Height is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.
Wherever it lands, the height makes a right angle with the base (or the base extended).
Formal definition and notation
For a polygon or solid with a chosen base, the height h is the length of a segment perpendicular to the base (or to the line or plane containing it) from the base to the opposite vertex, side or face. In a triangle this segment is the altitude; it may fall inside the triangle, along a side, or outside it. Height is a length, so it is measured in units such as cm or ft.
A box has a base 8 in by 6 in and a volume of 240 cubic inches. How tall is it?
Base area: B = 8 × 6 = 48 square inches.
V = Bh, so 240 = 48h.
h = 240 ÷ 48 = 5.
Answer: The box is 5 inches tall.
Common mistakes
✗ MistakeUsing a slanted side as the height: base 6, slanted side 5, so A = 12 × 6 × 5 = 15.
✓ InsteadUse the perpendicular height, 4: A = 12 × 6 × 4 = 12.
The slanted side leans, so it is longer than the straight-up distance. Only the perpendicular distance gives the area.
✗ MistakeThe height must be a vertical line on the page.
✓ InsteadThe height is perpendicular to whichever side you call the base, even if that base is tilted.
Turning a shape does not change it. Height is measured relative to the base, not to the edge of the paper.
✗ MistakeAn obtuse triangle has no height for its shorter sides because the line falls outside.
✓ InsteadExtend the base with a dashed line; the height meets that extension at a right angle.
Height is a distance to the line containing the base, so it can land outside the triangle.
Not to be confused with
Heightvs.Length
Length usually means how long something is from end to end, often the longest side. Height is how tall it is, measured straight up from its base.
Heightvs.Slant height
Slant height runs along a face of a pyramid or cone and is used for surface area. The height is straight down from the top and is used for volume.
Quick check: 3 questions
Question 1 of 3
A triangle has an area of 24 square feet and a base of 8 feet. What is its height?
Show answer and explanation
Answer: B.24 = 12 × 8 × h = 4h, so h = 6. A forgets the one half, C subtracts, and D multiplies.
Question 2 of 3
In a right triangle with legs 5 and 12 and hypotenuse 13, what is the height when the base is the leg of length 12?
Show answer and explanation
Answer: D. The two legs are perpendicular, so the other leg, 5, is the height to the leg of 12. The hypotenuse (13) is slanted relative to that base.
Question 3 of 3
A box has a volume of 100 cubic cm and a base 5 cm by 4 cm. How tall is it?
Show answer and explanation
Answer: A.100 ÷ (5 × 4) = 100 ÷ 20 = 5. B is the base area, C subtracts, and D multiplies.
Height: frequently asked questions
Is height the same as altitude?
In geometry, almost. The altitude of a triangle is the segment drawn from a vertex perpendicular to the opposite side, and the height is the length of that segment. Many books use the two words interchangeably. In everyday speech, altitude also means height above sea level, such as an airplane flying at an altitude of 30,000 feet.
Can the height of a triangle be outside the triangle?
Yes, in an obtuse triangle. If you use one of the two shorter sides next to the obtuse angle as the base, the perpendicular from the opposite vertex lands beyond the end of that side. Extend the base with a dashed line and measure to it. The area formula still works: one half times base times height.
PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5VolumeVolume measures the amount of space inside a three-dimensional figure, in cubic units such as cubic centimeters or cubic feet. You can find it by counting the unit cubes that fill the solid. For a rectangular prism, V = l × w × h; for any prism, it equals the area of the base multiplied by the height.Geometry · Grade 5, Grade 6, Grade 7InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4PyramidA pyramid is a solid figure with one polygon as its base and triangular faces that rise from the base edges and meet at a single top point, the apex. A pyramid is named by its base, such as a square pyramid. Its volume is one third of the base area times the height: V = 13Bh.Geometry · Grade 6, Grade 7, Grade 8ConeA cone is a three-dimensional solid with one flat circular base and a curved surface that narrows to a single point, the apex, like a party hat. Its volume is one third of a cylinder with the same base and height: V = 13π r2h, where r is the base radius and h the height.Geometry · Kindergarten, Grade 1, Grade 8Perpendicular LinesPerpendicular lines are lines that intersect to form right angles: their intersection makes a 90-degree angle. The symbol ⊥ means is perpendicular to. On a coordinate plane, two perpendicular lines that are not vertical have slopes that multiply to −1, such as 2 and −12.Geometry · Grade 4, Grade 5, Grade 6AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5ParallelogramA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Geometry · Grade 3, Grade 5, Grade 6TrapezoidA trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases, and the other two sides are the legs. Its area is the average of the bases times the height: A = b1 + b22 × h. In British English it is called a trapezium.Geometry · Grade 3, Grade 5, Grade 6WidthWidth is the distance across a shape or object from side to side. In a rectangle, width is one of the two side measurements used with length, usually the shorter one, and multiplying length by width gives the area of a rectangle. Width is measured in length units such as inches, feet, centimeters or meters.Measurement · Grade 3, Grade 4, Grade 5BaseA base is the number or expression that is repeatedly multiplied when using an exponent. In 45 the base is 4 and the exponent is 5, so 45 = 4 × 4 × 4 × 4 × 4 = 1024. The base is the big number written on the line; the exponent is the small raised number.Algebra · Grade 6, Grade 7, Grade 8