Circles, Area, Surface Area and Volume: practice solutions, Grade 7 – download the PDF

1 Radius or diameter? ★★★

The radius is half of the diameter, and the diameter is twice the radius.

1. \(r = 14 \div 2 = 7\) cm.
2. \(d = 2 \times 2.5 = 5\) in.
3. \(r = 9 \div 2 = 4.5\) ft.

2 Fencing a pond ★★★

The fence follows the circumference. \(C = \pi d \approx 3.14 \times 12 = 37.68\).

The fence is about 37.68 m long.

3 A small circle ★★★

\(A = \pi r^2 \approx 3.14 \times 5^2 = 3.14 \times 25 = 78.5\).

The area is about 78.5 cm².

4 Triangle and parallelogram ★★★

1. \(A = \dfrac{1}{2} \times 14 \times 9 = 7 \times 9 = 63\) in².
2. \(A = b \times h = 11 \times 6 = 66\) cm².

5 True or false? ★★★

1. False. For \(r = 3\): \(A = 3.14 \times 9 = 28.26\). For \(r = 6\): \(A = 3.14 \times 36 = 113.04\). Since \(113.04 \div 28.26 = 4\), the area is multiplied by 4, not 2.
2. True. \(C = \pi d\) and \(\pi \approx 3.14\) is a little more than 3.
3. False. The diameter is twice the radius.

6 Aquarium volume ★★★

\(V = \ell \times w \times h = 24 \times 10 \times 12 = 240 \times 12 = 2{,}880\).

The aquarium holds 2,880 in³.

7 Wrapping a cube ★★★

A cube has 6 identical square faces, each of area \(5 \times 5 = 25\) cm².

\(SA = 6 \times 25 = 150\). You need 150 cm² of paper.

8 Trapezoid garden bed ★★★

\(A = \dfrac{1}{2}(7 + 13) \times 5 = \dfrac{1}{2} \times 20 \times 5 = 10 \times 5 = 50\).

The area is 50 m².

9 Bicycle wheel ★★★

One turn covers the circumference: \(C = 3.14 \times 26 = 81.64\) in.

For 50 turns: \(81.64 \times 50 = 4{,}082\) in.

Since \(1 \text{ ft} = 12 \text{ in}\): \(4{,}082 \div 12 \approx 340.2\).

The bike travels 4,082 in, or about 340.2 ft.

10 From circumference to radius ★★★

From \(C = \pi d\) we get \(d = C \div \pi\).

\(d = 47.1 \div 3.14 = 15\) cm, because \(3.14 \times 15 = 47.1\).

The radius is \(15 \div 2 = 7.5\) cm.

The diameter is 15 cm and the radius is 7.5 cm.

11 Which pizza deal? ★★★

12-inch pizza: \(r = 6\), so \(A = 3.14 \times 6^2 = 3.14 \times 36 = 113.04\) in².

8-inch pizza: \(r = 4\), so \(A = 3.14 \times 4^2 = 3.14 \times 16 = 50.24\) in². Two of them: \(2 \times 50.24 = 100.48\) in².

\(113.04 - 100.48 = 12.56\).

The single 12-inch pizza is the better deal, with 12.56 in² more pizza.

12 Garden with a pond ★★★

Garden: \(15 \times 8 = 120\) m². Pond: \(3.14 \times 2^2 = 3.14 \times 4 = 12.56\) m².

\(120 - 12.56 = 107.44\).

107.44 m² of the garden is not covered by water.

13 Tent fabric ★★★

Two triangular ends: \(2 \times \dfrac{1}{2} \times 6 \times 4 = 24\) ft².

Floor: \(6 \times 9 = 54\) ft². Two slanted sides: \(2 \times (5 \times 9) = 90\) ft².

\(SA = 24 + 54 + 90 = 168\).

The tent needs 168 ft² of fabric.

14 Volume of a ramp ★★★

Base area: \(B = \dfrac{1}{2} \times 10 \times 6 = 30\) cm².

\(V = B \times h = 30 \times 15 = 450\).

The volume is 450 cm³.

15 A pyramid-shaped box ★★★

Base: \(10 \times 10 = 100\) cm².

One triangular face: \(\dfrac{1}{2} \times 10 \times 13 = 65\) cm². Four faces: \(4 \times 65 = 260\) cm².

\(SA = 100 + 260 = 360\). The surface area is 360 cm².

16 Find the missing measurement ★★★

1. \(A = b \times h\), so \(h = 84 \div 12 = 7\) cm.
2. \(A = \dfrac{1}{2} b h\), so \(45 = \dfrac{1}{2} \times b \times 9\), which gives \(45 = 4.5\,b\) and \(b = 10\) in.

17 The sports field ★★★

The two semicircles together make one full circle of diameter 50 m (radius 25 m).

1. Two straight sides: \(2 \times 80 = 160\) m. Circle: \(3.14 \times 50 = 157\) m. Distance around: \(160 + 157 = 317\) m.
2. Rectangle: \(80 \times 50 = 4{,}000\) m². Circle: \(3.14 \times 25^2 = 3.14 \times 625 = 1{,}962.5\) m². Area: \(4{,}000 + 1{,}962.5 = 5{,}962.5\) m².
3. \(3 \times 317 = 951\) m.

18 Path around a fountain ★★★

The outer circle has a radius of \(4 + 1.5 = 5.5\) m. Path area = outer circle \(-\) fountain.

Outer circle: \(3.14 \times 5.5^2 = 3.14 \times 30.25 = 94.985\). Fountain: \(3.14 \times 4^2 = 3.14 \times 16 = 50.24\).

\(94.985 - 50.24 = 44.745 \approx 44.7\).

The path has an area of about 44.7 m².

19 Filling a water tank ★★★

1. \(V = 2 \times 1.5 \times 1.2 = 3.6\) m³.
2. \(3.6 \times 1{,}000 = 3{,}600\) L.
3. \(3{,}600 \div 45 = 80\) minutes.

The tank holds 3,600 L and is full after 80 minutes.

20 A box without a lid ★★★

Bottom: \(30 \times 20 = 600\) cm².

Two long sides: \(2 \times (30 \times 15) = 900\) cm². Two short sides: \(2 \times (20 \times 15) = 600\) cm².

\(600 + 900 + 600 = 2{,}100\). The box uses 2,100 cm² of cardboard.

(A closed box would also count a top of 600 cm², giving 2,700 cm².)

21 Swimming pool ★★★

1. The parallel sides are the two depths: \(B = \dfrac{1}{2}(1 + 3) \times 20 = 2 \times 20 = 40\) m².
2. \(V = B \times h = 40 \times 10 = 400\) m³.
3. \(400 \times 1{,}000 = 400{,}000\) L.

The pool holds 400 m³, which is 400,000 L.

22 Find the error ★★★

1. The student squared the diameter. The formula needs the radius, which is \(10 \div 2 = 5\) cm.
2. \(A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5\) cm².
3. \(314 \div 78.5 = 4\). The answer was 4 times too large, because the diameter is twice the radius, and \(2^2 = 4\).