What is Geometry Basics?

Master area, perimeter, and volume formulas for common shapes, understand angle relationships, and solve problems involving geometric properties.

How can I study Geometry Basics effectively?

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 13 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Geometry Basics study guide free?

Yes — all study notes, flashcards, and practice problems for Geometry Basics on Study Mondo are 100% free. No account is needed to access the content.

What course covers Geometry Basics?

Geometry Basics is part of the SAT Prep course on Study Mondo, specifically in the Additional Topics in Math section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Geometry Basics?

Yes, this page includes 13 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Geometry Basics

Angle Relationships

Supplementary angles: Sum to 180°
Complementary angles: Sum to 90°
Vertical angles: Equal (formed by intersecting lines)
Linear pair: Supplementary and adjacent

Parallel Lines and Transversals

When a transversal crosses parallel lines:

Corresponding angles: Equal
Alternate interior angles: Equal
Alternate exterior angles: Equal
Co-interior (same-side interior) angles: Supplementary (sum to 180°)

Triangle Properties

Angle sum: 180°
Exterior angle = sum of two remote interior angles
Triangle Inequality: Sum of any two sides > third side

Special Triangles

Equilateral: All sides equal, all angles 60°

❓ Question:

In a triangle, two angles measure $45°$ and $70°$ . What is the measure of the third angle?

Solution:

Triangle angle sum: All angles add to

Shape	Formula
Triangle	$A = \frac{1}{2}bh$
Rectangle	$A = lw$
Parallelogram	$A = bh$
Trapezoid	$A = \frac{1}{2}(b_1 + b_2)h$
Circle	$A = \pi r^2$

Shape	Formula
Rectangular prism	$V = lwh$
Cylinder	$V = \pi r^2 h$
Cone	$V = \frac{1}{3}\pi r^2 h$
Sphere	$V = \frac{4}{3}\pi r^3$

❓ Question:

In a right triangle, one leg is 5 and the hypotenuse is 13. What is the length of the other leg?

Solution:

Use Pythagorean theorem: $a^2 + b^2 = c^2$

$5^2 + b^2 = 13^2$ $25 + b^2 = 169$

Answer: $12$

Recognition: This is the 5-12-13 Pythagorean triple!

SAT Tip: Knowing common triples (3-4-5, 5-12-13, 8-15-17) saves time!

❓ Question:

In a 30-60-90 triangle, the side opposite the $30°$ angle is 6. What is the length of the hypotenuse?

Solution:

30-60-90 ratio: $x : x\sqrt{3} : 2x$

Opposite $30°$ : $x$ (shortest)
Opposite $60°$ : $x\sqrt{3}$

Given: Side opposite $30°$ is 6 $x = 6$

Hypotenuse: $2x = 2(6) = 12$

Answer: $12$

SAT Tip: Memorize 30-60-90 ratios - they appear frequently! Side opposite 30° is half the hypotenuse!

❓ Question:

A rectangle has a length of 12 and a width of 5. What is its perimeter and area?

Perimeter: $P = 2l + 2w = 2(12) + 2(5) = 24 + 10 = 34$

Area: $A = lw = 12 \times 5 = 60$

Answer: Perimeter = 34, Area = 60

❓ Question:

A rectangle has a length of 12 and a width of 5. What is its perimeter and area?

Perimeter: $P = 2l + 2w = 2(12) + 2(5) = 24 + 10 = 34$

Area: $A = lw = 12 \times 5 = 60$

Answer: Perimeter = 34, Area = 60

❓ Question:

Two parallel lines are cut by a transversal. One of the angles formed is 115°. What are the measures of all eight angles?

When parallel lines are cut by a transversal, we get two types of angles:

The angle of 115° and its vertical angle are both 115°. The supplementary angles are $180° - 115° = 65°$ .

All eight angles are either 115° or 65°:

Four angles of 115° (the angle, its vertical angle, and corresponding angles)
Four angles of 65° (supplementary to the 115° angles)

Answer: Four angles are 115° and four are 65°.

Key relationships used: vertical angles, corresponding angles, supplementary angles.

❓ Question:

Two parallel lines are cut by a transversal. One of the angles formed is 115°. What are the measures of all eight angles?

When parallel lines are cut by a transversal, we get two types of angles:

The angle of 115° and its vertical angle are both 115°. The supplementary angles are $180° - 115° = 65°$ .

