📐 Geometry Practice
Recommended: 60–75 minutes. No calculator.
Problems
1.
Tags: Basic Geometry · Easy · source: Original (AMC-style)
What is the perimeter of a square with side length 4?
A) $8$ B) $12$ C) $16$ D) $20$ E) $24$
Answer & Solution
Answer: C
Perimeter = $4 \times$ side = $4 \times 4 = 16$.
2.
Tags: Triangle Perimeter · Easy · source: Original (AMC-style)
What is the perimeter of a triangle with sides 3, 4, and 5?
A) $10$ B) $12$ C) $15$ D) $18$ E) $20$
Answer & Solution
Answer: B
Perimeter = $3 + 4 + 5 = 12$.
3.
Tags: Circle Diameter · Easy · source: Original (AMC-style)
What is the diameter of a circle with radius 7?
A) $7$ B) $14$ C) $21$ D) $28$ E) $49$
Answer & Solution
Answer: B
Diameter = $2 \times$ radius = $2 \times 7 = 14$.
4.
Tags: Rectangle Perimeter · Easy · source: Original (AMC-style)
What is the perimeter of a rectangle with length 6 and width 4?
A) $10$ B) $16$ C) $20$ D) $24$ E) $28$
Answer & Solution
Answer: C
Perimeter = $2 \times$ (length + width) = $2 \times (6 + 4) = 20$.
5.
Tags: Basic Angles · Easy · source: Original (AMC-style)
What is the measure of a straight angle?
A) $90°$ B) $120°$ C) $150°$ D) $180°$ E) $360°$
Answer & Solution
Answer: D
A straight angle measures $180°$.
6.
Tags: Square Diagonal · Easy · source: Original (AMC-style)
What is the diagonal of a square with side length 1?
A) $1$ B) $\sqrt{2}$ C) $2$ D) $\sqrt{3}$ E) $3$
Answer & Solution
Answer: B
Diagonal = side × $\sqrt{2} = 1 \times \sqrt{2} = \sqrt{2}$.
7.
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 = $\frac{1}{2} \times 6 \times 4 = 12$.
8.
Tags: Circle Area · Easy · source: Original (AMC-style)
What is the area of a circle with diameter 8?
A) $8\pi$ B) $16\pi$ C) $32\pi$ D) $64\pi$ E) $128\pi$
Answer & Solution
Answer: B
Radius = $\frac{8}{2} = 4$, so area = $\pi r^2 = \pi \times 4^2 = 16\pi$.
9.
Tags: Basic Volume · Easy · source: Original (AMC-style)
What is the volume of a rectangular box with dimensions 2×3×4?
A) $9$ B) $12$ C) $18$ D) $24$ E) $36$
Answer & Solution
Answer: D
Volume = length × width × height = $2 \times 3 \times 4 = 24$.
10.
Tags: Angle Sum · Easy · source: Original (AMC-style)
What is the sum of angles in a quadrilateral?
A) $180°$ B) $270°$ C) $360°$ D) $450°$ E) $540°$
Answer & Solution
Answer: C
The sum of angles in any quadrilateral is $360°$.
11.
Tags: Similarity · Medium · source: Original (AMC-style)
If two rectangles are similar with ratio 2:3, and the smaller rectangle has area 8, what is the area of the larger rectangle?
A) $12$ B) $16$ C) $18$ D) $24$ E) $32$
Answer & Solution
Answer: C
Area ratio = (side ratio)² = $(2/3)^2 = 4/9$. So larger area = $8 \times (9/4) = 18$.
12.
Tags: Coordinate Geometry · Medium · source: Original (AMC-style)
What is the midpoint of the line segment from (1,3) to (5,7)?
A) $(2,4)$ B) $(3,5)$ C) $(4,6)$ D) $(5,7)$ E) $(6,8)$
Answer & Solution
Answer: B
Midpoint = $\left(\frac{1+5}{2}, \frac{3+7}{2}\right) = (3,5)$.
13.
Tags: Triangle Centers · Medium · source: Original (AMC-style)
In an equilateral triangle with side length 6, what is the height?
A) $3$ B) $3\sqrt{2}$ C) $3\sqrt{3}$ D) $6$ E) $6\sqrt{2}$
Answer & Solution
Answer: C
Height = $\frac{\sqrt{3}}{2} \times$ side = $\frac{\sqrt{3}}{2} \times 6 = 3\sqrt{3}$.
14.
Tags: Circle Properties · Medium · source: Original (AMC-style)
What is the circumference of a circle with area $25\pi$?
A) $5\pi$ B) $10\pi$ C) $15\pi$ D) $20\pi$ E) $25\pi$
Answer & Solution
Answer: B
Area = $\pi r^2 = 25\pi$, so $r = 5$. Circumference = $2\pi r = 2\pi \times 5 = 10\pi$.
