100 Geometry Problems Solutions

Calculate area using A = base height, where base and height are the sides forming the right angle.

For a right triangle with legs a and b and hypotenuse c, a + b = c.

Use the Pythagorean Theorem to solve for the unknown side.

Triangles are similar if corresponding angles are equal and corresponding sides are proportional.

The sum of interior angles in any triangle equals 180.

Base angles are equal, and the angle bisector, median, and altitude from the apex coincide.

All sides equal; all angles equal 60.

The sum of any two sides must exceed the third side.

Area = (s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter.

Area = ab sin C, where a and b are sides and C is the included angle.

An angle bisector divides the opposite side proportionally to adjacent sides.

The three medians intersect at the centroid, dividing each median in a 2:1 ratio.

The altitude from a vertex is perpendicular to the opposite side (or its extension).

R = abc/(4A), where a, b, c are side lengths and A is the area.

r = A/s, where A is the area and s is the semi-perimeter.

Opposite sides are parallel and equal; all angles are 90; diagonals are equal and bisect each other.

All sides equal; all angles 90; diagonals are equal, perpendicular, and bisect each other.

Opposite sides are parallel and equal; opposite angles are equal; diagonals bisect each other.

All sides equal; opposite angles equal; diagonals are perpendicular and bisect each other.

One pair of sides is parallel; area = (sum of parallel sides) height.

Area = length width.

Area = side or Area = diagonal.

Area = base height.

Area = (sum of parallel sides) height.

Area = product of diagonals.

Two pairs of adjacent sides equal; one diagonal is the perpendicular bisector of the other.

Area = product of diagonals.

The sum of interior angles of any quadrilateral is 360.

Divide into triangles or use Brahmagupta's formula: A = [(s-a)(s-b)(s-c)(s-d) - abcdcos(/2)].

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Area = r, where r is the radius.

Circumference = 2r = d, where r is radius and d is diameter.

Arc length = (/360) 2r, where is the central angle in degrees and r is the radius.

Area = (/360) r, where is the central angle in degrees and r is the radius.

Area = Area of sector - Area of triangle = (/360) r - r sin .

A tangent is perpendicular to the radius at the point of tangency. Two tangents from an external point have equal lengths.

Perpendicular from center to a chord bisects the chord. Equal chords subtend equal angles at the center.

Angle subtended by an arc at the center is twice the angle at any point on the remaining part of the circle.

Opposite angles are supplementary. The angle between a tangent and a chord equals the angle in the alternate segment.

If a tangent and a secant are drawn from an external point, the square of the tangent length equals the product of the whole secant and its external segment.

For any point P outside a circle, PA PB = PC PD, where A, B, C, D are intersection points of two lines through P with the circle.

Circles with the same center but different radii. The region between two concentric circles is an annulus with area (R-r).

Two circles intersect at points if the distance between centers d satisfies |r1-r2| < d < r1+r2.

If two chords AB and CD intersect at point E inside the circle, then AE EB = CE ED.

The measure of the angle formed by two chords intersecting inside a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Sum of interior angles = (n-2) 180, where n is the number of sides.

The sum of exterior angles of any polygon is 360.

Each interior angle = (n-2) 180/n, where n is the number of sides.

Each exterior angle = 360/n, where n is the number of sides.

Number of diagonals = n(n-3)/2, where n is the number of sides.

Area = perimeter apothem = n s a, where n is sides, s is side length, and a is apothem.

Area = (5s/4) cot(/5), where s is the side length.

Area = (33/2) s, where s is the side length.

Area = 2(1+2) s, where s is the side length.

Regular polygons that tessellate are equilateral triangles, squares, and regular hexagons, as their interior angles divide 360 evenly.

Divide into triangles and rectangles, or use the shoelace formula: |x1y2 + x2y3 + ... + xny1 - (y1x2 + y2x3 + ... + ynx1)|.

In a regular n-gon with side s, diagonal length = 2s sin(k/n), where k is the number of vertices between the endpoints.

