Free Geometry Calculator with Steps
Calculate area, perimeter, volume, surface area, and arc length step-by-step for 17 shapes. Learn geometry formulas with instant feedback. Perfect for GCSE and A-Level maths.
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What is Geometry?
Geometry is the branch of mathematics concerned with the properties and relationships of points, lines, surfaces, and solids. The word comes from the Greek geo (earth) and metron (measure) — literally "earth measurement".
The field is broadly divided into plane geometry (2D shapes like triangles, circles, and polygons) and solid geometry (3D shapes like spheres, cylinders, and prisms). Both are central to GCSE and A-Level mathematics, as well as real-world applications in architecture, engineering, design, and nature.
The ancient Greek mathematician Euclid — often called the "father of geometry" — systematised geometric knowledge in his work Elements around 300 BC. Many of the formulas you'll use in this calculator trace their origins back to Euclid's proofs.
Types of 2D Shapes
Triangles
- Scalene — all sides different
- Isosceles — two sides equal
- Equilateral — all sides equal (60° each)
- Right-angled — one 90° angle
Quadrilaterals
- Rectangle — 4 right angles
- Square — all sides and angles equal
- Parallelogram — opposite sides parallel
- Rhombus — all sides equal
- Trapezium — one pair of parallel sides
Circles & Sectors
- Circle — all points equidistant from centre
- Sector — "pizza slice" of a circle
- Arc — part of the circumference
- Ellipse — stretched circle, two foci
Regular Polygons
- Pentagon — 5 sides (108° each)
- Hexagon — 6 sides (120° each)
- Octagon — 8 sides (135° each)
- Decagon — 10 sides (144° each)
Types of 3D Shapes
Prisms
Uniform cross-section along their length.
- Cube — V = s³, SA = 6s²
- Cuboid — V = lwh, SA = 2(lw+wh+hl)
- Triangular Prism — V = ½bh × l
Pyramids & Cones
Taper to a point from a base.
- Pyramid — V = ⅓ × base area × h
- Cone — V = ⅓πr²h, SA = πr(r+l)
Curved Solids
Curved surfaces, no flat faces (sphere) or mixed (cylinder).
- Sphere — V = ⁴⁄₃πr³, SA = 4πr²
- Cylinder — V = πr²h, SA = 2πr(r+h)
Key Formulas Reference
2D Shapes
3D Shapes
The Pythagorean Theorem
For any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. This is one of the most useful results in all of geometry.
When to use it:
- Finding the diagonal of a rectangle: d = √(l² + w²)
- Finding the diagonal of a square: d = s√2
- Finding the slant height of a cone: l = √(r² + h²)
- Finding the slant height of a pyramid
- Checking whether a triangle is right-angled
- Finding the side of a rhombus from its diagonals: s = √((d₁/2)² + (d₂/2)²)
Sectors and Arcs
A sector is a "slice" of a circle, defined by two radii and an arc. The key idea is that a sector is a fraction of the full circle, where the fraction is θ/360 (with θ in degrees).
Sector Area
A = (θ/360) × πr²
Take the full circle area (πr²) and multiply by the fraction (θ/360) to get the sector area.
Arc Length
Arc = (θ/360) × 2πr
Take the full circumference (2πr) and multiply by the same fraction (θ/360) to get the arc length.
Worked Example
A sector has radius 8 cm and angle 135°. Find the area and arc length.
Area = (135/360) × π × 8² = 0.375 × 64π = 24π ≈ 75.40 cm²
Arc = (135/360) × 2π × 8 = 0.375 × 16π = 6π ≈ 18.85 cm
Perimeter = arc + 2r = 18.85 + 16 = 34.85 cm
Common Mistakes to Avoid
Using diameter instead of radius in πr²
Always halve the diameter first: r = d/2
Forgetting to halve when calculating triangle area (using b×h instead of ½bh)
Triangle area = ½ × base × height
Confusing area (cm²) and perimeter (cm) — mixing up square and linear units
Area = space inside (cm²), Perimeter = distance around (cm)
Using arc length formula for sector area (or vice versa)
Area uses πr², arc length uses 2πr — both multiplied by θ/360
Not including the two radii when calculating sector perimeter
Sector perimeter = arc length + 2r (the curved part plus two straight edges)
Using 2πr for circumference but writing the answer as area
Circumference is a length (cm), area is πr² (cm²)
Forgetting the ⅓ factor for cones and pyramids
Cone and pyramid volumes are ⅓ of the corresponding prism/cylinder volume
Using the vertical height instead of slant height for cone surface area
Lateral SA of a cone uses slant height l = √(r²+h²), not the vertical height h
Worked Examples
Circle Area
Find the area of a circle with diameter 14 cm.
r = 14 ÷ 2 = 7 cm
A = πr² = π × 7² = 49π
A ≈ 153.94 cm²
Sector Area
Find the area of a sector with radius 10 cm and angle 72°.
