The radius of the sector equals the slant height of the cone \(l\).

The arc length of the sector equals the circumference of the base:

\(\dfrac{90^\circ}{360^\circ}\cdot 2\pi l=2\pi r\).

Hence \(\dfrac{\pi l}{2}=2\pi r\Rightarrow l=4r\).

Therefore, \(l:r=4:1\).

Answer: \(4:1\).

In the axial section, the base equals the diameter and the equal sides are the slant heights.

Hence \(2r=6\Rightarrow r=3\text{ cm}\), \(l=6\text{ cm}\).

The total surface area is

\(S=\pi r^2+\pi rl\).

For \(\pi=3\): \(S=3\cdot 3^2+3\cdot 3\cdot 6=27+54=81\text{ cm}^2\).

Answer: \(81\text{ cm}^2\).

From \(\pi r^2=9\pi\), we get \(r=3\text{ cm}\).

The area of the axial section equals \(rh\), so \(3h=12\Rightarrow h=4\text{ cm}\).

The slant height is \(l=\sqrt{r^2+h^2}=\sqrt{9+16}=5\text{ cm}\).

Answer: \(5\text{ cm}\).

We have \(2r=4\Rightarrow r=2\text{ cm}\), and the slant height is \(l=2\sqrt{10}\text{ cm}\).

The height is \(h=\sqrt{l^2-r^2}=\sqrt{40-4}=6\text{ cm}\).

The volume is \(V=\dfrac13\pi r^2h\). For \(\pi=3\): \(V=\dfrac13\cdot 3\cdot 4\cdot 6=24\text{ cm}^3\).

Answer: \(24\text{ cm}^3\).

The axial section of a cone is an isosceles triangle. If it is right-angled, then the base radius equals the height: \(r=h\).

The area of the axial section is \(rh\), so \(r^2=36\), hence \(r=h=6\text{ cm}\).

The volume is \(V=\dfrac13\pi r^2h\). For \(\pi=3\): \(V=\dfrac13\cdot 3\cdot 36\cdot 6=216\text{ cm}^3\).

Answer: \(216\text{ cm}^3\).

From \(\pi r^2=64\pi\), we get \(r=8\text{ cm}\).

From \(\pi rl=80\pi\), we get \(8l=80\Rightarrow l=10\text{ cm}\).

The height is \(h=\sqrt{l^2-r^2}=\sqrt{100-64}=6\text{ cm}\).

Answer: \(6\text{ cm}\).

A section through the apex and a chord of the base is an isosceles triangle with base \(10\text{ cm}\).

Half of the chord is \(5\text{ cm}\), and the distance from the center of the base to the chord is

\(d=\sqrt{13^2-5^2}=12\text{ cm}\).

The height of the cone is \(9\text{ cm}\), so the distance from the cone apex to this chord in the section plane is \(\sqrt{9^2+12^2}=15\text{ cm}\).

The area of the section is \(\dfrac12\cdot 10\cdot 15=75\text{ cm}^2\).

Answer: \(75\text{ cm}^2\).

From the volume formula, \(27\pi=\dfrac13\pi r^2\cdot 3=\pi r^2\), so \(r^2=27\).

Then the slant height is \(l=\sqrt{r^2+h^2}=\sqrt{27+9}=6\text{ cm}\).

Answer: \(6\text{ cm}\).

From the condition on the arc length of the sector,

\(\dfrac{120^\circ}{360^\circ}\cdot 2\pi l=2\pi r\Rightarrow l=3r\).

Then

\(\dfrac{S_{\text{lat}}}{S_{\text{base}}}=\dfrac{\pi rl}{\pi r^2}=\dfrac{l}{r}=3\).

Answer: \(3:1\).

The base of the axial section equals the diameter of the cone base: \(2r=6\sqrt3\Rightarrow r=3\sqrt3\text{ cm}\).

The height of an equilateral triangle with side \(6\sqrt3\) is \(\dfrac{6\sqrt3\cdot \sqrt3}{2}=9\text{ cm}\). This is the cone height.

The volume is \(V=\dfrac13\pi r^2h\). For \(\pi=3\): \(V=\dfrac13\cdot 3\cdot 27\cdot 9=243\text{ cm}^3\).

Answer: \(243\text{ cm}^3\).

From \(\pi r^2=25\pi\), it follows that \(r=5\text{ cm}\).

The area of the axial section equals \(rh\), so \(5h=60\Rightarrow h=12\text{ cm}\).

The slant height is \(l=\sqrt{5^2+12^2}=13\text{ cm}\).

Answer: \(13\text{ cm}\).

We have \(2r=6\Rightarrow r=3\text{ cm}\), and the slant height is \(l=\sqrt{34}\text{ cm}\).

The height is \(h=\sqrt{34-9}=5\text{ cm}\).

The volume is \(V=\dfrac13\pi r^2h\). For \(\pi=3\): \(V=\dfrac13\cdot 3\cdot 9\cdot 5=45\text{ cm}^3\).

Answer: \(45\text{ cm}^3\).

As in the case of a right axial section, we have \(r=h\).

The area of the axial section equals \(rh\), so \(r^2=81\Rightarrow r=h=9\text{ cm}\).

Then \(V=\dfrac13\pi r^2h\). For \(\pi=3\): \(V=\dfrac13\cdot 3\cdot 81\cdot 9=729\text{ cm}^3\).

Answer: \(729\text{ cm}^3\).

From \(\pi r^2=25\pi\), we have \(r=5\text{ cm}\).

From \(\pi rl=65\pi\), we get \(5l=65\Rightarrow l=13\text{ cm}\).

The height is \(h=\sqrt{13^2-5^2}=12\text{ cm}\).

Answer: \(12\text{ cm}\).

The base of the section is a chord of length \(16\text{ cm}\), so half of it is \(8\text{ cm}\).

The distance from the center of the base to the chord is \(\sqrt{17^2-8^2}=15\text{ cm}\).

Then the altitude of the section to the chord is \(\sqrt{20^2+15^2}=25\text{ cm}\).

The area of the section is \(\dfrac12\cdot 16\cdot 25=200\text{ cm}^2\).

Answer: \(200\text{ cm}^2\).

From the volume formula: \(26\pi=\dfrac13\pi r^2\cdot 6=2\pi r^2\).

Hence \(r^2=13\).

The slant height is \(l=\sqrt{r^2+h^2}=\sqrt{13+36}=7\text{ cm}\).

Answer: \(7\text{ cm}\).