Pyramid Calculator - Volume Calculator & Surface Area Calculator
Free pyramid calculator & volume calculator. Calculate pyramid volume, surface area, and lateral area with step-by-step solutions. Our calculator handles pyramid geometry with comprehensive analysis.
Last updated: October 26, 2024
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Length of the base of the pyramid
Width of the base (leave empty for square pyramid)
Height of the pyramid from base to apex
This is a square pyramid with equal base dimensions.
Square pyramids have equal length and width, creating four identical triangular faces. They are commonly found in architecture and geometry problems.
How to Use:
- • Enter the base length and height of the pyramid
- • Optionally enter base width for rectangular pyramids
- • Select pyramid type (square, rectangular, etc.)
- • Calculator applies pyramid formulas automatically
- • Use common examples for quick testing
Pyramid Calculator Types & Functions
Formula
V = (1/3) × Base × Height
Uses base area and height to calculate volume
Formula
SA = Base + Lateral Area
Includes base area plus triangular face areas
Type
Square Base
Four identical triangular faces
Type
Rectangular Base
Two pairs of identical triangular faces
Process
Step-by-Step
Provides detailed step-by-step process for calculations
Analysis
Complete Analysis
Applies surface area formulas with comprehensive geometric analysis
Quick Example Result
For square pyramid (10×10×15):
Volume
500.00 cubic units
Surface Area
400.00 square units
Lateral Area
300.00 square units
How Our Pyramid Calculator Works
Our pyramid calculator uses fundamental geometric formulas adapted for pyramid analysis. The calculation applies volume, surface area, and lateral area formulas to ensure accurate pyramid geometry analysis with comprehensive mathematical explanations.
The Pyramid Analysis Process
1. Identify base dimensions and height2. Calculate base area3. Apply volume formula4. Calculate slant height5. Determine surface areasThis process ensures accurate pyramid analysis by systematically applying geometric formulas and providing comprehensive mathematical explanations.
Shows base, height, slant height, and triangular faces
Mathematical Foundation
Pyramid calculations are based on solid geometry principles. The volume formula V = (1/3) × Base Area × Height comes from the relationship between pyramids and their corresponding prisms. Surface area calculations involve the base area plus the lateral area of triangular faces, requiring slant height calculations.
- Volume formula: V = (1/3) × Base Area × Height
- Surface area formula: SA = Base Area + Lateral Area
- Slant height formula: s = √((base/2)² + height²)
- Lateral area formula: LA = (1/2) × Perimeter × Slant Height
- Applications: architecture, construction, packaging, design
- Verification: check geometric relationships and known examples
Sources & References
- Elementary Geometry for College Students - Alexander, KoeberleinStandard reference for solid geometry and pyramid calculations
- Geometry: A Comprehensive Course - Dan PedoeComprehensive coverage of geometric formulas and solid shapes
- Khan Academy - GeometryEducational resources for understanding solid geometry concepts
Need help with other geometry calculations? Check out our sphere calculator and area of parallelogram calculator.
Get Custom Calculator for Your PlatformPyramid Calculator Examples
Pyramid Properties:
- Type: Square pyramid
- Base: 10 × 10 square units
- Height: 15 units
- Base Area: 100 square units
Calculation Steps:
- Calculate base area: 10 × 10 = 100 square units
- Apply volume formula: V = (1/3) × 100 × 15
- Volume: V = 500 cubic units
- Calculate slant height: s = √((10/2)² + 15²) = √(25 + 225) = √250
- Calculate lateral area: LA = 2 × 10 × √250 = 316.23 square units
- Total surface area: SA = 100 + 316.23 = 416.23 square units
Result: Volume = 500 cubic units, Surface Area = 416.23 square units
This square pyramid has a volume of 500 cubic units and requires 416.23 square units of material to construct.
Rectangular Pyramid
8×12×20: Volume = 640 cubic units
Surface Area = 512 square units
Small Pyramid
5×5×8: Volume = 66.67 cubic units
Surface Area = 89.44 square units
Frequently Asked Questions
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