Spheres, Cones and Pyramids

Work out the area of the square base first.

Work out 3^3 first, then multiply by \frac{4}{3}.

Use the vertical height, not the slant height.

Square the radius first, then multiply by 4 and by \pi.

Add the area of the square base to the areas of the four triangles.

The curved surface uses the slant height, not the vertical height.

1. r=5 and l=13 (the slant height).
2. Curved surface area =\pi\times5\times13=65\pi\text{ cm}^2.

The base is a rectangle, so its area is length times width.

1. Area of base =8\times5=40\text{ cm}^2.
2. Volume =\frac{1}{3}\times40\times9=40\times3=120\text{ cm}^3.

The formula uses the radius, which is half the diameter.

1. Radius =14\div2=7 cm.
2. Volume =\frac{4}{3}\times\pi\times7^3=\frac{4}{3}\times\pi\times343=1436.75...\text{ cm}^3.
3. Correct to 3 significant figures: 1440\text{ cm}^3.

A solid cone has a curved surface and a flat circular base. Add the two areas.

1. Curved surface area =\pi\times3\times7=21\pi\text{ cm}^2.
2. Area of the base =\pi\times3^2=9\pi\text{ cm}^2.
3. Total =30\pi=94.24...\text{ cm}^2, which is 94.2\text{ cm}^2 to 1 decimal place.

Work out the volume of the whole sphere first.

1. Whole sphere: \frac{4}{3}\times\pi\times6^3=\frac{4}{3}\times216\pi=288\pi\text{ cm}^3.
2. Hemisphere: 288\pi\div2=144\pi\text{ cm}^3.

Work out the area of the base, then think about what you multiply it by to get the volume.

1. Area of base =6\times4=24\text{ cm}^2.
2. \frac{1}{3}\times24\times height =80, so 8\times height =80.
3. Vertical height =80\div8=10 cm.

The curved part is half of the surface of a whole sphere. The flat part is a circle.

Part (a)

1. Whole sphere: 4\times\pi\times5^2=100\pi\text{ cm}^2.
2. Curved surface of the hemisphere =100\pi\div2=50\pi\text{ cm}^2.

Part (b)

1. Flat face =\pi\times5^2=25\pi\text{ cm}^2.
2. Total surface area =50\pi+25\pi=75\pi\text{ cm}^2.

Find the volume of the box and the volume of the ball, then subtract.

1. Volume of the box =8\times8\times8=512\text{ cm}^3.
2. Volume of the ball =\frac{4}{3}\times\pi\times4^3=268.08...\text{ cm}^3.
3. Space =512-268.08...=243.91...\text{ cm}^3, which is 243.9\text{ cm}^3 to 1 decimal place.

Work out the volume in cubic metres first, then change it to litres.

1. Volume =\frac{4}{3}\times\pi\times0.6^3=\frac{4}{3}\times\pi\times0.216=0.9047...\text{ m}^3.
2. In litres: 0.9047...\times1000=904.77... litres.
3. To the nearest litre: 905 litres.

The vertical height goes straight down to the centre of the base. The height of a triangular face runs along the face.

Part (a)

1. Area of base =10\times10=100\text{ cm}^2.
2. Volume =\frac{1}{3}\times100\times12=400\text{ cm}^3.

Part (b)

1. Area of one triangle =\frac{1}{2}\times10\times13=65\text{ cm}^2, so the four triangles give 260\text{ cm}^2.
2. Total surface area =100+260=360\text{ cm}^2.

Find the volume of the cylinder and the volume of the hemisphere separately, then add them.

1. Cylinder: \pi\times4^2\times10=160\pi=502.65...\text{ cm}^3.
2. Whole sphere of radius 4 cm: \frac{4}{3}\times\pi\times4^3=268.08...\text{ cm}^3, so the hemisphere is 134.04...\text{ cm}^3.
3. Total =502.65...+134.04...=636.69...\text{ cm}^3, which is 636.7\text{ cm}^3 to 1 decimal place.

Find the volume of the jug and the volume of one cup, then divide.

1. Jug: \pi\times5^2\times20=500\pi=1570.79...\text{ cm}^3.
2. One cup: \frac{1}{3}\times\pi\times3^2\times10=30\pi=94.24...\text{ cm}^3.
3. 1570.79...\div94.24...=16.66...
4. Only whole cups count, so 16 cups can be filled completely.

Work out the surface area of the cuboid, then use it in the formula and work backwards to find r.

1. Surface area of the cuboid =2\times(10\times8+10\times5+8\times5)=2\times170=340\text{ cm}^2.
2. 4\pi r^2=340, so r^2=340\div(4\pi)=27.05...
3. r=\sqrt{27.05...}=5.201... cm.
4. Correct to 1 decimal place: 5.2 cm.

The formula needs the slant height. The radius, the vertical height and the slant height make a right-angled triangle.

1. Slant height: l^2=9^2+12^2=81+144=225, so l=15 cm.
2. One hat =\pi\times9\times15=135\pi=424.11...\text{ cm}^2.
3. 10 hats =4241.1...\text{ cm}^2.
4. 4241.1...\div1200=3.53..., so 3 pots are not enough and she needs 4 pots.

Find the volume of the cube, then work backwards to find the area of the square base.

1. Volume of the cube =6\times6\times6=216\text{ cm}^3.
2. \frac{1}{3}\times area of base \times8=216, so area of base =216\times3\div8=81\text{ cm}^2.
3. The base is a square and 9\times9=81, so each side is 9 cm.