AMC 12 Geometry & Measurement 2

Question 1

What is the area of a triangle with side lengths 13, 14, and 15?

Accepted Answer

84

Answer

Heron's formula with s = 21 gives sqrt(21·8·7·6) = sqrt(7056) = 84.

Question 2

What is the circumradius of a triangle with side lengths 13, 14, and 15?

Accepted Answer

65/8

Answer

R = abc/(4K) = 2730/(4·84) = 65/8.

Question 3

What is the inradius of a right triangle with legs 3 and 4?

Accepted Answer

1

Answer

r = Area/s = 6/6 = 1.

Question 4

What is the area of a regular hexagon with side length 2?

Accepted Answer

6√3

Answer

A regular hexagon is six equilateral triangles of area √3 each, so the total is 6√3.

Question 5

A sphere is inscribed in a cube with edge length 6. What is the volume of the sphere?

Accepted Answer

36π

Answer

The radius is 3, so V = (4/3)π(27) = 36π.

AMC 12 - - American Mathematics Contest 12 Practice Test