🔺 Pyramid Angle Calculator
Calculate angles in a square pyramid
How to Use This Calculator
Enter Base Length and Height
Input the length of one side of the square base and the perpendicular height of the pyramid. Make sure both values are positive numbers.
Click Calculate
Press the "Calculate Angles" button to compute the apex angle (angle at the top) and base edge angle (angle between lateral edges at the base).
Review Results
View the calculated apex angle, base edge angle, and lateral edge length. These angles describe the geometry of the pyramid.
Formulas
Lateral Edge = √((base/2)² + h²)
Length of edge from apex to base corner
Apex Angle (α) = 2 × arctan((base/2)/h)
Angle at the apex between opposite lateral edges
Base Edge Angle (β) = 2 × arcsin(base/(2 × lateral edge))
Angle between lateral edges at the base
Where:
- base = length of one side of the square base
- h = perpendicular height of the pyramid
- α = apex angle (angle at the top point)
- β = base edge angle (angle between edges at base)
Example 1: Square pyramid with base = 10 units, height = 12 units
Lateral Edge = √((10/2)² + 12²) = √(25 + 144) = √169 = 13 units
Apex Angle = 2 × arctan(5/12) ≈ 45.24°
Base Edge Angle = 2 × arcsin(10/(2×13)) ≈ 45.24°
Example 2: Square pyramid with base = 8 units, height = 6 units
Lateral Edge = √((8/2)² + 6²) = √(16 + 36) = √52 ≈ 7.21 units
Apex Angle = 2 × arctan(4/6) ≈ 67.38°
Base Edge Angle = 2 × arcsin(8/(2×7.21)) ≈ 67.38°
About Pyramid Angle Calculator
A square pyramid is a 3D shape with a square base and four triangular faces meeting at an apex. This calculator determines key angles in the pyramid: the apex angle (angle at the top point between opposite lateral edges) and the base edge angle (angle between lateral edges at the base), based on the base length and height.
When to Use This Calculator
- Architecture: Calculate angles for pyramid-shaped buildings or structures
- Engineering: Design pyramid-shaped components with specific angle requirements
- Mathematics Education: Teach students about 3D geometry and trigonometry in pyramids
- 3D Modeling: Create accurate pyramid models in CAD or graphics software
- Construction: Determine cutting angles for pyramid roof structures
- Design: Plan pyramid shapes with specific geometric properties
Why Use Our Calculator?
- ✅ Multiple Angles: Calculates both apex and base edge angles
- ✅ Instant Results: Get angle measurements immediately
- ✅ Lateral Edge Length: Also computes the lateral edge dimension
- ✅ Step-by-Step Display: See the formulas applied with your values
- ✅ 100% Accurate: Uses precise trigonometric formulas
- ✅ Completely Free: No registration required
Understanding Pyramid Angles
A square pyramid has several important angles:
- Apex Angle (α): The angle at the top point between two opposite lateral edges. This describes how "sharp" or "flat" the pyramid is at the top.
- Base Edge Angle (β): The angle between two lateral edges meeting at a base corner. This describes the slope of the pyramid sides.
- Lateral Edge: The length from the apex to any corner of the base, calculated using the Pythagorean theorem.
- Face Angle: The angle within each triangular face, different from the apex and base angles.
Real-World Applications
Architecture: Designing a pyramid roof with base 20 m and height 15 m. The apex angle is 2 × arctan(10/15) ≈ 67.38°, and lateral edge ≈ 18.03 m. These angles determine the roof's appearance and structural requirements.
Engineering: A pyramid-shaped component needs specific angles for manufacturing. With base 8 cm and height 12 cm, apex angle ≈ 45.24° and lateral edge ≈ 13.27 cm help determine cutting angles and assembly requirements.
Mathematics: Understanding how base size and height affect pyramid angles helps solve geometry problems. A tall, narrow pyramid (small base, large height) has smaller apex angles, while a short, wide pyramid has larger apex angles.
Frequently Asked Questions
What is the apex angle?
The apex angle is the angle at the top point of the pyramid, formed between two opposite lateral edges. It's calculated as 2 × arctan((base/2)/h), where the base is the square base side length and h is the height.
How does height affect the angles?
Increasing the height (with fixed base) decreases the apex angle, making the pyramid more "pointy." Decreasing the height increases the apex angle, making it flatter. The base edge angle follows a similar relationship.
Can I use this for rectangular pyramids?
This calculator is specifically designed for square pyramids (base with equal sides). For rectangular pyramids, the angles differ along different axes, requiring more complex calculations.
What is the lateral edge?
The lateral edge is the line segment from the apex (top point) to any corner of the square base. It's one of the four edges that form the triangular faces. It's calculated using the Pythagorean theorem: √((base/2)² + h²).
Are apex angle and base edge angle always equal?
Not necessarily. In a square pyramid, these angles can be different depending on the ratio of base to height. However, for certain ratios, they may be equal or similar.
How is this different from a tetrahedron?
A square pyramid has a square base and four triangular faces. A tetrahedron (triangular pyramid) has a triangular base and four triangular faces (all triangular). The angle calculations are different for tetrahedra.