15.
Tags: Volume · Medium · source: Original (AMC-style)
What is the volume of a cylinder with radius 3 and height 4?
A) $12\pi$ B) $24\pi$ C) $36\pi$ D) $48\pi$ E) $72\pi$
Answer & Solution
Answer: C
Volume = $\pi r^2 h = \pi \times 3^2 \times 4 = 36\pi$.
16.
Tags: Angle Properties · Medium · source: Original (AMC-style)
If two parallel lines are cut by a transversal, and one angle is 60°, what is the measure of its corresponding angle?
A) $30°$ B) $60°$ C) $120°$ D) $180°$ E) Cannot be determined
Answer & Solution
Answer: B
Corresponding angles are equal, so the corresponding angle is also $60°$.
17.
Tags: Area of Sector · Medium · source: Original (AMC-style)
What is the area of a sector with central angle 90° in a circle of radius 4?
A) $\pi$ B) $2\pi$ C) $4\pi$ D) $8\pi$ E) $16\pi$
Answer & Solution
Answer: C
Area = $\frac{90°}{360°} \times \pi r^2 = \frac{1}{4} \times \pi \times 16 = 4\pi$.
18.
Tags: Triangle Inequality · Medium · source: Original (AMC-style)
Can a triangle have sides of length 2, 3, and 6?
A) Yes B) No C) Sometimes D) Cannot be determined E) Depends on angle
Answer & Solution
Answer: B
No, because $2 + 3 = 5 < 6$, violating the triangle inequality.
19.
Tags: Pythagorean Applications · Medium · source: Original (AMC-style)
What is the distance from the origin to the point (3,4)?
A) $3$ B) $4$ C) $5$ D) $6$ E) $7$
Answer & Solution
Answer: C
Distance = $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
20.
Tags: Regular Polygons · Medium · source: Original (AMC-style)
What is the measure of each interior angle in a regular pentagon?
A) $72°$ B) $90°$ C) $108°$ D) $120°$ E) $144°$
Answer & Solution
Answer: C
Each interior angle = $\frac{(n-2) \times 180°}{n} = \frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108°$.
21.
Tags: Advanced Geometry · Hard · source: Original (AMC-style)
In a circle, if a chord of length 8 is 3 units from the center, what is the radius?
A) $4$ B) $5$ C) $6$ D) $7$ E) $8$
Answer & Solution
Answer: B
Using the formula: $r^2 = d^2 + (\frac{c}{2})^2 = 3^2 + 4^2 = 9 + 16 = 25$, so $r = 5$.
22.
Tags: Complex Area · Hard · 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: C
The region is bounded by $x = -2$ to $x = 2$. Area = $\int_{-2}^{2} (4 - x^2) dx = [4x - \frac{x^3}{3}]_{-2}^{2} = 16 - \frac{16}{3} = \frac{32}{3}$.
23.
Tags: Advanced Triangles · Hard · source: Original (AMC-style)
In triangle ABC, if angle A = 30°, side AB = 6, and side AC = 8, what is the area?
A) $12$ B) $16$ C) $20$ D) $24$ E) $32$
Answer & Solution
Answer: A
Area = $\frac{1}{2} \times AB \times AC \times \sin A = \frac{1}{2} \times 6 \times 8 \times \sin 30° = 24 \times \frac{1}{2} = 12$.
24.
Tags: Circle Tangents · Hard · source: Original (AMC-style)
Two circles with radii 2 and 3 are externally tangent. What is the length of their common external tangent?
A) $2\sqrt{6}$ B) $3\sqrt{2}$ C) $4\sqrt{3}$ D) $5$ E) $6$
Answer & Solution
Answer: A
The common external tangent length = $\sqrt{d^2 - (r_1 + r_2)^2} = \sqrt{5^2 - (2+3)^2} = \sqrt{25 - 25} = 0$. Wait, let me recalculate: distance between centers = $2+3 = 5$, so tangent length = $\sqrt{5^2 - (3-2)^2} = \sqrt{25-1} = \sqrt{24} = 2\sqrt{6}$.
25.
Tags: Advanced Volume · Hard · source: Original (AMC-style)
What is the volume of a sphere with surface area $36\pi$?
A) $9\pi$ B) $18\pi$ C) $27\pi$ D) $36\pi$ E) $54\pi$
Answer & Solution
Answer: D
Surface area = $4\pi r^2 = 36\pi$, so $r = 3$. Volume = $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi \times 27 = 36\pi$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | C | B | B | C | D | B | B | B | D | C | C | B | C | B | C | B | C | B | C | C | B | C | A | A | D |