Geometry Calculator
Use our free Geometry Calculator to quickly compute the area, perimeter, volume, and surface area of common geometric shapes. Whether you're working on homework, engineering tasks, or simple daily measurements, this tool saves time and eliminates mistakes.
How to Use This Geometry Calculator
Our geometry calculator handles the most common 2D and 3D shapes. Follow these steps to get instant results:
- Select a shape from the dropdown menu: circle, square, rectangle, triangle, trapezoid, cylinder, or sphere.
- Enter the required dimensions in the input fields that appear. Each shape requires different measurements (radius, side length, width, height, etc.).
- View your results instantly. For 2D shapes, you will see the area and perimeter. For 3D shapes, you will see the volume and surface area.
- Switch between shapes at any time. The calculator will update the input fields to match the new shape.
Understanding Geometry
Fundamental Concepts
Geometry is the branch of mathematics concerned with the properties and relationships of points, lines, surfaces, and solids. Two of the most fundamental measurements in geometry are area (the amount of space inside a flat shape) and perimeter (the total distance around the outside of a shape). For three-dimensional objects, we also measure volume (the space enclosed within a solid) and surface area (the total area of all outer surfaces).
The Pythagorean Theorem
One of the most important results in geometry is the Pythagorean theorem. For any right triangle with legs a and b and hypotenuse c:
This theorem is used extensively for calculating distances, checking if a triangle is right-angled, and solving real-world problems in construction, navigation, and physics.
Pi and Circular Shapes
The constant pi (approximately 3.14159) is the ratio of a circle's circumference to its diameter. It appears in every formula involving circles, cylinders, and spheres. Pi is an irrational number, meaning its decimal representation goes on forever without repeating.
Area and Perimeter Formulas for 2D Shapes
| Shape | Area | Perimeter |
|---|---|---|
| Circle | pi * r^2 | 2 * pi * r |
| Square | s^2 | 4s |
| Rectangle | l * w | 2(l + w) |
| Triangle | 0.5 * b * h | a + b + c |
| Trapezoid | 0.5 * (a + b) * h | a + b + c + d |
| Parallelogram | b * h | 2(a + b) |
| Ellipse | pi * a * b | approx. 2*pi*sqrt((a^2+b^2)/2) |
Volume and Surface Area Formulas for 3D Shapes
| Shape | Volume | Surface Area |
|---|---|---|
| Cube | s^3 | 6s^2 |
| Rectangular Prism | l * w * h | 2(lw + lh + wh) |
| Cylinder | pi * r^2 * h | 2*pi*r*(r + h) |
| Sphere | (4/3)*pi*r^3 | 4*pi*r^2 |
| Cone | (1/3)*pi*r^2*h | pi*r*(r + sqrt(r^2+h^2)) |
| Pyramid (square base) | (1/3)*s^2*h | s^2 + 2*s*l |
Worked Example: Composite Shape (Rectangle + Semicircle)
Find the total area and perimeter of a shape made by attaching a semicircle to the top of a rectangle with width 10 cm and height 6 cm.
Step 1: Identify the components. The shape consists of a rectangle (10 cm x 6 cm) and a semicircle with diameter 10 cm (radius = 5 cm).
Step 2: Calculate the rectangle area. Area = l * w = 10 * 6 = 60 cm^2
Step 3: Calculate the semicircle area. Area = (1/2) * pi * r^2 = 0.5 * 3.14159 * 25 = 39.27 cm^2
Step 4: Add the areas. Total Area = 60 + 39.27 = 99.27 cm^2
Step 5: Calculate the perimeter. The perimeter includes the bottom of the rectangle (10 cm), two sides (6 cm each), and the semicircular arc (pi * r = 3.14159 * 5 = 15.71 cm). The top edge of the rectangle is not part of the perimeter because it is inside the shape.
Step 6: Add the perimeter segments. Perimeter = 10 + 6 + 6 + 15.71 = 37.71 cm
Geometry Tips and Tricks
- Always include units. Area is measured in square units (cm^2, m^2), perimeter in linear units (cm, m), and volume in cubic units (cm^3, m^3). Forgetting units is a very common mistake.
- Break complex shapes into simpler ones. A composite shape can be divided into rectangles, triangles, and circles. Calculate each part separately, then add (or subtract) the areas.
- Use the Pythagorean theorem for diagonals. The diagonal of a rectangle with sides a and b is sqrt(a^2 + b^2). This is useful for construction and design problems.
- Remember pi is approximately 3.14159. For quick mental estimates, you can use 3.14 or even 22/7 as approximations.
- Double-check 3D formulas. Volume formulas for cones and pyramids include a 1/3 factor that is easy to forget. A cone is exactly one-third the volume of a cylinder with the same base and height.
- Use symmetry to simplify problems. If a shape has a line of symmetry, you can calculate one half and double the result, which often reduces complexity.