📐 Geometry Area And Length Classics
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Triangle Area · Easy · source: Original (AMC-style)
What is the area of a triangle with base 6 and height 4?
A) $10$ B) $12$ C) $15$ D) $18$ E) $24$
Answer & Solution
Answer: B
Area = $\frac{1}{2} \times$ base × height = $\frac{1}{2} \times 6 \times 4 = 12$.
2.
Tags: Rectangle Area · Easy · source: Original (AMC-style)
What is the area of a rectangle with length 8 and width 5?
A) $13$ B) $26$ C) $35$ D) $40$ E) $45$
Answer & Solution
Answer: D
Area = length × width = $8 \times 5 = 40$.
3.
Tags: Circle Area · Easy · source: Original (AMC-style)
What is the area of a circle with radius 3?
A) $3\pi$ B) $6\pi$ C) $9\pi$ D) $12\pi$ E) $18\pi$
Answer & Solution
Answer: C
Area = $\pi r^2 = \pi \times 3^2 = 9\pi$.
4.
Tags: Square Area · Easy · source: Original (AMC-style)
What is the area of a square with side length 7?
A) $14$ B) $28$ C) $35$ D) $49$ E) $56$
Answer & Solution
Answer: D
Area = side² = $7^2 = 49$.
5.
Tags: Parallelogram Area · Easy · source: Original (AMC-style)
What is the area of a parallelogram with base 9 and height 4?
A) $18$ B) $26$ C) $32$ D) $36$ E) $40$
Answer & Solution
Answer: D
Area = base × height = $9 \times 4 = 36$.
6.
Tags: Trapezoid Area · Easy · source: Original (AMC-style)
What is the area of a trapezoid with bases 5 and 7 and height 3?
A) $15$ B) $18$ C) $21$ D) $24$ E) $27$
Answer & Solution
Answer: B
Area = $\frac{1}{2} \times$ (base1 + base2) × height = $\frac{1}{2} \times (5+7) \times 3 = 18$.
7.
Tags: Heron’s Formula · Medium · source: Original (AMC-style)
What is the area of a triangle with sides 3, 4, and 5?
A) $6$ B) $8$ C) $10$ D) $12$ E) $15$
Answer & Solution
Answer: A
Using Heron's formula: $s = \frac{3+4+5}{2} = 6$, so area = $\sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6$.
8.
Tags: Circle Sector · Medium · source: Original (AMC-style)
What is the area of a sector with central angle 60° in a circle of radius 6?
A) $\pi$ B) $2\pi$ C) $3\pi$ D) $6\pi$ E) $12\pi$
Answer & Solution
Answer: D
Area = $\frac{60°}{360°} \times \pi r^2 = \frac{1}{6} \times \pi \times 36 = 6\pi$.
9.
Tags: Regular Polygon · Medium · source: Original (AMC-style)
What is the area of a regular hexagon with side length 2?
A) $3\sqrt{3}$ B) $6\sqrt{3}$ C) $9\sqrt{3}$ D) $12\sqrt{3}$ E) $18\sqrt{3}$
Answer & Solution
Answer: B
Area = $\frac{1}{2} \times$ perimeter × apothem = $\frac{1}{2} \times 12 \times \sqrt{3} = 6\sqrt{3}$.
10.
Tags: Complex Area · Medium · source: Original (AMC-style)
What is the area of the region bounded by $y = x^2$ and $y = 4$?
A) $\frac{8}{3}$ B) $\frac{16}{3}$ C) $\frac{32}{3}$ D) $\frac{64}{3}$ E) $\frac{128}{3}$
Answer & Solution
Answer: B
The region is bounded by $x = -2$ to $x = 2$. Area = $\int_{-2}^{2} (4 - x^2) dx = [4x - \frac{x^3}{3}]_{-2}^{2} = 16 - \frac{16}{3} = \frac{32}{3}$.
11.
Tags: Advanced Area · Hard · source: Original (AMC-style)
What is the area of the region bounded by the lines $y = x$, $y = -x$, $x = 1$, and $x = -1$?
A) $1$ B) $2$ C) $3$ D) $4$ E) $5$
Answer & Solution
Answer: B
This forms a square with vertices at (1,1), (1,-1), (-1,-1), and (-1,1). The side length is $\sqrt{2}$, so area = $(\sqrt{2})^2 = 2$.
12.
Tags: Circle Properties · Hard · source: Original (AMC-style)
In a circle with radius 5, what is the area of a sector with arc length $\pi$?
A) $\frac{\pi}{2}$ B) $\pi$ C) $\frac{3\pi}{2}$ D) $2\pi$ E) $\frac{5\pi}{2}$
Answer & Solution
Answer: E
Arc length = $r\theta$, so $\pi = 5\theta$ and $\theta = \frac{\pi}{5}$. Area = $\frac{1}{2}r^2\theta = \frac{1}{2} \times 25 \times \frac{\pi}{5} = \frac{5\pi}{2}$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | B | D | C | D | D | B | A | D | B | B | B | E |