Spheres, Cones, Pyramids and Frustums

Use the sphere volume formula with the radius cubed.

1. The volume is \frac43\pi r^3=\frac43\pi\times3^3.
2. This is 36\pi\text{ cm}^3.

A cone has one third of the volume of the matching cylinder.

1. The volume is \frac13\pi r^2h=\frac13\pi\times4^2\times9.
2. This equals 48\pi\text{ cm}^3.

Multiply the base area by the perpendicular height and divide by three.

1. The base area is 6^2=36\text{ cm}^2.
2. The volume is \frac13\times36\times10=120\text{ cm}^3.

Use half of the surface area of the whole sphere.

1. The sphere surface area is 4\pi\times5^2=100\pi\text{ cm}^2.
2. The hemisphere curved area is half of this: 50\pi\text{ cm}^2.

Find the radius before using the sphere surface-area formula.

1. The radius is 7\text{ cm}.
2. The surface area is 4\pi\times7^2=196\pi\text{ cm}^2.

Add the circular base to the curved surface area.

1. The curved area is \pi rl=\pi\times5\times13=65\pi\text{ cm}^2.
2. The base area is 25\pi\text{ cm}^2.
3. The total surface area is 90\pi\text{ cm}^2.

A solid hemisphere has a curved surface and a circular flat face.

1. The curved surface is half of a sphere's surface: \frac12\times4\pi r^2=2\pi r^2=2\pi\times4^2=32\pi\text{ cm}^2.
2. The flat face is a circle: \pi r^2=\pi\times4^2=16\pi\text{ cm}^2.
3. The total is 32\pi+16\pi=48\pi\text{ cm}^2.

Rearrange the pyramid volume formula after finding the base area.

1. The base area is 5\times8=40\text{ cm}^2.
2. The height satisfies \frac13\times40\times h=120.
3. Therefore h=360\div40=9\text{ cm}.

Find the square of the radius using the cone volume formula.

1. Put the values into the formula: \frac13\pi r^2\times9=75\pi.
2. \frac13\times9=3, so 3\pi r^2=75\pi.
3. Divide both sides by 3\pi: r^2=25.
4. The radius is positive, so r=\sqrt{25}=5\text{ cm}.

Find the radius cubed before taking a cube root.

1. Put the volume into the formula: \frac43\pi r^3=288\pi.
2. Divide both sides by \pi: \frac43r^3=288. Multiply by \frac34: r^3=288\times\frac34=216.
3. The radius is \sqrt[3]{216}=6\text{ cm}.
4. The diameter is twice the radius: 2\times6=12\text{ cm}.

The volume uses perpendicular height, which you can find using Pythagoras.

1. The perpendicular height is \sqrt{17^2-8^2}=15\text{ cm}.
2. The volume is \frac13\pi\times8^2\times15.
3. This equals 320\pi\text{ cm}^3.

Compare volumes rather than radii or surface areas.

1. The volume ratio is \frac{\frac43\pi\times6^3}{\frac43\pi\times2^3}.
2. This is \left(\frac62\right)^3=27.
3. Exactly 27 small spheres can be made.

Find the cylinder and cone volumes separately, then add.

1. The cylinder volume is \pi\times3^2\times8=72\pi\text{ cm}^3.
2. The cone volume is \frac13\pi\times3^2\times4=12\pi\text{ cm}^3.
3. The total is 84\pi\text{ cm}^3.

Subtract the removed cone volume from the original cone volume.

1. The original cone volume is \frac13\pi\times6^2\times15=180\pi\text{ cm}^3.
2. The removed cone volume is \frac13\pi\times2^2\times5=\frac{20\pi}{3}\text{ cm}^3.
3. The frustum volume is 180\pi-\frac{20\pi}{3}=\frac{520\pi}{3}\text{ cm}^3.

Use half a base side and the vertical height to find the height of each triangular face.