Fraction = 72/360 = 0.2
A = 0.2 × π × 10² = 0.2 × 100π
A = 20π ≈ 62.83 cm²
Cone Volume & Surface Area
Find the volume and total surface area of a cone with r = 6 cm, h = 8 cm.
l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm
V = ⅓ × π × 36 × 8 = 96π ≈ 301.59 cm³
SA = π × 6 × (6 + 10) = 96π ≈ 301.59 cm²
Regular Octagon Interior Angle
Find the interior angle of a regular octagon.
n = 8
Interior angle = (n − 2) × 180° / n
= (8 − 2) × 180 / 8 = 6 × 180 / 8
= 135°
Exterior angle = 360/8 = 45°. Check: 135 + 45 = 180°.
Real-World Applications
Architecture
Architects use geometry to design buildings, calculate floor areas, roof angles, and structural load distributions. The Pythagorean theorem ensures walls are perfectly square.
Engineering
Engineers calculate volumes for tanks and pipes, surface areas for heat transfer, and cross-sections for structural strength. Sector geometry is vital in gear and turbine design.
Design
Graphic and industrial designers rely on geometry for layouts, packaging, and product design. Regular polygons create tessellations; circles and arcs create smooth curves.
Nature
Geometry appears everywhere in nature: hexagonal honeycombs, spherical bubbles, spiral shells (logarithmic spirals), and the fractal patterns of snowflakes and ferns.
Frequently Asked Questions
How do I calculate the area of a circle?
Use the formula A = πr². Square the radius and multiply by π (approximately 3.14159). For a radius of 5, area = π × 25 ≈ 78.54 square units. Remember to halve the diameter first if that is what you are given.
How do I calculate the area of a sector?
Use A = (θ/360) × πr². The fraction θ/360 tells you what proportion of the full circle the sector represents. A 90° sector is ¼ of a circle, so its area is ¼ × πr².
What is the formula for arc length?
Arc length = (θ/360) × 2πr. It is the same fraction of the circumference as the sector angle is of a full turn (360°). The perimeter of a sector = arc length + 2r.
How do I find the interior angle of a regular polygon?
Use the formula: interior angle = (n − 2) × 180° / n, where n is the number of sides. The exterior angle is 360° / n. Interior + exterior = 180°.
What is the Pythagorean theorem?
a² + b² = c², where c is the hypotenuse of a right-angled triangle. Use it to find diagonals of rectangles (d = √(l²+w²)), slant heights of cones (l = √(r²+h²)), and to check if a triangle is right-angled.
How do I calculate the volume of a sphere?
Use V = (4/3)πr³. Cube the radius, multiply by π, then multiply by 4/3. A sphere with radius 3 has volume = (4/3) × π × 27 = 36π ≈ 113.1 cubic units.
What is the difference between volume and surface area?
Volume measures the 3D space inside a shape (cm³) — how much it can hold. Surface area measures the total outer area of all faces (cm²) — how much material covers it.
Why do cones and pyramids have a ⅓ factor?
A cone is exactly ⅓ of the volume of a cylinder with the same base and height. Similarly, a pyramid is ⅓ of the corresponding prism. This can be proved by integration or Cavalieri's principle.
How do I find the surface area of a cylinder?
SA = 2πr² + 2πrh. The two circles (top and bottom) contribute 2πr², and the curved surface "unrolls" into a rectangle of width 2πr and height h, giving 2πrh.
Is this calculator suitable for GCSE maths?
Yes! Covers all GCSE and A-Level geometry topics: circles, sectors, arc length, all 2D and 3D shapes, interior angles, diagonals, symmetry, and the Pythagorean theorem — with step-by-step working.
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