 6 = 9 $. The height is 9 m.

9 Mulch for a flower bed ★★★

Area: $ 3.5 \times 2 = 7 $ m².

Bags: $ 7 \div 1.75 = 4 $.

You need 4 bags of mulch.

10 A trapezoid sail ★★★

$ A = \dfrac{4 + 6}{2} \times 3.5 = 5 \times 3.5 = 17.5 $ ft².

Cost: $ 17.5 \times 4 = 70 $.

The fabric costs $70.

11 Carpeting a room ★★★

Main room: $ 9 \times 6 = 54 $ m². Alcove: $ 4 \times 3 = 12 $ m². Total: $ 54 + 12 = 66 $ m².

Cost: $ 66 \times 12 = 792 $.

The carpet costs $792.

12 Triangle on the coordinate plane ★★★

Take AB as the base: it is horizontal, so $ b = 7 - 1 = 6 $. The height is the vertical distance from C to the line $ y = 2 $: $ h = 8 - 2 = 6 $.

$ A = \dfrac{1}{2} \times 6 \times 6 = 18 $. The area is 18 square units.

13 Wrapping a cube ★★★

A cube has 6 identical square faces, each of area $ 7 \times 7 = 49 $ cm².

$ SA = 6 \times 49 = 294 $. You need 294 cm² of paper.

14 A box with mixed numbers ★★★

$ V = \dfrac{7}{2} \times 2 \times \dfrac{3}{2} = 7 \times \dfrac{3}{2} = \dfrac{21}{2} = 10.5 $.

The volume is 10.5 in³.

15 Gift box surface area ★★★

Face areas: $ 12 \times 5 = 60 $, $ 12 \times 4 = 48 $, $ 5 \times 4 = 20 $. $ SA = 2(60 + 48 + 20) = 2 \times 128 = 256 $ cm².
The lid and bottom are the two $ 12 \times 5 $ faces: $ 2 \times 60 = 120 $. Painted area: $ 256 - 120 = 136 $ cm².

The surface area is 256 cm² and the painted area is 136 cm².

16 A square pyramid ★★★

One triangular face: $ \dfrac{1}{2} \times 10 \times 13 = 65 $ cm². Four faces: $ 4 \times 65 = 260 $ cm².
Base: $ 10 \times 10 = 100 $ cm². $ SA = 100 + 260 = 360 $ cm².

The lateral area is 260 cm² and the surface area is 360 cm².

17 The camping tent ★★★

Each triangle: $ \dfrac{1}{2} \times 6 \times 4 = 12 $ ft², so the two ends are 24 ft². Each sloping side: $ 5 \times 9 = 45 $ ft², so 90 ft² for two. Fabric: $ 24 + 90 = 114 $ ft².
The floor is $ 6 \times 9 = 54 $ ft². $ SA = 114 + 54 = 168 $ ft².

The fabric area is 114 ft² and the total surface area is 168 ft².

18 A swallowtail banner ★★★

Rectangle: $ 18 \times 10 = 180 $ cm². Notch: $ \dfrac{1}{2} \times 10 \times 4 = 20 $ cm².

$ 180 - 20 = 160 $. The banner has an area of 160 cm².

19 An L-shaped polygon on the grid ★★★

AB $ = 7 - 1 = 6 $; BC $ = 4 - 1 = 3 $; CD $ = 7 - 4 = 3 $; DE $ = 7 - 4 = 3 $; EF $ = 4 - 1 = 3 $; FA $ = 7 - 1 = 6 $.
Perimeter: $ 6 + 3 + 3 + 3 + 3 + 6 = 24 $ units.
Split at $ y = 4 $: the bottom rectangle is $ 6 \times 3 = 18 $ and the top rectangle is $ 3 \times 3 = 9 $. $ A = 18 + 9 = 27 $ square units.

The perimeter is 24 units and the area is 27 square units.

20 Water in a fish tank ★★★

$ V = \dfrac{5}{2} \times \dfrac{3}{2} \times 2 = \dfrac{15}{2} = 7.5 $ ft³.
The water height is $ \dfrac{3}{4} \times 2 = \dfrac{3}{2} = 1.5 $ ft. $ V = \dfrac{5}{2} \times \dfrac{3}{2} \times \dfrac{3}{2} = \dfrac{45}{8} = 5.625 $ ft³. (Check: $ \dfrac{3}{4} \times 7.5 = 5.625 $.)

The tank holds 7.5 ft³ and the water takes up 5.625 ft³ (that is $ 5\dfrac{5}{8} $ ft³).

21 Same base, same height ★★★

Both triangles use the segment from $ (0, 0) $ to $ (8, 0) $ as a base, so $ b = 8 $. The third vertex of each triangle is 5 units above the $ x $-axis, so both heights equal 5, even though the shapes lean differently.

$ A = \dfrac{1}{2} \times 8 \times 5 = 20 $ for each one.

Both triangles have an area of 20 square units.