Geometry Calculators
Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. Each calculator below shows you every step of the solution using your numbers, then explains the general method with worked examples.
Available Geometry Calculators
- Pythagorean Theorem Calculator — find the missing side of a right triangle given any two sides
- Circle Area Calculator — calculate the area of a circle from its radius or diameter
- Circumference Calculator — find the circumference (perimeter) of a circle
- Triangle Area Calculator — find the area of any triangle using base and height
- Rectangle Area Calculator — find the area and perimeter of a rectangle or square
- Cylinder Volume Calculator — find the volume and surface area of a cylinder
- Distance Between Two Points Calculator — find the straight-line distance between two coordinates
Geometry Reference Guide
Every geometry problem starts with a formula. Here is a quick reference for the most common shapes, with examples worked out step by step.
Right Triangles and the Pythagorean Theorem
A right triangle has one 90° angle. The two shorter sides are called legs (labeled $a$ and $b$) and the longest side — opposite the right angle — is the hypotenuse (labeled $c$).
The Pythagorean theorem states:
$$a^2 + b^2 = c^2$$Example: A triangle has legs $a = 3$ and $b = 4$. Find the hypotenuse.
$$c^2 = 3^2 + 4^2 = 9 + 16 = 25$$ $$c = \sqrt{25} = 5$$You can also solve for a leg if the hypotenuse is known:
$$a = \sqrt{c^2 - b^2}$$Circles
Two key measurements describe a circle:
- Radius ($r$) — the distance from the center to the edge
- Diameter ($d$) — the distance straight across, equal to $2r$
Area — the space inside the circle:
$$A = \pi r^2$$Circumference — the distance around the circle:
$$C = 2\pi r = \pi d$$Example: A circle has a radius of 5.
$$A = \pi \times 5^2 = 25\pi \approx 78.54$$ $$C = 2\pi \times 5 = 10\pi \approx 31.42$$Triangles
The area of any triangle is half the product of its base and height (where height is perpendicular to the base):
$$A = \frac{1}{2} \times b \times h$$Example: A triangle has a base of 10 and a height of 6.
$$A = \frac{1}{2} \times 10 \times 6 = 30$$Rectangles and Squares
A rectangle has two pairs of equal sides called length ($l$) and width ($w$).
$$\text{Area} = l \times w$$ $$\text{Perimeter} = 2l + 2w$$A square is a rectangle where $l = w$, so:
$$\text{Area} = s^2 \qquad \text{Perimeter} = 4s$$Example: A rectangle is 8 wide and 3 tall.
$$A = 8 \times 3 = 24 \qquad P = 2(8) + 2(3) = 22$$Cylinders
A cylinder has a circular base with radius $r$ and a height $h$.
$$\text{Volume} = \pi r^2 h$$ $$\text{Lateral Surface Area} = 2\pi r h$$ $$\text{Total Surface Area} = 2\pi r h + 2\pi r^2$$Example: A cylinder with radius 3 and height 7.
$$V = \pi \times 3^2 \times 7 = 63\pi \approx 197.92$$ $$\text{Total SA} = 2\pi(3)(7) + 2\pi(3^2) = 42\pi + 18\pi = 60\pi \approx 188.50$$Distance Between Two Points
The straight-line distance between points $(x_1, y_1)$ and $(x_2, y_2)$ comes from the Pythagorean theorem:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$Example: Find the distance between $(1, 2)$ and $(4, 6)$.
$$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$Frequently Asked Questions
What is $\pi$ (pi)?
Pi ($\pi$) is the ratio of a circle's circumference to its diameter. It is irrational — its decimal form never ends or repeats. For calculations, use $\pi \approx 3.14159$.
What is the difference between area and perimeter?
Area measures the two-dimensional space inside a shape, expressed in square units (cm², ft², etc.). Perimeter (or circumference for circles) measures the distance around the outside, expressed in linear units.
What is volume?
Volume measures the three-dimensional space a solid occupies, expressed in cubic units (cm³, in³, etc.).
Does the Pythagorean theorem work for all triangles?
No — only right triangles. For other triangles, use the Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$, where $C$ is the angle opposite side $c$.