Volume and Surface Area of Prisms and Cylinders

Multiply the triangle area by the prism length.

1. The cross-section area is \frac12\times3\times4=6\text{ cm}^2.
2. The volume is 6\times7=42\text{ cm}^3.

Halve the diameter before using the cylinder volume formula.

1. The radius is 4\text{ cm}.
2. The volume is \pi\times4^2\times5=80\pi\text{ cm}^3.

Divide the volume by the product of the two known dimensions.

1. The width satisfies 9\times4\times w=252.
2. Therefore w=252\div36=7\text{ cm}.

There are two faces of each of three different sizes.

1. The three face areas are 3\times5=15, 3\times8=24 and 5\times8=40\text{ cm}^2.
2. The total surface area is 2(15+24+40)=158\text{ cm}^2.

Add the curved surface and both circular ends.

1. The curved area is 2\pi rh=2\pi\times3\times7=42\pi\text{ cm}^2.
2. The two ends total 2\pi\times3^2=18\pi\text{ cm}^2.
3. The total is 60\pi\text{ cm}^2.

Only one circular end is present.

1. The curved surface area is 2\pi\times5\times6=60\pi\text{ cm}^2.
2. The base area is \pi\times5^2=25\pi\text{ cm}^2.
3. The material area is 85\pi\text{ cm}^2.

Subtract the corner area before multiplying by the prism length.

1. The cross-section area is 6\times5-2\times3=24\text{ cm}^2.
2. The volume is 24\times9=216\text{ cm}^3.

Use the trapezium area formula for the cross-section.

1. The cross-section area is \frac12(4+10)\times5=35\text{ cm}^2.
2. The volume is 35\times6=210\text{ cm}^3.

Find the volume in cubic centimetres before converting to litres.

1. The capacity is \pi\times20^2\times50=20000\pi\text{ cm}^3.
2. Since 1000\text{ cm}^3=1 litre, this is 20\pi=62.8318\ldots litres.
3. To 3 significant figures, the capacity is 62.8 litres.

Add the two triangular ends and the three rectangular faces.

1. The two ends total 2\times\frac12\times3\times4=12\text{ cm}^2.
2. The rectangular faces total (3+4+5)\times10=120\text{ cm}^2.
3. Unfolding the faces shows the two triangular ends and three rectangles separately.

4. The total surface area is 132\text{ cm}^2.

The water volume stays the same, but the horizontal cross-section changes.

1. The water volume is \pi\times3^2\times10=90\pi\text{ cm}^3.
2. The cuboid base area is 9\times5=45\text{ cm}^2.
3. The depth is 90\pi\div45=2\pi\text{ cm}.

Divide the volume by the circular base area.

1. The height satisfies 25\pi h=120\pi.
2. Hence h=120\div25=4.8\text{ cm}.

Use curved surface area to find the radius first.

1. The equation is 2\pi r\times9=54\pi.
2. Thus r=3\text{ cm}.
3. The diameter is 6\text{ cm}.

Find both the area and the perimeter of the cross-section.

1. Total surface area of a closed prism = two end faces + the rectangles round the sides.
2. Each end face (the L-shape) has area 8\times6-3\times2=42\text{ cm}^2.
3. Go round the L-shape: the full sides are 8\text{ cm} and 6\text{ cm}, the notch edges are 3\text{ cm} and 2\text{ cm}, and the two shortened sides are 8-3=5\text{ cm} and 6-2=4\text{ cm}. The perimeter is 8+6+3+2+5+4=28\text{ cm}.
4. The two shortened edges are 5 cm and 4 cm. The side faces can be unfolded into a strip with width equal to the cross-section perimeter.

5. Each edge of the L-shape makes a rectangle 5\text{ cm} long, so the side faces total 28\times5=140\text{ cm}^2.
6. The total surface area is 2\times42+140=224\text{ cm}^2.

The added volume raises the surface by the added volume divided by the base area.

1. The base area is 25\pi\text{ cm}^2.
2. The rise is \frac{150}{25\pi}=\frac6\pi\text{ cm}.
3. The new depth is 8+\frac6\pi=9.90985\ldots\text{ cm}, or 9.91\text{ cm} to 3 significant figures.

Work in centimetres throughout, find the volume, change it to litres, then divide by the rate.

1. The cross-section is a trapezium: area =\frac12(a+b)h=\frac12(60+40)\times30=1500\text{ cm}^2.
2. The length is 2\text{ m}=200\text{ cm}, so the volume is 1500\times200=300\,000\text{ cm}^3.
3. Since 1000\text{ cm}^3=1 litre, the trough holds 300\,000\div1000=300 litres.
4. At 6 litres per minute, it takes 300\div6=50 minutes to fill.
5. (Using 2 for the length while the other lengths are in cm would give an answer 100 times too small.)

Find the missing triangle side before using the volume.

1. The other perpendicular side is \sqrt{10^2-6^2}=8\text{ cm}.
2. The cross-section area is \frac12\times6\times8=24\text{ cm}^2.
3. The prism length is 162\div24=6.75\text{ cm}.

Write the area of each of the six faces in terms of x and form a quadratic equation.

1. The top and bottom are squares, each with area x^2\text{ cm}^2. The four sides are rectangles x by 5, each with area 5x\text{ cm}^2.
2. Group the six faces by their areas before forming the equation.

3. The total surface area gives 2x^2+4\times5x=192, so 2x^2+20x-192=0.
4. Divide every term by 2: x^2+10x-96=0.
5. Factorise: (x+16)(x-6)=0, so x=-16 or x=6.
6. A length must be positive, so x=6. (Check: 2\times36+20\times6=72+120=192.)

Use cross-section area, then volume, then density, and finally convert grams to kilograms.

1. The cross-section area is \frac12(8+12)\times5=50\text{ cm}^2.
2. The volume is 50\times15=750\text{ cm}^3.
3. The mass is 750\times2.4=1800\text{ g}.
4. This is 1.8\text{ kg}.

Remove the two circular openings and add the curved wall inside the hole.

1. The original cuboid surface area is 2(10\times8+10\times6+8\times6)=376\text{ cm}^2.
2. The two openings remove 2\pi\times2^2=8\pi\text{ cm}^2.
3. The hole wall adds 2\pi\times2\times6=24\pi\text{ cm}^2.
4. The openings remove area, while the cylindrical wall adds exposed area.

5. The exposed area is (376-8\pi+24\pi)=(376+16\pi)\text{ cm}^2.