R = s/(2 sin(/n)), where s is the side length and n is the number of sides.

r = s/(2 tan(/n)), where s is the side length and n is the number of sides.

Area = n s r = perimeter inradius, where n is sides, s is side length, and r is inradius.

Volume = s, where s is the side length.

Surface area = 6s, where s is the side length.

Volume = length width height = l w h.

Surface area = 2(lw + lh + wh), where l, w, h are dimensions.

Volume = (4/3)r, where r is the radius.

Surface area = 4r, where r is the radius.

Volume = rh, where r is the radius and h is the height.

Surface area = 2r + 2rh = 2r(r+h), where r is the radius and h is the height.

Volume = (1/3)rh, where r is the radius and h is the height.

Surface area = r + rl = r(r+l), where r is the radius, h is the height, and l = (r + h) is the slant height.

Volume = (1/3)Bh, where B is the base area and h is the height.

Surface area = B + (1/2)P l, where B is the base area, P is the perimeter of the base, and l is the slant height of each face.

Volume = Bh, where B is the base area and h is the height.

Surface area = 2B + Ph, where B is the base area, P is the perimeter of the base, and h is the height.

Volume = 2Rr, where R is the major radius and r is the minor radius.

Distance between points (x,y) and (x,y) is [(x-x) + (y-y)].

Midpoint of points (x,y) and (x,y) is ((x+x)/2, (y+y)/2).

Slope between points (x,y) and (x,y) is m = (y-y)/(x-x).

Line with slope m and y-intercept b: y = mx + b. Or general form: Ax + By + C = 0.

Lines are parallel if their slopes are equal.

Lines are perpendicular if the product of their slopes is -1.

Distance from point (x,y) to line Ax + By + C = 0 is |Ax + By + C|/(A + B).

Area = |x(y-y) + x(y-y) + x(y-y)|.

Circle with center (h,k) and radius r: (x-h) + (y-k) = r.

Ellipse with center (h,k), horizontal axis 2a, vertical axis 2b: (x-h)/a + (y-k)/b = 1.

Parabola with vertex (h,k): (y-k) = a(x-h) (vertical) or (x-h) = a(y-k) (horizontal).

Hyperbola with center (h,k): (x-h)/a - (y-k)/b = 1 (horizontal) or (y-k)/a - (x-h)/b = 1 (vertical).

Find the circle equation by solving the system formed by equating the distance from center to each point to the radius.

Solve the system of two linear equations representing the lines to find their intersection point.

tan = |(m-m)/(1+mm)|, where m and m are the slopes of the lines and is the acute angle between them.

The intersection point of medians (lines from vertices to midpoints of opposite sides). Coordinates of centroid = ((x+x+x)/3, (y+y+y)/3).

Intersection of perpendicular bisectors of sides. Equidistant from all three vertices. Center of circumscribed circle.

Intersection of angle bisectors. Equidistant from all three sides. Center of inscribed circle.

Intersection of altitudes (perpendiculars from vertices to opposite sides).

In a non-equilateral triangle, the centroid, circumcenter, and orthocenter are collinear, lying on the Euler line.

The circle that passes through nine significant points: the midpoint of each side, the foot of each altitude, and the midpoint of each segment from a vertex to the orthocenter.

For concurrent cevians AD, BE, CF of triangle ABC: (BD/DC)(CE/EA)(AF/FB) = 1.

For points D, E, F on sides (or extensions) BC, CA, AB of triangle ABC: (AF/FB)(BD/DC)(CE/EA) = -1 if and only if D, E, F are collinear.

For a cyclic quadrilateral ABCD: AC BD = AB CD + AD BC.

Area = ab, where a and b are the semi-major and semi-minor axes.

Geometry provides the foundation for understanding spatial relationships and is essential in fields ranging from architecture and engineering to computer graphics and physics. By mastering these 100 fundamental problems and solutions, you've built a solid foundation in geometric principles that will serve you well across mathematical disciplines and practical applications.