All eight angles are either 115° or 65°:

Four angles of 115° (the angle, its vertical angle, and corresponding angles)
Four angles of 65° (supplementary to the 115° angles)

Answer: Four angles are 115° and four are 65°.

Key relationships used: vertical angles, corresponding angles, supplementary angles.

❓ Question:

A rectangular box has dimensions 3 × 4 × 12. What is the length of the longest diagonal inside the box?

3D diagonal formula: $d = \sqrt{l^2 + w^2 + h^2}$

$d = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$

Answer: The space diagonal is 13.

Alternatively: First find the diagonal of the base: $\sqrt{3^2 + 4^2} = 5$ (3-4-5 triple), then use that with the height: (5-12-13 triple).

❓ Question:

A rectangular box has dimensions 3 × 4 × 12. What is the length of the longest diagonal inside the box?

3D diagonal formula: $d = \sqrt{l^2 + w^2 + h^2}$

$d = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$

Answer: The space diagonal is 13.

Alternatively: First find the diagonal of the base: $\sqrt{3^2 + 4^2} = 5$ (3-4-5 triple), then use that with the height: (5-12-13 triple).

❓ Question:

The volume of a cylinder is $100\pi$ cubic cm and its height is 4 cm. What is the total surface area?

Step 1: Find the radius using the volume formula: $V = \pi r^2 h$ $100\pi = \pi r^2 (4)$ $r^2 = 25$ $r = 5 \text{ cm}$

Step 2: Calculate total surface area (two circles + lateral surface): $SA = 2\pi r^2 + 2\pi rh$ $= 2\pi(25) + 2\pi(5)(4)$

Answer: $90\pi$ cm² (approximately 282.74 cm²)

❓ Question:

The volume of a cylinder is $100\pi$ cubic cm and its height is 4 cm. What is the total surface area?

Step 1: Find the radius using the volume formula: $V = \pi r^2 h$ $100\pi = \pi r^2 (4)$ $r^2 = 25$ $r = 5 \text{ cm}$

Step 2: Calculate total surface area (two circles + lateral surface): $SA = 2\pi r^2 + 2\pi rh$ $= 2\pi(25) + 2\pi(5)(4)$

Answer: $90\pi$ cm² (approximately 282.74 cm²)

❓ Question:

Two similar triangles have corresponding sides in the ratio $3:5$ . If the area of the smaller triangle is 27 cm², what is the area of the larger triangle?

Key property of similar figures: If corresponding sides have ratio $k$ , then areas have ratio $k^2$ .

Side ratio: $\frac{3}{5}$

Area ratio: $\left(\frac{3}{5}\right)^2 = \frac{9}{25}$

$\frac{27}{A_{\text{large}}} = \frac{9}{25}$

Answer: 75 cm²

Remember: Side ratio = $k$ , Area ratio = $k^2$ , Volume ratio = $k^3$ .

❓ Question:

Two similar triangles have corresponding sides in the ratio $3:5$ . If the area of the smaller triangle is 27 cm², what is the area of the larger triangle?

Key property of similar figures: If corresponding sides have ratio $k$ , then areas have ratio $k^2$ .

Side ratio: $\frac{3}{5}$

Area ratio: $\left(\frac{3}{5}\right)^2 = \frac{9}{25}$

$\frac{27}{A_{\text{large}}} = \frac{9}{25}$

Answer: 75 cm²

Remember: Side ratio = $k$ , Area ratio = $k^2$ , Volume ratio = $k^3$ .

Geometry Basics

Try the Interactive Version!

Geometry Basics

Angle Relationships

Parallel Lines and Transversals

Triangle Properties

Special Triangles

📚 Practice Problems

1Problem 1easy

❓ Question:

Practice Lab

Practice with Flashcards

Browse All Topics

📌 Related Topics in Additional Topics in Math

❓ Frequently Asked Questions

Area Formulas

Volume Formulas

SAT Tips

2Problem 2medium

❓ Question:

3Problem 3hard

❓ Question:

4Problem 4easy

❓ Question:

5Problem 5easy

❓ Question:

6Problem 6medium

❓ Question:

7Problem 7medium

❓ Question:

8Problem 8medium

❓ Question:

9Problem 9medium

❓ Question:

10Problem 10hard

❓ Question:

11Problem 11hard

❓ Question:

12Problem 12expert

❓ Question:

13Problem 13expert

❓ Question: