Practice (63)

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3082

The polygon shown here is constructed from two squares and six equilateral triangles, each of side length 6 units. This polygon may be folded into a polyhedron by creasing along the dotted lines and joining adjacent edges as indicated by the arrows. What is the volume of the resulting polyhedron? Express your answer in simplest radical form.

3098
A 12-sided game die has the shape of a hexagonal bipyramid, which consists of two pyramids, each with a regular hexagonal base of side length 1 cm and with height 1 cm, glued together along their hexagons. When this game die is rolled and lands on one of its triangular faces, how high off the ground is the opposite face? Express your answer as a common fraction in simplest radical form.
3105

A plane passing through the vertex $A$ and the center of its inscribed sphere of a tetrahedron $ABCD$ intersects its edge $BC$ and $CD$ at point $E$ and $F$, as shown. If $AEF$ divides this tetrahedron into two equal volume parts: $A-BDEF$ and $A-CEF$, what is the relationship between these two parts' surface areas $S_1$ and $S_2$ where $S_1 = S_{A-BDEF}$ and $S_1=S_{A-CEF}$? $(A) S_1 < S_2\quad(B) S_1 > S_2\quad (C) S_1 = S_2 \quad(D) $ cannot determine

3144

In tetrahedron $ABCD$, $\angle{ADB} = \angle{BDC} = \angle{CDA} = 60^\circ$, $AD=BD=3$, and $CD=2$. Find the radius of $ABCD$'s circumsphere.

3145
In tetrahedron $P-ABC$, $AB=BC=CA$ and $PA=PB=PC$. If $AB=1$ and the altitude from $P$ to $ABC$ is $\sqrt{2}$, find the radius of $P-ABC$'s inscribed sphere.
3783
A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
4107

A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).
PlaneGeometry AreaMethod RotationMethod SolidGemoetry CoordinatedGeometry LawOfCosines

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