1. The apex is above the centre of the base, which is 4\text{ cm} (half of 8) from the middle of each base edge.
2. The vertical height 3\text{ cm} and this 4\text{ cm} form a right-angled triangle. Its hypotenuse is the height of a triangular face: \sqrt{3^2+4^2}=5\text{ cm}.
3. Extract the right-angled section to find the face height, then use it in one triangular face.

4. Each triangular face has area \frac12\times8\times5=20\text{ cm}^2, so the four faces total 80\text{ cm}^2.
5. The square base has area 8^2=64\text{ cm}^2.
6. The total surface area is 80+64=144\text{ cm}^2.

Use the volume to find the perpendicular height, then use Pythagoras for the slant height.

1. The volume equation \frac13\pi\times9^2h=324\pi gives h=12\text{ cm}.
2. The slant height is \sqrt{9^2+12^2}=15\text{ cm}.
3. The curved area is \pi\times9\times15=135\pi\text{ cm}^2.
4. Adding the base gives 135\pi+81\pi=216\pi\text{ cm}^2.

Extend the sloping sides to form two similar cones and find their heights.

1. Extend the sloping sides to complete the cone. The small cone on top and the full cone are similar, with radii 4 and 8, so every length in the full cone is 2 times the matching length in the small cone.
2. Let the small cone's height be h. The full cone's height is 2h, and the frustum's height is the difference: 2h-h=9, so h=9\text{ cm} and the full cone's height is 18\text{ cm}.
3. Complete the cone and compare matching radii in its central section.

4. Full cone: \frac13\pi\times8^2\times18=384\pi\text{ cm}^3. Small cone: \frac13\pi\times4^2\times9=48\pi\text{ cm}^3.
5. The frustum's volume is 384\pi-48\pi=336\pi\text{ cm}^3.

Extend the frustum to two similar cones and subtract their curved areas before adding the circular faces.

1. Extend the sloping side to complete the cone. The small cone on top and the full cone are similar, with radii 4 and 8, so the full cone's slant height is 2 times the small cone's.
2. Let the small cone's slant height be l. Then the full cone's is 2l, and 2l-l=5, so l=5\text{ cm} and the full cone's slant height is 10\text{ cm}.
3. Compare the two similar cones along the same sloping side.

4. The frustum's curved surface is the full cone's minus the small cone's: \pi\times8\times10-\pi\times4\times5=80\pi-20\pi=60\pi\text{ cm}^2.
5. The circular faces have areas \pi\times8^2=64\pi and \pi\times4^2=16\pi\text{ cm}^2.
6. The total surface area is 60\pi+64\pi+16\pi=140\pi\text{ cm}^2.

Subtract the hemisphere's volume to find the cylinder's volume; remember the hemisphere adds one radius to the height.

1. A hemisphere is half a sphere, so its volume is \frac12\times\frac43\pi r^3=\frac23\pi\times3^3=18\pi=56.548\ldots\text{ m}^3.
2. The cylinder holds the rest: 200-56.548\ldots=143.451\ldots\text{ m}^3.
3. Volume of a cylinder =\pi r^2h, so \pi\times3^2\times h=143.451\ldots, giving h=\frac{143.451\ldots}{9\pi}=5.0735\ldots\text{ m}.
4. The hemisphere on top is 3\text{ m} high (its radius), so the total height is 5.0735\ldots+3=8.0735\ldots\text{ m}.
5. To 3 significant figures, the silo is 8.07\text{ m} tall.

Use similarity to find the small pyramid base before subtracting its volume.

1. Volume of a pyramid =\frac13\times base area \times height. The full pyramid has volume \frac13\times10^2\times12=400\text{ cm}^3.
2. The small top pyramid is similar to the full one. Its height is 6 out of 12, half, so its base side is half of 10: 5\text{ cm}.
3. A section through the apex shows why the small base side is half the full base side.

4. The small pyramid's volume is \frac13\times5^2\times6=50\text{ cm}^3.
5. The remaining volume is 400-50=350\text{ cm}^3.